Je Remercie

Cordialement Messieurs

Steen Rohde

Fredrik Viklund D'

Matthieu Ciements À Messieurs

Julien Astorg

Athanasios Barral

Bertrand Batakis

Duplantier

Jin Xiong

Fredrik Viklund

Marie-Laurence Poncet

Hieu Anh Lan

Keywords: Whole-plane SLE, logarithmic moments, Beliaev-Smirnov equation, generalized spectrum, McMullen's asymptotic variance, Grunsky coefficients

Ce manuscrit conclut trois ans de travail, et je tiens en ces quelques lignes à exprimer ma reconnaissance envers tous ceux qui de près ou de loin y ont contribué.

En premier lieu, je tiens à remercier Michel Zinsmeister, mon directeur de thèse pour son encadrement, ses conseils, ainsi que pour ses encouragements, son enthousiasme, sa conance et sa patience. Je lui suis reconnaissante pour son aide précieuse dans la relecture et la correction de ma thèse.

Contents

1.3

Expectation of log f 0 (z) z

Introduction

The starting point of this thesis was to study the logarithmic coecients of conformal maps associated with the interior whole-plane SLE κ process. The starting motivation was to revisit Bieberbach's conjecture in the framework of SLE κ theory.

In this section, we recall Bieberbach's conjecture as well as the history of the proof.

We will also give denition for SLE κ . Let f (z) = n≥0 a n z n be a holomorphic function in the unit disk D. Bieberbach proved in 1916 [3] that if f is further assumed to be injective, then

|a 2 | ≤ 2|a 1 |,
and he conjectured that

|a n | ≤ n|a 1 |
for all n > 2. This famous conjecture has been nally proved in 1984 by de Branges [4]. His proof was made possible by the addition of a new idea (an inequality of Askey and Gasper) to a series of methods and results developed in almost a century.

The earliest important contribution to the proof of Bieberbach's conjecture is the proof [14] by Charles Loewner in 1923 for n = 3. De Branges' one indeed used Loewner's idea in a essential way.

Let γ : [0, ∞) → C be a simple curve such that |γ(t)| → +∞ as t → +∞ and such that γ(t) = 0, t ≥ 0. Dene then for each t > 0, the slit domain Ω t = C \ γ([t, ∞)) being a simply connected domain containing 0 and we can thus consider the Riemann mapping f t : D = {|z| < 1} → Ω t characterized by f t (0) = 0, f t (0) > 0. By the Caratheodory convergence theorem, f t converges as t → 0 to f := f 0 , the Riemann mapping of Ω 0 . Assuming without loss of generality that f 0 (0) = 1 and changing the time t if necessary, we may choose the normalization f t (0) = e t , t ≥ 0.

The key idea of Loewner is to observe that the sequence of domains Ω t is increasing, which translates into the fact that ∂ft ∂t /z ∂ft ∂z > 0 or, equivalently, that this quantity is the Poisson integral of a positive measure on the unit circle, actually a probability measure, due to the above normalization. Since the domains Ω t are slit domains, this probability measure must be a Dirac mass at λ(t) = f -1 t (γ(t)) ∈ ∂D. We notice that λ is a continuous function from [0, ∞) to the unit circle. Loewner has shown that the Loewner chain (f t ) associated with (Ω t ) is driven by the function λ, in the sense that (f t ) satises the following PDE:

∂ ∂t f t (z) = z ∂ ∂z f t (z) λ(t) + z λ(t) -z . (0.0.1) INTRODUCTION ( ) ( )= ( )= ( ) ( )= z t = f 1 t ( ) 8
( ) t 0 f t 0 f t 1 0 f 1 0 0 ( ) t f t t
Figure 1: Loewner map z → f t (z) from D to the slit domain Ω t = C\γ([t, ∞)) (here slit by a single curve γ([t, ∞)) for SLE κ≤4 ). One has f t (0) = 0, ∀t ≥ 0. At t = 0, the driving function λ(0) = 1, so that the image of z = 1 is at the tip γ(0) = f 0 (1) of the curve (Fig. 1 in [6]).

INTRODUCTION that (a 3 ) ≤ 3. From the remaining equation and applying the Cauchy-Schwarz inequality, one can complete the proof (see Appendix A of Ref. [6] for details). Besides Loewner's theory of growth processes, de Branges' proof also relied much on the consideration, developed by Grunsky [7] and later Lebedev and Milin [10], of logarithmic coecients. More precisely, if f : D → C is holomorphic and injective with f (0) = 0, we may consider the power series,

log f (z) z = 2 n≥1 γ n z n .
The purpose of introducing this logarithm was to prove Robertson's conjecture [20],

which was known to imply Bieberbach's. Let f be in the class S of schlicht functions, i.e., holomorphic and injective in the unit disk, and normalized as f (0) = 0, f (0) = 1. There is a branch f [START_REF] Beliaev | Harmonic Measure and SLE[END_REF] of z → f (z 2 ) which is an odd function in S. We then write f [START_REF] Beliaev | Harmonic Measure and SLE[END_REF] it is apparent that Robertson's conjecture implies Bieberbach's. For n = 2, Robertson's conjecture is the same as |a 2 | ≤ 2. Using Loewner's method, Robertson proved in 1936 that the conjecture is true for n = 3.

Lebedev and Milin approached Robertson's conjecture with observing that log f [START_REF] Beliaev | Harmonic Measure and SLE[END_REF] ( √ z)

√ z = 1 2 log f (z) z ,
and consequently that

∞ n=0 b 2n+1 z n = exp ∞ n=1 γ n z n .
They proved an inequality that is now called the second Lebedev-Milin inequality, a relation between the coecients of any power series to those of its exponential, namely ∀n ≥ 0, (0.0.4)

INTRODUCTION

From the above inequality, Milin [16] naturally conjectured that ∀f ∈ S, ∀n ≥ 1,

n m=1 m k=1 k|γ k | 2 - 1 k ≤ 0.
Milin's conjecture, proved in 1984 by de Branges, implies Robertson's, hence Bieberbach's conjecture.

We now return to Loewner's theory. It is remarkable that Loewner's method has a converse: given any continuous function λ : [0, ∞) → C with |λ(t)| = 1 for t ≥ 0, then the Loewner equation (0.0.1), supplemented by the boundary (initial) condition lim t→+∞ f t (e -t z) = z, has a solution f t (z), such that (f t (z)) t≥0 is a chain of Riemann maps onto simply connected domains (Ω t ) that are increasing with t.

However, Loewner's ideas go far beyond Bieberbach's conjecture: In 1999, Oded Schramm [21] introduced into the Loewner equation (0.0.1) the random driving function, λ(t) := e i √ κBt , (0.0.5) where B t is standard one dimensional Brownian motion and κ a non-negative parameter, thereby making Eq. (0.0.1) a stochastic PDE and creating the celebrated Schramm-Loewner Evolution (or Stochastic Loewner Evolution) with parameter κ (SLE κ ). There exist several variants of SLE κ known as chordal, radial.

The associated conformal maps f t from D to C \ γ([t, ∞)), obeying (0.0.1) for (0.0.5), dene the interior whole-plane SLE κ . Their coecients a n (t), which are random variables, are dened by the normalized series expansion: f t (z) = e t z + n≥2 a n (t)z n , (0.0.6)

The starting point of this thesis was to study their logarithmic coecients γ n (t), which are also random variables, dened as log e -t f t (z) z = 2 n≥1 γ n (t)z n .

(0.0.7)

Besides, we also consider some other problems such as McMullen's asymptotic variance, or Grunsky coecients matrix. For the reader's convenience, we describe briey the content of each chapter in this thesis.

In chapter one, we rst recall a result obtained in Ref. [6], that is an explicit expression for the expectations of the coecients a n := a n (0) of the expansion (0.0.6). By the same method as used in [6], we will give an explicit expression for the expectations of the logarithmic coecients γ n := γ n (0) of the expansion (0.0.7). We end the chapter by introducing an algorithm to compute E(|γ n (0)| 2 ), together with computations.

Chapter 2 is devoted to introduce the main results obtained in our article [5]. In particular, we give expressions in closed form for the mixed moments, E (f (z)) p/2 (f (z)) q/2 ; E |f (z)| p |f (z)| q (here f (z) := f 0 (z)),

INTRODUCTION

along an integrability curve R, which is a parabola in the (p, q) plane depending on the SLE parameter κ.

If p = 2 and q = 0 then, due to Parseval's formula, explicit expression for E (|f (z)| 2 ) yields E (|a n | 2 ). For example, as given in Ref. [6], E (|a n | 2 ) = 1 for SLE 6 and E (|a n | 2 ) = n for SLE 2 . Similarly, if one knows E (|f (z)/f (z)| 2 ) then E (|γ n | 2 ) is known. In particular, we prove that E (|γ n | 2 ) = 1/2n 2 for SLE 2 .

In chapter 3, we rst give expressions for the SLE κ functions,

F (z) := E (f (z)) p/2 (f (z)/z) q/2 ; G(z, z) := E |f (z)| p |f (z)/z| q ,
for some special parameters (p, q). The rest of this chapter is devoted to the study on the generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the mixed moments E (|f (z)| p /|f (z)| q ). This is also an important part in our work [5].

In chapter 4, we study the (expected) McMullen asymptotic variance of SLE 2 . More precisely, we determine explicit expression for E(| log f (z)| 2 ) for SLE 2 , whereby we prove that this function satises the following formula:

lim p→0 2 β(p, f ) p 2 = lim r→1 - 1 4π| log(1 -r)| |z|=r E(| log f (z)| 2 )|dz|,
where β(p, f ) is the average integral means spectrum of the (time 0) interior wholeplane SLE 2 map f .

In chapter 5, we study the Grunsky coecients, in expectation, for interior whole-plane SLE κ . In particular, with the support of MAPLE, we can compute the coecients d n,m dened by the power series,

G(z, ξ) := E (z -ξ) q f (z) p f (ξ) p (f (z) -f (ξ)) q = ∞ n,m=0
d n,m z n ξ m , (z, ξ) ∈ D × D, for all p, q ∈ R. We will give explicit expressions for G(z, ξ) for special values of (p, q; κ).

Chapter 1

Radial, whole-plane SLE κ processes 1.1 Mathematical background Loewner equation involves Riemann mapping theorem for simply connected domains. Therefore, we begin this section with a brief introduction to this subject.

Simply connected domains

An arc in a metric space X is a continuous mapping γ : [a, b] ⊂ R → X. Such an arc is said to be closed if γ(a) = γ(b). Two arcs γ 1 , γ 2 dened on the same interval [a, b] ∈ Ω, Ind(z, γ) = 0.

We recall that Ind(z, γ) stands for the variation of the argument (measured in number of turns) of γ(t) -z along [a, b]. When γ is piecewise C 1 this quantity is also equal to Let Ω be a simply connected proper subdomain of C and w ∈ C. Then there exists a unique biholomorphic map g : Ω → D such that g(ω) = 0, g (ω) > 0.

An equivalent statement is that there exists a unique holomorphic bijection f : D → Ω sending 0 to z 0 ∈ Ω and f (0) > 0. The specic map f will be called the Riemann map for z 0 .

1.1.2 Caratheodory convergence theorem Denition 1.1.4. Let U n be a sequence of open sets in C containing 0. Let V n be the connected component of the interior of k≤n U k containing 0. The kernel of the sequence is dened to be the union of the V n 's, provided that it is non-empty; otherwise it is dened to be {0}. Thus the kernel is either a connected open set containing 0 or the one point set {0}.

The sequence is said to converge to a kernel if each subsequence has the same kernel. We now recall the Caratheodory convergence theorem.

Theorem 1.1.5. (Caratheodory convergence theorem). Let (f n ) be a sequence of holomorphic univalent functions on the unit disk D, normalized so that f n (0) = 0 and f n (0) > 0. Then f n converges uniformly on compacta in D to a function f if and only if U n = f n (D) converges to its kernel and this kernel is not C. If this kernel is {0}, then f = 0. Otherwise the kernel is a connected open set U , f is univalent on D and f (D) = U .

1.1.3 Brownian motion -Itô formula Denition 1.1.6. A standard, one-dimentional Brownian motion is a continuoustime stochastic process (B t ) t≥0 characterised by the following properties:

i . B 0 = 0 and B t is continuous in t (a.s.);

ii . Stationarity: if 0 ≤ s ≤ t, the B t -B s has the same law as B t-s ;

iii . Markov property:

B t 1 , B t 2 -B t 1 , • • • , B t k -B t k-1 are independent for all 0 ≤ t 1 < t 2 < • • • < t k .
We say that B t has independent increments. iv . Gaussianity: B t has a normal distribution with mean 0 and variance t.

Y (t, ω) = g(t, X t )
is again an Itô process, whose component number k, Y k , is given by

dY k = ∂g k ∂t (t, X)dt + i ∂g k ∂x i (t, X)dX i + 1 2 i,j ∂ 2 g k ∂ x i ∂ x j (t, X)dX i dX j ,
where dB

(i) t • dB (j) t = δ ij dt, dB (i)
t • dt = dt • dB (i) t = 0.
Readers can see chapter 4 of Ref. [17] to know the denitions of 1-dimensional and multi-dimensional Itô processes.

Radial, whole-plane stochastic Loewner evolution

The stochastic Loewner evolution (or Schramm-Loewner evolution) with parameter κ (SLE κ ), discoverd by Oded Schramm (2000), is a family of random planar curves that have been proven to be a scaling limit of a variety of two-dimentional lattice models in statistical mechanics. Given a parameter κ and a domain in the complex plane U , it gives a family of random curves in U with κ controlling how much the curve turns. SLE κ is the Loewner process driven by the function λ(t) = e i √ κBt in the whole-plane and radial cases, and

λ(t) = √ κB t 1.2. EXPECTATION OF F 0 (Z)
in the chordal case. κ ∈ [0, ∞) and B t is a standard, one-dimensional Brownian motion.

SLE κ is conjectured or proved to describe the scaling limit of various stochastic processes in the plane, the most famous one is critical percolation. The characteristic function of the process √ κB t has the form E(e iξ √ κBt ) = e -tκξ 2 /2

(1.1.1)

We end the rst section with recalling the denitions of (standard, inner) radial SLE κ and the (interior) whole-plane SLE κ version which we study in this work.

Denition 1.1.10. Let λ(t) = exp(i √ κB t ) be a two-sided Brownian motion on the unit circle. The standard inner radial SLE κ process is the the family of conformal maps (g t ) t≥0 satisfying

∂ t g t (z) = g t (z) λ(t) + g t (z) λ(t) -g t (z) , z ∈ D \ K t , (1.1.2)
with initial condition g 0 (z) = z.

Here, (K t ) t≥0 (so-called SLE hulls) is a random increasing family of subsets of the unit disk that grows towards the origin 0. The map g t is the unique conformal map from D \ K t onto D, such that g t (0) = 0 and g t (0) = e t . It can be continued to negative times (via the two-sided Brownian motion B t in the driving function λ(t)).

Denition 1.1.11. The interior whole-plane SLE κ process driven by λ(t) = e i √ κBt is the family of conformal maps (f t ) t≥0 , from D onto the slit domains

(Ω t ), satisfying ∂ ∂t f t (z) = z ∂ ∂z f t (z) λ(t) + z λ(t) -z , z ∈ D, with initial condition lim t→+∞ f t (e -t z) = z.
1.2 Expectation of f 0 (z)

In Ref. [6], the authors give an explicit expression for the expectations of the coecients a n (0) of the expansion (0.0.6) of the interior whole-plane SLE κ map (at time zero) f 0 , thereby obtaining the expectation of the map, E[f 0 (z)]. We now recall this result.

Theorem 1.2.1. For n ≥ 3, setting a n := a n (0) and a 2 := a 2 (0),

a n (0) (law) = e i(n-1) √ κBt a n (t), E(a n ) = -2 n-2 k=1 ( κ 2 k 2 -k -2) n-1 k=1 ( κ 2 k 2 + k) , E(a 2 ) = - 4 κ + 2 . 1.3. EXPECTATION OF LOG F 0 (Z) Z Corollary 1.2.2. The expected conformal map E[f 0 (z)] of the interior whole-plane SLE κ is polynomial of degree k + 1 if κ 2 k 2 = k + 2.
In the same way, we also obtain an explicit expression for the expectations of the logarithmic coecients γ n (0) of the expansion (0.0.7). This result is introduced in the next section.

1.3 Expectation of log f 0 (z) z Dierentiating both sides of (0.0.7) with respect to t and using Loewner equation

(0.0.1) lead to 2 ∞ k=1 γk (t)z k + 1 = z f t (z) f t (z) λ(t) + z λ(t) -z . (1.3.1)
By expanding the right hand side of the quation (1.3.1) as power series, and identifying coecients, one gets the set of equations

γ1 (t) -γ 1 (t) = λ(t), (1.3.2)
γn (t) -nγ n (t) = λ(t) n + 2 n-1 k=1 kλ(t) n-k γ k (t), n ≥ 2;
(

where the dot means a t-derivative and λ(t) = 1/λ(t) = e -i √ κBt .

From Eq. (1.3.2), we have

γ 1 (t) = -e t +∞ t
e -s λ(s)ds,

and hence

E(γ 1 (0)) = - 2 κ + 2 . (1.3.4)
We now use the auxiliary function

β n (t) := e -nt γ n (t), (1.3.5)
and the shorthand notation

X t := e -t-i √ κBt . (1.3.6)
The dierential recursion (1.3.3) then becomes (1.3.9)

βn (t) = 2 n-1 k=1 kX n-k t β k (t) + X n t , n ≥ 2, (1.3.7) with β 1 (t) = e -t γ 1 (t) = - +∞ t X s ds.
Because Eq. (1.3.9) coincides with Eq. ( 64) used for proving Theorem 3.1 in [6] (denoted there by u n (t) with u 1 (t) = 1), we therefore use the same method and arguments for our problem. The proof of Theorem 1.

α n (t) = X t α n-1 (t) -2(n -1) +∞ t X s α n-1 (s)ds = X α n-1 (t) + (n -1)J α n-1 (t) = [X + (n -1)J ]α n-1 (t), (1.3.11)
where the operators X and J are dened as

X α(t) := X t α(t), (1.3.12)
J α(t) := -2 +∞ t X s α(s)ds.

(1.3.13)

Owing to (1.3.10), (1.3.11) and (1.3.13), we obtain for n ≥ 2,

β n = J • [X + (n -1)J ] • • • • • [X + J ]α 1 = 1 2 J n-1 k=1 •[X + kJ ]1, (1.3.14)
where 1 is the constant function equal to 1 on R + .

On the other hand, because of the strong Markov property of Brownian motion

(∀s ≥ t, B s (law)
= B t + Bs-t with Bs being an independent copy of the Brownian motion B s ), we have the identity in law,

X s (law) = X t Xs-t , ∀s ≥ t, (1.3.15)
with Xs := e -s -i √ κ Bs (s ≥ 0) being an independent copy of the process X s (1.3.6).

Using (1.3.15) in the operator J (1.3.13) yields the indentity in law, J α(t)

(law) = -2X t +∞ 0 Xu α(u + t)du = X • J α(t), (1.3
β n (law) = 1 2 J n-1 k=1 •[X (1 + k J [k] )]1, n ≥ 2, (1.3.17)
where the operators

J [k] , k = 1, • • • , n -1, involve successive independent copies, X[k] s k , k = 1, • • • , n -1, of the original process X s (i.e. J [k] α(t) := -2 +∞ 0 X[k] s k α(s k + t)ds k ). Moreover, owing to X s k +t (law) = X t X[k] s k with k = 1, n -1, we obtain J [k] X k-1 t = -2 +∞ 0 X[k] s k (X s k +t ) k-1 ds k (law) = -2X k-1 t +∞ 0 X[k] s k k ds k ,
and thus,

[X (1 + k J [k] )]X k-1 t (law) = X k t 1 -2k +∞ 0 X[k] s k k ds k , k = 1, • • • , n -1. (1.3.18)
The indentities in law (1.3.17

β n (t) (law) = - +∞ t X n s n-1 k=1 1 -2k +∞ 0 X[k] s k k ds k ds. ( 1
e in √ κBt γ n (t) = (X t ) -n β n (t) (law) = - +∞ 0 Xn s n-1 k=1 1 -2k +∞ 0 X[k] s k k ds k ds (law) = β n 0 = γ n (0), (1.3.20)
which implies that the logarithm of the conjugate whole-plane Schramm Loewner evolution e -i √ κBt f t (e i √ κBt z) has the same law as log f 0 (z)

z . Since Xs , X[k] s k , k = 1, • • • , n -1, are successive independent copies of the original process X s , we can then compute E[γ n (0)] by the identity (1.3.20) (also recall that E[( Xs ) k ] = e -( κ 2 k 2 +k)s ). In particular, E[γ n (0)] = - +∞ 0 E[ Xn s ] n-1 k=1 1 -2k +∞ 0 E[( X[k] s k ) k ]ds k ds = - 1 κ 2 n 2 + n n-1 k=1 1 - 2k κ 2 k 2 + k . ( 1
:= γ n (0) and γ 1 (0) := γ 1 , γ n (0) (law) = e in √ κBt γ n (t), E(γ n ) = - n-1 k=1 ( κ 2 k 2 -k) n k=1 ( κ 2 k 2 + k) , (1.3.22) E(γ 1 ) = - 2 κ + 2 . Corollary 1.3.2. The expected conformal map E log f 0 (z) z of the logarithm of the interior whole-plane SLE κ is polynomial of degree k if κ = 2 k . Proof. From Theorem 1.3.1, one sees that E log f 0 (z) z is polynomial if there exists k ∈ N * such that κ 2 k 2 = k, i.e. κ = 2/k, as all E(γ n ) then vanish for n ≥ k + 1.
Corollary 1.3.3. In the SLE 2 case,

E(γ n ) = -1/2, n = 1, 0, n ≥ 2.
The formula (1.3.22) gives for the rst terms:

E(γ 2 ) = - κ -2 2(κ + 1)(κ + 2) , E(γ 3 ) = - 2(κ -1)(κ -2) 3(κ + 1)(κ + 2)(3κ + 2) , E(γ 4 ) = - (κ -1)(κ -2)(3κ -2) 4(κ + 1)(κ + 2)(3κ + 2)(2κ + 1) , E(γ 5 ) = - 2(κ -1)(κ -2)(3κ -2)(2κ -1) 5(κ + 1)(κ + 2)(3κ + 2)(2κ + 1)(5κ + 2) , E(γ 6 ) = - (κ -1)(κ -2)(3κ -2)(2κ -1)(5κ -2) 6(κ + 1)(κ + 2)(3κ + 2)(2κ + 1)(5κ + 2)(3κ + 1)
.

Logarithmic coecient quadratic expectations

In 2010, we computed E(|γ n | 2 ) for small n (see [9]). In particular, we got

E(|γ 1 | 2 ) = 2 κ + 2 , E(|γ 2 | 2 ) = κ 2 + 16κ + 12 4(κ + 1)(κ + 2)(κ + 6)
.

Therefore, as a continuation of this work, we nd an algorithm that we have implemented on MATLAB to compute E(|γ n | 2 ). This section is devoted to present the algorithm and the results which we obtain for γ 3 to γ 9 .

Computational experiments

This algorithm is the same as the one used for computing E(|a n | 2 ) in [6]. In particular, the algorithm is divided into two parts: the rst encodes the computation of γ n , while the second uses it to compute E(|γ n | 2 ).

For the encoding of γ n , we observe that they are linear combinations of successive integrals of the form

∞ t ds 1 ∞ s 1 ds 2 . . . ∞ s k-1 e -(iα 1 √ κBs 1 +β 1 s 1 )-(iα 2 √ κBs 2 +β 2 s 2 )-...-(iα k √ κBs k +β k s k ) ds k . (1.4.1)
Their expectations are encoded as

(α 1 , β 1 ) . . . (α k , β k ) (1 ≤ k ≤ n), (1.4.2)
and are explicitly computed by using as above the strong Markov property and the Gaussian characteristic function (1.1.1):

(α 1 , β 1 ) . . . (α k , β k ) = k-1 j=0 [β k + β k-1 + . . . + β k-j + η(α k + α k-1 + . . . + α k-j )] -1 .
where η(ξ) := κξ 2 /2. Next, in order to compute E(|γ n | 2 ), we have to evaluate the expectation of products of integrals such as (1.4.1) with complex conjugate of others, that we symbolically denote by

[(α 1 , β 1 ) . . . (α k , β k ); (-α 1 , β 1 ) . . . (-α , β )] (1 ≤ k, ≤ n).
(1.4.

3)

The product integrals may be written as a sum of k+ k ordered integrals with k + variables: the k rst ones and the last ones are ordered and the number of ordered integrals corresponds to the number of ways of shuing k cards in the left hand with cards in the right hand. This sum is quite large and, in order to systematically compute it, we write its expectation as the sum of expectations of integrals of the form (1.4.2) that begin with a term of type (α 1 , β 1 ) or with a term of type (-α 1 , β 1 ), thus reducing the work to a computation at lower order.

Computations of coecients for higher orders

Using dynamic programing, we perform computations (formal up to n = 9 and numerical up to n = 16) on a usual computer. Here are the results for γ 3 to γ 9 : + 10856675483136000κ + 444609285120000

E |γ 3 | 2 = 2 9 κ 4 + 31κ 3 + 302κ 2 + 356κ + 120 (2 + 3κ)(1 + κ)(2 + κ)(6 + κ)(10 + κ) ; E |γ 4 | 2 = 1 16
3κ 7 + 154κ 6 + 3105κ 5 + 28534κ 4 + 91464κ 3 + 106672κ 2 + 54288κ + 10080 (14 + κ)(1 + κ)(2 + 3κ)(2 + κ)(6 + κ)(10 + κ)(1 + 2κ)(3 + κ) ; E |γ 5 | 2 = 2 25 6κ 9 + 431κ 8 + 12818κ 7 + 198509κ 6 + 1583120κ 5 + 5077844κ 4 + 6741152κ 3 + 4456176κ 2 + 1442304κ + 181440 / (18 + κ)(14 + κ)(1 + κ)(2 + 3κ)(2 + κ)(6 + κ)(10 + κ) (1 + 2κ)(3 + κ)(2 + 5κ) ; E |γ 6 | 2 = 1 36 30κ
/ (22 + κ)(5 + κ)(1 + 3κ)(18 + κ)(2 + 5κ)(14 + κ)(2 + κ)(1 + κ) (6 + κ)(2 + 3κ)(10 + κ)(1 + 2κ)(3 + κ) ; E |γ 7 | 2 = 2 49 90κ 14 + 11565κ
/ (1 + κ)(1 + 2κ)(2 + κ)(1 + 3κ)(3 + κ)(1 + 4κ)(3 + 2κ)(2 + 3κ)(5 + κ)(2 + 5κ) (6 + κ)(7 + κ)(2 + 7κ)(10 + κ)(14 + κ)(18 + κ)(22 + κ)(26 + κ)(30 + κ) ; E |γ 9 | 2 = 2 81 15120κ 21 + 3130020κ
/ (1 + κ)(1 + 2κ)(2 + κ)(1 + 3κ)(3 + κ)(1 + 4κ)(3 + 2κ)(2 + 3κ)(5 + κ) (2 + 5κ)(6 + κ)(7 + κ)(2 + 7κ)(2 + 9κ)(10 + κ)(10 + 3κ)(14 + κ) (18 + κ)(22 + κ)(26 + κ)(30 + κ)(34 + κ) .
These results lead us to three comments:

i . All the coecients of the polynomial expansions in κ are positive;

ii . For κ → ∞, these expectations vanish as 1 κ .

iii . For all explicitly computed coecients (n ≤ 9), and for all numerically computed ones (n ≤ 16), that the conclusion E(|γ n | 2 ) = then, for κ = 2,

E(γ n ) = -1/2, n = 1, 0, n ≥ 2.
Dierentiating both sides of (2.1.1), we get

z f (z) f (z) = 1 + 2 n≥1 nγ n z n . (2.1.2)
We now consider the SLE one-point function

F (z) := E z f (z) f (z) , (2.1.3)
and, following Ref. [2], aim at nding a partial dierential equation satised by F .

Before we enter the details of the proof, let us, for the benet of the reader not familiar with Ref. [2], detail the strategy of that paper that we will apply in various contexts here.

The starting point is to consider the standard inner radial SLE κ (g t ) t≥0 as dened in Denition 1.1.10. We denote by g -1 t the inverse map of g t , which maps D onto D \ K t . The whole-plane SLE κ map f is rather related to the map g -1 t , but this last function satises, by Loewner's theory, a PDE not well-suited to Itô calculus. To overcome this diculty, one notices that g -1 t and g -t have the same law up to the conjugation by e i √ κBt . This is the purpose of the following lemma (Lemma 1 in Ref.

[2]).

Lemma 2.1.2. Let g t be a radial SLE κ , then for all t ∈ R the map z → g -t (z) has the same distribution as the map z → g -1 t (zλ(t))/λ(t).

We therefore redene a radial SLE κ ft , t ≥ 0, as the continuation g -t of the standard inner radial SLE process g t to negative times, as follows.

Denition 2.1.3. The (conjugate, inverse) radial SLE κ process ft is dened, for

t ∈ R, as ft (z) := g -t (z) (law) = g -1 t (zλ(t))/λ(t), (2.1.4)
thus mapping D onto D \ K t .

Lemma 2.1.2 implies in particular that ft is a solution to the ODE:

∂ t ft (z) = ft (z) ft (z) + λ(t) ft (z) -λ(t) , f0 (z) = z, (2.1.5)
which is the right property needed to apply Itô calculus. In order to apply stochastic calculus, we use Lemma 2 in Ref. [2], which is a version of the SLE's Markov property, ft (z) = λ(s) ft-s ( fs (z)/λ(s)), s ≤ t.

(2.1.6)

To nish, one has to connect the whole-plane SLE with the (modied) radial one.

This is done through Lemma 3 in Ref. [2], which is in our present setting (with a change of an e -t convergence factor there to an e t factor here, when passing from the exterior to the interior of the unit disk D):

Lemma 2.1.4. The limit in law, lim t→+∞ e t ft (z), exists, and has the same law as the (time zero) interior whole-plane random map f 0 (z):

lim t→+∞ e t ft (z) (law) = f 0 (z).
Let us now return to the proof of Theorem 2.1.1.

Proof. Let us introduce the auxiliary, time-dependent, radial variant of the SLE one-point function F (z) (2.1.3) above,

F (z, t) := E z f t (z) ft (z) , (2.1.7)
where ft is a modied radial SLE map at time t as in Denition 2.1.3. Owing to Lemma (2.1.4), we have

lim t→+∞ F (z, t) = F (z). (2.1.8)
We then use a martingale technique to obtain an equation satised by F (z, t).

For s ≤ t, dene M s := E f t (z) ft(z)
|F s , where F s is the σ-algebra generated by {B u , u ≤ s}. (M s ) s≥0 is by construction a martingale. Because of the Markov property of SLE, we have

M s = E f t (z) ft (z) |F s = E f s (z) λ(s) f t-s ( fs (z)/λ(s)) ft-s ( fs (z)/λ(s)) |F s = f s (z) λ(s) E f t-s ( fs (z)/λ(s)) ft-s ( fs (z)/λ(s)) |F s = f s (z) fs (z) F (z s , τ ),
where z s := fs (z)/λ(s), and τ := t -s.

We have from Eq. (2.1.5)

∂ s log f s = ∂ z fs fs+λ(s) fs-λ(s) f s = fs + λ(s) fs -λ(s) - 2λ(s) fs ( fs -λ(s)) 2 (2.1.9) = 1 - 2 (1 -z s ) 2 , ∂ s log fs = ∂ s fs fs = z s + 1 z s -1 , (2.1.10) dz s = z s z s + 1 z s -1 - κ 2 ds -iz s √ κdB s . (2.1.11)
The coecient of the ds-drift term of the Itô derivative of M s is obtained from the above as,

f s (z) fs (z) - 2z s (1 -z s ) 2 + z s z s + 1 z s -1 - κ 2 ∂ z -∂ τ - κ 2 z 2 s ∂ 2 z F (z s , τ ), (2.1.12)
and vanishes by the (local) martingale property. Because fs is univalent, f s does not vanish in D, therefore the bracket above vanishes.

Owing to the existence of the limit (2.1.8), we can now take the τ → +∞ limit in the above, and get the ODE satised by F (z),

P(∂)[F (z)] := - 2z (1 -z) 2 F (z) + z z + 1 z -1 - κ 2 F (z) - κ 2 z 2 F (z) (2.1.13) = - 2z (1 -z) 2 + z z + 1 z -1 ∂ z - κ 2 (z∂ z ) 2 F (z) = 0.
Following Ref. [6], we now look for solutions to (2.1.13) of the form ϕ γ (z) := (1-z) γ .

We have

P(∂)[ϕ γ ] = A(2, 2, γ)ϕ γ + B(2, γ)ϕ γ-1 + C(2, γ)ϕ γ-2 ,
where, in anticipation of the notation that will be introduced in Section 2.2 below,

A(2, 2, γ) := γ - κ 2 γ 2 , B(2, γ) := 2 -3 + κ 2 γ + κγ 2 , C(2, γ) := -2 + 2 + κ 2 γ - κ 2 γ 2 , with, identically, A + B + C = 0. The linear independence of ϕ γ , ϕ γ-1 , ϕ γ-2 thus shows that P(∂)[ϕ γ ] = 0 is equivalent to A = B = C = 0, which yields κ = 2, γ = 1, and F (z) = 1 -z. From Denition (2.1.
3), we thus get Lemma 2.1.5. Let f (z) = f 0 (z) be the interior whole-plane SLE 2 map at time 0, we then have

E z f (z) f (z) = 1 -z.
Theorem 2.1.1 follows from Lemma 2.1.5 and the series expansion (2.1.2).

SLE one-point function

We can generalize the result of Lemma 2.1.5 by considering SLE one-point function with general exponents. In particular, we prove the following theorem.

Theorem 2.2.1. Let f (z) = f 0 (z) be the interior whole-plane SLE κ map at time zero. Consider the curve R, dened parametrically by

p = - κ 2 γ 2 + 2 + κ 2 γ, 2p -q = 1 + κ 2 γ, γ ∈ R. (2.2.1)
On R, the whole-plane SLE κ one-point function has the integrable form,

E (f (z)) p 2 (f (z)/z) q 2 = (1 -z) γ .
Remark 2.2.1. Eq. (2.2.1) describes a parabola in the (p, q) plane (see Fig. 2.1), which is given in Cartesian coordinates by

2κ 2p -q 2 + κ 2 -(4 + κ) 2p -q 2 + κ + p = 0, (2.2.2)
with two branches,

γ = γ ± 0 (p) := 1 2κ 4 + κ ± (4 + κ) 2 -8κp , p ≤ (4 + κ) 2 8κ , q = 2p -1 + κ 2 γ ± 0 (p). (2.2.3)
or, equivalently,

2p = q + 2 + κ 8κ 6 + κ ± (6 + κ) 2 -16κq , q ≤ (6 + κ) 2 16κ .
(2.2.4)

-0.5 0.5 1 1.5 2 -8 -6 -4 -2 2
p p ( )

q Figure 2.1: Integral curves R of Theorem 2.2.
1, for κ = 2 (blue), κ = 4 (red), and κ = 6 (green). In addition to the origin, the q = 0 intersection point with the p-axis is at p(κ) := (6 + κ)(2 + κ)/8κ, with p(2) = p(6) = 2 [6, 11].

Proof. Our aim is to derive an ODE satised by the whole-plane SLE κ one-point function,

F (z) := E z q 2 (f (z)) p 2 (f (z)) q 2 , (2.2.5)
which, by construction, stays nite at the origin and such that F (0) = 1.

We introduce the shorthand notation,

X t (z) := ( f t (z)) p 2 ( ft (z)) q 2 , (2.2.6)
where ft is the conjugate, reversed radial SLE process in D, as introduced in Denition 2.1.3, and such that by Lemma 2.1.4, the limit, lim t→+∞ e t ft (z), is the same in law as the whole-plane map at time zero. Applying the same method as in the previous section, we consider the auxiliary, time-dependent function

F (z, t) := E z q 2 X t (z) , (2.2.7) such that lim t→+∞ exp p -q 2 t F (z, t) = F (z).
(2.2.8)

Consider now the martingale (M s ) t≥s≥0 , dened by

M s = E(X t (z)|F s ).
By the SLE Markov property we get, setting z s := fs (z)/λ(s),

M s = X s (z) F (z s , τ ), τ := t -s.
(2.2.9)

As before, the partial dierential equation satised by F (z s , τ ) is obtained by expressing the fact that the ds-drift term of the Itô dierential of Eq. (2.2.9),

dM s = F dX s + X s d F ,
vanishes. The dierential of X s is simply computed from Eqs. (2.1.9) and (2.1.10) above as:

dX s (z) = X s (z)F 1 (z s )ds, F 1 (z) := p 2 1 - 2 (1 -z) 2 - q 2 1 - 2 1 -z .
(2.2.10)

The Itô dierential d F brings in the ds terms proportional to ∂ zs F , ∂ 2 zs F , and ∂ τ F ; therefore, in the PDE satised by F , the latter terms are exactly the same as in the PDE (2.1.12). We therefore directly arrive at the vanishing condition of the overall drift term coecient in dM s ,

X s (z) F 1 (z s ) + z s z s + 1 z s -1 - κ 2 ∂ z -∂ τ - κ 2 z 2 s ∂ 2 z F (z s , τ ) = 0. (2.2.11)
Since X s (z) does not vanish in D, the bracket in (2.2.11) must identically vanish:

F 1 (z s ) + z s z s + 1 z s -1 ∂ z -∂ τ - κ 2 (z s ∂ z ) 2 F (z s , τ ) = 0, (2.2.12)
where we used z∂ z + z 2 ∂ 2 z = (z∂ z ) 2 .

To derive the ODE satised by F (z) (2.2.5), we rst recall its expression as the limit (2.2.8), which further implies

lim τ →+∞ exp p -q 2 τ ∂ τ F (z, τ ) = - p -q 2 F (z).
Multiplying the PDE (2.2.11) satised by F by exp( p-q 2 τ ) and letting τ → +∞, we obtain

P(∂)[F (z)] := - κ 2 (z∂ z ) 2 - 1 + z 1 -z z∂ z + F 1 (z) + p -q 2 F (z) = - κ 2 (z∂ z ) 2 - 1 + z 1 -z z∂ z - p (1 -z) 2 + q 1 -z + p -q F (z) = 0.
(2.2.13)

We now look specically for solutions to (2.2.13), together with the boundary condition F (0) = 1, of the form ϕ γ (z) = (1 -z) γ . This function satises the simple dierential operator algebra [6] P

(∂)[ϕ γ ] = A(p, q, γ)ϕ γ + B(q, γ)ϕ γ-1 + C(p, γ)ϕ γ-2 , (2.2.14)
where A(p, q, γ)

:= p -q + γ - κ 2 γ 2 , ( 2

SLE two-point function

In this section, we aim at proving the important property mentioned at the end of chapter 1. That is the following theorem.

Theorem 2.3.1. Let f (z) := f 0 (z) be the time 0 interior whole-plane SLE κ map, in the same setting as in Theorem 2.1.1, then, for κ = 2,

E(|γ n | 2 ) = 1 2n 2 , ∀n ≥ 1.
The proof of Theorem 2.3.1 is a rst motivation of our article [5]. The idea behind the proof, which is a development of Parseval's formula, is to use the series expansion (2.1.2) and to compute E z f (z)

f (z) 2
. We indeed prove:

Theorem 2.3.2. Let f be the interior whole-plane SLE κ map, in the same setting as in Theorem 2.3.1; then for κ = 2,

E z f (z) f (z) 2 = (1 -z)(1 -z) 1 -z z .
We now establish systematically the proof of Theorem 2.3.2.

BeliaevSmirnov type equations

In order to get Theorem 2.3.2, we will determine the moduli one-point function,

E |z| q |f (z)| p |f (z)| q ,
for (p, q) belonging to the same parabola R as in Theorem 2.2.1, and where f = f 0 is the (time zero) interior whole-plane SLE κ map.

In contradistinction to the method used in Refs. [2, 6] for writing a PDE obeyed by E(|f (z)| p ), we shall use here a slightly dierent approach, building on the results obtained in Section 2.2. We shall study the SLE two-point function for z 1 , z 2 ∈ D,

G(z 1 , z2 ) := E z q 2 1 (f (z 1 )) p 2 (f (z 1 )) q 2 z q 2 2 (f (z 2 )) p 2 (f (z 2 )) q 2 . (2.3.1)
As before, we dene a time-dependent, auxiliary two-point function,

G(z 1 , z2 , t) := E   z q 2 1 ( f t (z 1 )) p 2 ( ft (z 1 )) q 2 z q 2 2 ( f t (z 2 )) p 2 ( ft (z 2 )) q 2   = E z q 2 1 X t (z 1 )z q 2 2 X t (z 2 ) , (2.3.2)
where as above ft is the reverse radial SLE κ process 2.1.3, and where we used the shorthand notation (2.2.6). This time, the two-point function (2.3.1) is the limit

lim t→+∞ e (p-q)t G(z 1 , z2 , t) = G(z 1 , z2 ). (2.3.3)
Let us dene the two-point martingale (M s ) t≥s≥0 , with

M s := E(X t (z 1 )X t (z 2 )|F s ).
By the Markov property of SLE,

E X t (z 1 )X t (z 2 )|F s = X s (z 1 )X s (z 2 ) G(z 1s , z2s , τ ), τ := t -s, (2.3.4)
where z 1s := fs (z 1 )/λ(s); z2s := fs (z 2 )/λ(s) = fs (z 2 )λ(s).

(2.3.5)

Their Itô dierentials, dz 1s and dz 2s , are as in (2.1.11),

dz 1s = z 1s z 1s + 1 z 1s -1 - κ 2 ds -i √ κ z 1s dB s , dz 2s = z2s z2s + 1 z2s -1 - κ 2 ds + i √ κ z2s dB s . (2.3.6)
As before, the partial dierential equation satised by G(z 1s , z2s , τ ) is obtained by expressing the fact that the ds-drift term of the Itô dierential of Eq. (2.3.4),

dM s = [dX s (z 1 )X s (z 2 ) + X s (z 1 )dX s (z 2 )] G + X s (z 1 )X s (z 2 ) d G, (2.3.7)
vanishes.

The dierentials of X s , X s are as in Eq. (2.2.10) above:

dX s (z 1 ) = X s (z 1 )F 1 (z 1s )ds, dX s (z 2 ) = X s (z 2 )F 1 (z 2s )ds, F 1 (z) := p 2 - q 2 - p (1 -z) 2 + q 1 -z . (2.3.8)
We thus obtain the simple expression

dM s = X s (z 1 )X s (z 2 ) [F 1 (z 1s ) + F 1 (z 2s )] G ds + d G , (2.3.9)
and the vanishing of the ds-drift term in dM s requires that of the drift term in the right-hand side bracket in (2.3.9), since X s (z) does not vanish in D.

- κ 2 z 2 1s ∂ 2 1 G ds - κ 2 z2 2s ∂2 2 G ds + κz 1s z2s ∂ 1 ∂2 G ds,
where use was made of the shorthand notations, ∂ 1 := ∂ z 1 and ∂2 := ∂ z2 . We observe that the only coupling between the z 1s , z2s variables arises in the last term of (2.3.10), the other terms simply resulting from the independent contributions of the z 1s and z2s parts.

Using again the Itô dierentials (2.3.6), we can rewrite (2.3.10) as

d G = -i √ κ z 1s ∂ 1 -z2s ∂2 G dB s (2.3.11) + z 1s + 1 z 1s -1 z 1s ∂ 1 G ds + z2s + 1 z2s -1 z2s ∂2 G ds -∂ τ G ds - κ 2 (z 1s ∂ 1 -z2s ∂2 ) 2 G ds,
where we used the obvious formal identity

(z 1 ∂ 1 ) 2 + (z 2 ∂2 ) 2 -2z 1 ∂ 1 z2 ∂2 = (z 1 ∂ 1 -z2 ∂2 ) 2 .
(2.3.12)

At this stage, comparing the computations (2.3.9) and (2.3.11) above with those in the one-point martingale study in Section 2.2, it is clear that the PDE obeyed by G = G(z 1s , z2s , τ ) is obtained as two duplicates of Eq. (2.2.12), completed as in (2.3.12) by the derivative coupling between variables z 1s , z2s :

F 1 (z 1s ) + z 1s z 1s + 1 z 1s -1 ∂ 1 + F 1 (z 2s ) + z2s z2s + 1 z2s -1 ∂2 -∂ τ - κ 2 (z 1s ∂ 1 -z2s ∂2 ) 2 G = 0.
(2.3.13)

The existence of the limit (2.3.3) further implies that of

lim τ →∞ e (p-q)τ ∂ τ G(z 1 , z2 , τ ) = -(p -q)G(z 1 , z2 ).
Multiplying the PDE (2.3.13) satised by G by exp((p -q)τ ) and letting τ → +∞, then gives the expected PDE for G(z 1 , z2 ). It can be most compactly written in terms of the ODE (2.2.13) as

P(∂ 1 ) + P( ∂2 ) + κz 1 ∂ 1 z2 ∂2 G(z 1 , z2 ) = 0, (2.3.14)
and its fully explicit expression is

P(D)[G(z 1 , z2 )] = - κ 2 (z 1 ∂ 1 -z2 ∂2 ) 2 G - 1 + z 1 1 -z 1 z 1 ∂ 1 G - 1 + z2 1 -z2 z2 ∂2 G (2.3.15) + - p (1 -z 1 ) 2 - p (1 -z2 ) 2 + q 1 -z 1 + q 1 -z2 + 2p -2q G = 0.

Moduli one-point function

We note that one can take the z 1 = z 2 = z case in Denition (2.3.1) above, thereby obtaining the whole-plane SLE κ moduli one-point function,

G(z, z) = E |z| q |f (z)| p |f (z)| q . ( 2
P(D)[G(z, z)] = - κ 2 (z∂ -z ∂) 2 G - 1 + z 1 -z z∂G - 1 + z 1 - z z ∂G (2.3.17) + - p (1 -z) 2 - p (1 -z) 2 + q 1 -z + q 1 - z + 2p -2q G = 0,
which is the generalization to q = 0 of the BeliaevSmirnov equation studied in Refs.

[6] and [11].

Integrable case

We begin with a Lemma concerning the uniqueness of the solutions to Eq.(2.3.17).

Lemma 2.3.3. The space of formal series F (z 1 , z2 ) = k, ∈N a k, z k 1 z 2 , with complex coecients and that are solutions of the PDE (2.3.15), is one-dimensional.

Proof. We assume that F is a solution to (2.3.15) with F (0, 0) = 0; it suces to prove that, necessarily, F = 0. We argue by contradiction: If not, consider the minimal (necessarily non constant) term a k,l z k z in the series of F , with a k, = 0 and k + minimal (and non vanishing). Then P(D)[F ] (2.3.15) will have a minimal term, equal to -a k, κ

2 (k -) 2 + k + z k 1 z 2 , which is non-zero, contradicting the fact that P(D)[F ] vanishes.
As a second step, following Ref. [6], let us consider the action of the operator P(D) of (2.3.15) on a function of the factorized form ϕ(z 1 )ϕ(z 2 )P (z 1 , z2 ), which we write, in a shorthand notation, as ϕ φP . By Leibniz's rule, it is given by

P(D)[ϕ φP ] = - κ 2 ϕ φ(z 1 ∂ 1 -z2 ∂2 ) 2 P -κ(z 1 ∂ 1 -z2 ∂2 )(ϕ φ)(z 1 ∂ 1 -z2 ∂2 )P + κ(z 1 ∂ 1 ϕ)(z 2 ∂2 φ)P -ϕ φ 1 + z 1 1 -z 1 z 1 ∂ 1 P -ϕ φ 1 + z2 1 -z2 z2 ∂2 P - κ 2 φ(z 1 ∂ 1 ) 2 ϕ + κ 2 ϕ(z 2 ∂2 ) 2 φ + φ 1 + z 1 1 -z 1 z 1 ∂ 1 ϕ + ϕ 1 + z2 1 -z2 z2 ∂2 φ P + - p (1 -z 1 ) 2 - p (1 -z2 ) 2 + q 1 -z 1 + q 1 -z2 + 2p -2q ϕ φP.
Note that the operator z 1 ∂ 1 -z2 ∂2 is antisymmetric with respect to z 1 , z2 ; therefore, if we choose a symmetric function, P (z 1 , z2 ) = P (z 1 z2 ), the rst line of P(D)[ϕ φP ] above identically vanishes.

One then looks for solutions to (2.3.15) of the particular form,

G(z 1 , z2 ) = ϕ γ (z 1 )ϕ γ (z 2 )P (z 1 z2 ),
where, as before, ϕ γ (z) = (1 -z) γ . The action of the dierential operator then takes the simple form,

P(D)[ϕ γ φγ P ] =z 1 z2 ϕ γ-1 φγ-1 κγ 2 P -2(1 -z 1 z2 )P + P(∂ 1 )[ϕ γ ] φγ P + P( ∂2 )[ φγ ]ϕ γ P,
where P is the derivative of P with respect to z 1 z2 , and P(∂) is the so-called boundary operator (2.2.13) [6].

The ODE, κγ 2 P (x) -2(1 -x)P (x) = 0 with x = z 1 z2 and P (0) = 1, has for solution P (z 1 z2 ) = (1 -z 1 z2 ) -κγ 2 /2 . It is then sucient to pick for γ the value γ = γ ± 0 (p) (2.2.3) such that P(∂)[ϕ γ ] = 0, as obtained in the proof of Theorem 2.2.1, to get a solution of the PDE, P(D)[ϕ γ φγ P ] = 0 (2.3.15). By uniqueness of the solution with G(0, 0) = 1, it gives the explicit form of the SLE two-point function,

G(z 1 , z2 ) = ϕ γ (z 1 )ϕ γ (z 2 )(1 -z 1 z2 ) -κγ 2 /2 .
We thus get: Theorem 2.3.4. Let f (z) = f 0 (z) be the interior whole-plane SLE κ map in the setting of Theorem (2.1.1); then, for (p, q) belonging to the parabola R dened in Theorem 2.2.1 by Eqs. (2.2.1) or (2.2.2) or (2.2.3), and for any pair

(z 1 , z 2 ) ∈ D×D, E z q 2 1 (f (z 1 )) p 2 (f (z 1 )) q 2 z q 2 2 (f (z 2 )) p 2 (f (z 2 )) q 2 = (1 -z 1 ) γ (1 -z2 ) γ (1 -z 1 z2 ) β , β = κ 2 γ 2 .
Corollary 2.3.5. In the same setting as in Theorem 2.3.4, we have for z ∈ D,

E |z| q |f (z)| p |f (z)| q = (1 -z) γ (1 -z) γ (1 -z z) β , β = κ 2 γ 2 , for γ = γ ± 0 (p) := 1 2κ 4 + κ ± (4 + κ) 2 -8κp , p ≤ (4 + κ) 2 8κ , q = 2p -1 + κ 2 γ ± 0 (p).
Let us stress some particular cases of interest. First, the p = 0 case gives the integral means of f . Corollary 2.3.6. The interior whole-plane SLE κ map has the integrable moment

E   f (z 1 ) z 1 (2+κ)(4+κ) 4κ f (z 2 ) z2 (2+κ)(4+κ) 4κ   = (1 -z 1 ) 4+κ κ (1 -z2 ) 4+κ κ (1 -z 1 z2 ) (4+κ) 2 2κ , E f (z) z (2+κ)(4+κ) 2κ = (1 -z) 4+κ κ (1 -z) 4+κ κ (1 -z z) (4+κ) 2 2κ
.

Second, taking p = q gives the logarithmic integral means we started with:

Corollary 2.3.7. The interior whole-plane

SLE κ map f (z) = f 0 (z) has the inte- grable logarithmic moment E   z 1 f (z 1 ) f (z 1 ) 2+κ 2κ z2 f (z 2 ) f (z 2 ) 2+κ 2κ   = (1 -z 1 ) 2 κ (1 -z2 ) 2 κ (1 -z 1 z2 ) 2 κ , E z f (z) f (z) 2+κ κ = (1 -z) 2 κ (1 -z) 2 κ (1 -z z) 2 κ
.

Theorem 2.3.2 is the κ = 2 case of the latter result.

z f (z) f (z) 2 = 1 + 2 n≥1 nγ n (z n + zn ) + n≥1 m≥1
nmγ n γm z n zm .

(2.3.18)

On the other hand, by Theorem 2.3.2,

E z f (z) f (z) 2 = (1 -z)(1 -z) (1 -z z) = 1 - n≥0 z n+1 zn - n≥0 z n zn+1 + 2 n≥1 z n zn .
Identifying the latter with the expectation of (2.3.18), we get the expected coecients

E(γ 1 ) = -1/2, E(γ n ) = 0, n ≥ 2, E(|γ n | 2 ) = 1 2n 2 , n ≥ 1, E(γ n γn+1 ) = - 1 n(n + 1) , E(γ n γn+k ) = 0, n ≥ 1, k ≥ 2,
k|γ k | 2 - 1 k = - 1 2 n m=1 m k=1 1 k = - n + 1 2 n+1 k=2 1 k ,
which gives an example of the validity in expectation" of the Milin conjecture.

Recalling Denition (0.0.2), we also obtain, in expectation, a check of Robertson's conjecture (0.0.3):

E log n k=0 |b 2k+1 | 2 n + 1 ≤ 1 n + 1 E n m=1 m k=1 k|γ k | 2 - 1 k = - 1 2 n+1 k=2 1 k , owing to the second Lebedev-Milin inequality (0.0.4), ∀n ≥ 0, n k=0 |b 2k+1 | 2 ≤ (n + 1) exp 1 n + 1 n m=1 m k=1 k|γ k | 2 - 1 k .
Chapter 3

Generalized spectrum

The starting point of this chapter was to study a new parabola along which we obtain the general expressions for the whole-plane SLE κ functions (2.2.5) and (2.3.16). Next, we determine all parameters (p, q) such that the SLE κ one-point function (2.2.5) is a polynomial of degree n (for any positive integer n), and consider the p = q = -2 -κ case. We end this chapter by studying the generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the mixed moments E |f (z)| p /|f (z)| q , in the whole parameter space (p, q) ∈ R 2 .

A new parabola

In Theorem 2.2.1, by looking for solutions to (2.2.13) of the form ϕ γ (z) = (1-z) γ , we obtained the parabola R dened by (2.2.1) or (2.2.2). In the following theorem, we will give a new parabola, denoted by R 1 , by looking for solutions to (2.2.13) of the particular form,

(1 -α)ϕ γ-1 (z) + αϕ γ (z), γ, α ∈ R. (3.1.1) Theorem 3.1.1. Let f (z) = f 0 (z) be the interior whole-plane SLE κ map at time zero. Consider the curve R 1 , dened parametrically by p = - κ 2 γ 2 + 2 + 3κ 2 γ -κ -2, 2p -q = 1 + 3κ 2 γ -κ -2, γ ∈ R. (3.1.2)
On R 1 , the whole-plane SLE κ one-point function has the integrable form,

E (f (z)) p 2 (f (z)/z) q 2 = 2 - 2κ 2 + κ γ (1 -z) γ-1 + 2κ 2 + κ γ -1 (1 -z) γ . (3.1.3) Remark 3.1.1. Eq. (3.1.
2) describes a parabola in the (p, q) plane (see Fig. 3.1), which is given in Cartesian coordinates by

2κ 2p -q + κ + 2 2 + 3κ 2 -(4 + 3κ) 2p -q + κ + 2 2 + 3κ + κ + 2 + p = 0, (3.1.4) 3.1. A NEW PARABOLA
with two branches,

γ = γ ± 1 (p) := 1 2κ 4 + 3κ ± (4 + κ) 2 -8κp , p ≤ (4 + κ) 2 8κ , q = 2p -1 + 3κ 2 γ ± 1 (p) + κ + 2, (3.1.5)
or, equivalently, . The p = 0 intersection points with the q-axis are at q 1 (κ) := -2(2 + κ)(1 + κ)/κ, with q 1 (2) = -12, and q 2 (κ) := (2 -κ)/2 (not marked).

2p = q+ 2 + 3κ 8κ 6 + 3κ ± -7κ 2 + 4κ(1 -4q) + 36 -κ-2, q ≤ -7κ 2 + 4κ + 36 16κ . ( 3
Proof. As mentioned above, let us look specically for solutions to the ODE (2.2.13), together with the boundary condition F (0) = 1, of the particular form (3.1.1).

According to (2.2.14), the function (3.1.1) satises the simple dierential operator algebra

P(∂)[(1 -α)ϕ γ-1 + αϕ γ ] = αA(p, q, γ)ϕ γ + [αB(q, γ) + (1 -α)A(p, q, γ -1)]ϕ γ-1 + [αC(p, γ) + (1 -α)B(q, γ -1)]ϕ γ-2 + (1 -α)C(p, γ -1)ϕ γ-3 , (3.1.7)
where, as dened by (2.2.15), (2.2.16) and (2.2.17),

A(p, q, γ)

:= p -q + γ - κ 2 γ 2 , B(q, γ) := q -3 + κ 2 γ + κγ 2 , C(p, γ) := -p + 2 + κ 2 γ - κ 2 γ 2 .
Owing to A + B + C = 0, identically, it follows that the sum of the coecients of ϕ γ , ϕ γ-1 , ϕ γ-2 , ϕ γ-3 in (3.1.7) identically equals 0. Because ϕ γ , ϕ γ-1 , ϕ γ-2 , ϕ γ-3 are linearly independent, the condition P

(∂)[(1 -α)ϕ γ-1 + αϕ γ ] is equivalent to the system A(p, q, γ) = C(p, γ -1) = αB(q, γ) + (1 -α)A(p, q, γ -1) = 0, hence C(p, γ -1) = -p - κ 2 γ 2 + 2 + 3κ 2 γ -κ -2 = 0, A(p, q, γ) -C(p, γ -1) = 2p -q -1 + 3κ 2 γ + k + 2 = 0, and α = 2κ 2 + κ γ -1.
It yields precisely the parabola parametrization (3.1.2) and the expression (3.1.3) given in Theorem 3.1.1.

Next, we will determine the SLE κ moduli one-point function G(z, z) (2.3.16) for (p, q) belonging to the parabola R 1 as in Theorem 3.1.1. Relying on some results which we obtained for some specic values of p, q and κ, we aim at looking for solutions to the PDE (2.3.17) with (p, q) given by Eq. (3.1.2) of the particular form,

(1 -z) γ-1 (1 -z) γ-1 (P 0 (z z) + (z + z)P 1 (z z)), (3.1.9)
where P 0 and P 1 are symmetric functions, i.e. P 0 (z, z) = P 0 (z z), P 1 (z, z) = P 1 (z z).

Substituting (3.1.9) into the corresponding PDE yields the following system,

4u(u -1)P 1 (u) + ((2γ 2 κ -κ -2)u -κ -2)P 1 (u) + (2 + κ -2γκ)P 0 (u) = 0, 4u(u -1)P 0 (u) + 2(γ 2 κ -2)uP 0 (u) + 8(κ + 1 -γκ)uP 1 (u) = 0, (3.1.10)
where u := z z. Solving the above system by MAPLE and using the boundary condition G(0, 0) = 1, we nd the particular solution (P 0 , P 1 ) dened as

P 0 (u) = 1 (1 -u) β(γ,κ) hypergeom 1 -(γ -1)κ, 3 2 - 4γ -3 4 κ , 1 2 + κ 4 , u , (3.1.11) P 1 (u) = (4γ -3)κ -6 2(κ + 2)(1 -u) β(γ,κ) (u -1) hypergeom 2 -(γ -1)κ, 5 2 - 4γ -3 4 κ , 3 2 + κ 4 , u - 1 2(1 -u) β(γ,κ) hypergeom 1 -(γ -1)κ, 3 2 - 4γ -3 4 κ , 1 2 + κ 4 , u , (3.1.12) with β(γ, κ) := κ 2 γ 2 + κγ -κ -2.
The hypergeom(n, d, z) calling sequence is the generalized hypergeometric function F (n, d, z), as introduced in Appendix A.

We thus get: Theorem 3.1.2. Let f (z) = f 0 (z) be the interior whole-plane SLE κ map at time zero; then, for (p, q) belonging to the parabola R 1 dened in Theorem 3.1.1 by Eqs.

E |z| q |f (z)| p |f (z)| q = (1 -z) γ-1 (1 -z) γ-1 P 0 (z z) + (z + z)P 1 (z z) ,
where the functions P 0 and P 1 are dened by Eqs. (3.1.11) and (3.1.12), respectively.

From Theorem 3.1.2, together with the support of MAPLE, we obtain the following expressions for G(z, z) for some specic values of (κ, γ). [START_REF] Lebedev | On the coecients of certain classes of univalent functions[END_REF] .

• In the (κ = 6, γ = 1) case, E f (z) z 2 = 2 -(z + z) 2(1 -z z) . • In the (κ = 10, γ = 1) case, E f (z) z 4 = 3 -z z + (z + z)(z z -2) 3(1 -z z) 3 . • In the (κ = 14, γ = 1) case, E f (z) z 6 = 2(z z) 2 -10z z + 20 + (z + z)(-3(z z) 2 + 12z z -15) 20(1 -z z) 5 . • In the (κ = 3, γ = 2) case, E |f (z)| 2 f (z) z 2 = (1 -z)(1 -z) 5(z z) 2 + 18z z + 5 -7(z + z)(1 + z z) 5(1 -z z) 7 . • In the (κ = 4, γ = 2) case, E |f (z)| 2 f (z) z 4 = (1 -z)(1 -z) 3(z z) 3 + 21(z z) 2 + 21z z + 3 -(z + z)(5(z z) 2 + 14z z + 5) 3(1 -z z)
• In the (κ = 2, γ = 3) case, E |f (z)| 2 f (z) z 4 = (1 -z) 2 (1 -z) 2 P 0 (z z) + (z + z)P 1 (z z) ,
where P 0 (z z), P 1 (z z) are dened as

P 0 (z z) = (z z) 3 + 9(z z) 2 + 9(z z) + 1 (1 -z z) 11 , P 1 (z z) = - 2(z z) 2 + 6(z z) + 2 (1 -z z) 11 . • In the (κ = 1, γ = 4) case, E |f (z)| 3 f (z) z = (1 -z) 3 (1 -z) 3 3(z z) 2 + 14z z + 3 -5(z + z)(z z + 1) 3(1 -z z) 9 .
Although Eqs. (3.1.11) and (3.1.12) can't give us the explicit expressions for P 0 and P 1 when γ = 1 and κ ∈ {4, 8, 12, 16, 20}, however, by solving the system (3.1.10) corresponding to each specic value of (γ, κ), together with the condition G(0, 0) = 1, we still obtain the explicit expressions for P 0 and P 1 (given in the polar coordinate system (r, θ)) in these case. For instance, we have: [START_REF] De Branges | A proof of the Bieberbach conjecture[END_REF] ,

• In the (κ = 4, γ = 1) case, E f (z) z = ln 1+r 1-r 2r + 2r -(1 + r 2 ) ln 1+r 1-r 4r 2 cos(θ). • In the (κ = 8, γ = 1) case, E f (z) z 3 = 6r(1 + r 2 ) -3(1 -r 2 ) 2 ln 1+r 1-r 16r 3 (1 -r 2 ) 2 + 9(r 6 -r 4 -r 2 + 1) ln 1+r 1-r -18r 5 + 12r 3 -18r 32r 4 (1 -r 2 ) 2 cos(θ). • In the (κ = 12, γ = 1) case, E f (z) z 5 = 5 512
Q 1 (r) + Q 2 (r) ln 1+r 1-r + Q 3 (r) + Q 4 (r) ln 1+r 1-r cos(θ) r 6 (1 -r 2 )
where the polynomials Q i (i = 1, 4) are dened as

Q 1 (r) = -12r 2 + 44r 4 + 44r 6 -12r 8 , Q 2 (r) = 6(r -4r 3 + 6r 5 -4r 7 + r 9 ), Q 3 (r) = 30r -80r 3 + 36r 5 -80r 7 + 30r 9 , Q 4 (r) = -15 + 45r 2 -30r 4 -30r 6 + 45r 8 -15r 10 .
Remark 3.1.2. In the polar coordinate system (r, θ), the PDE (2.3.17) satised by

G becomes P(D)[G(re iθ , re iθ )] = κ 2 ∂ 2 θ,θ G + r(r 2 -1) r 2 -2r cos(θ) + 1 ∂ r G - 2r sin(θ) r 2 -2r cos(θ) + 1 ∂ θ G + p r 4 + 4r 2 (1 -r cos(θ)) -1 (r 2 -2r cos(θ) + 1) 2 + 1 + q 2r cos(θ) -2r 2 r 2 -2r cos(θ) + 1 G = 0.
One can thus veries the accuracy of the solutions above.

The case F (z) polynomial

In this section, our aim is to answer the following question: Let n be an arbitrary positive integer, what cases will the SLE κ one-point fuction F (z) (2.2.5) be a polynomial n?

We approach to this question consisted in observing that: F (z) is an arbitrary polynomial of degree n if, and only if, F (z) is a linear combination of the functions ϕ i (z) = (1 -z) i with i = 0, n. As a result, we now look for solutions to (2.2.13), together with the boundary condition F (0) = 1, of the particular form,

F (z) = α 0 ϕ γ (z) + α 1 ϕ γ-1 (z) + • • • + α k ϕ γ-k (z), (3.2.1)
where k is a given nonnegative integer, and α 0 , • • • , α k are real numbers such that

α 0 + • • • + α k = 1.
Recall that Theorem 2.2.1 and Theorem 3.1.1 are obtained by considering the cases of k = 0 and k = 1, respectively. According to (2.2.14), the function (3.2.1) satises the dierential operator algebra

P(∂) [α 0 ϕ γ (z) + α 1 ϕ γ-1 (z) + • • • + α k ϕ γ-k (z)] (3.2.2) = k i=0 α i [A(p, q, γ -i)ϕ γ-i + B(q, γ -i)ϕ γ-i-1 + C(p, γ -i)ϕ γ-i-2 ] = α 0 A(p, q, γ)ϕ γ + [α 0 B(q, γ) + α 1 A(p, q, γ -1)]ϕ γ-1 + [α 0 C(p, γ) + α 1 B(q, γ -1) + α 2 A(p, q, γ -2)]ϕ γ-2 + • • • + α k C(p, γ -k)ϕ γ-k-2 ,
where, as before, A(p, q, γ), B(q, γ), C(p, γ) are dened by (2.2.15), (2.2.16) and (2.2.17), such that, identically, A

+ B + C = 0. Because ϕ γ , ϕ γ-1 , • • • , ϕ γ-k-2 are linearly independent, one has A(p, q, γ) = C(p, γ -k) = 0, hence C(p, γ -k) = -p - κ 2 γ 2 + 2 + (2k + 1) κ 2 γ -k(k + 1) κ 2 -2k = 0, A(p, q, γ) -C(p, γ -k) = 2p -q -1 + (2k + 1) κ 2 γ + k(k + 1) κ 2 + 2k = 0, which are equivalent to p = - κ 2 γ 2 + 2 + (2k + 1) κ 2 γ -k(k + 1) κ 2 -2k, 2p -q = 1 + (2k + 1) κ 2 γ -k(k + 1) κ 2 -2k. (3.2.3) Remark 3.2.1. Eq. (3.2.
3) describes a parabola, denoted by R k , in the (p, q) plane, which is given in Cartesian coordinates by

2κ 2p -q + k(k + 1) κ 2 + 2k 2 + (2k + 1)κ 2 -(4 + (2k + 1)κ) 2p -q + k(k + 1) κ 2 + 2k 2 + (2k + 1)κ (3.2.4) + k(k + 1) κ 2 + 2k + p = 0,
with two branches,

γ = γ ± k (p) := 1 2κ 4 + (2k + 1)κ ± (4 + κ) 2 -8κp , p ≤ (4 + κ) 2 8κ , q = 2p -1 + (2k + 1) κ 2 γ ± k (p) + k(k + 1) κ 2 + 2k, (3.2.5)
or, equivalently,

γ = γ ± k (q) := 1 4κ 6 + (2k + 1)κ ± (1 -4k(k + 1))κ 2 + 4κ(3 -2k -4q) + 36 , q ≤ (1 -4k(k + 1))κ 2 + 4κ(3 -2k) + 36 16κ , 2p = q + 1 + (2k + 1) κ 2 γ ± k (q) -k(k + 1) κ 2 -2k.
Obviously, one regets the parabola R (2.2.2) for k = 0, and the parbola R 1 (3.1.4) for k = 1.

Eq. (3.2.3) can be rewritten in a simplier form,

p = 2(γ -k) - κ 2 (γ -k)(γ -k -1), q = p + γ - κ 2 γ 2 . (3.2.6)
Solving the ODE (2.2.13) with (p, q) given by Eq. (3.2.6), together with the boundary condition F (0) = 1, we obtain

F (z) = (1 -z) γ-k hypergeom -k, (2γ -k)κ -2 κ , κ + 2 κ , z = (1 -z) γ-k 1 + ∞ m=1 1 m! • m-1 j=0 (k -j) (k -2γ -j)κ + 2 m j=1 (jκ + 2) z m =    (1 -z) γ if k = 0, (1 -z) γ-k 1 + k m=1 1 m! • m-1 j=0 (k-j)((k-2γ-j)κ+2) m j=1 (jκ+2) z m if k ≥ 1. (3.2.7)
Eq. (3.2.7) shows the determination of real numbers α i (i = 0, k) in (3.2.1), and the fact that F (z) is a polynomial of degree n when γ = n and k ≤ n,

F (z) =    (1 -z) n if k = 0, (1 -z) n-k 1 + k m=1 1 m! • m-1 j=0 (k-j)((k-2n-j)κ+2) m j=1 (jκ+2) z m if 1 ≤ k ≤ n.
From this, we get the following proposition.

:= {p = (4 + κ) 2 /8κ} and ∆ 1 := {(p, q) ∈ R 2 , q = p + 1/
p = 2(n -k) - κ 2 (n -k)(n -k -1), q = p + n - κ 2 n 2 , k = 0, n.
In other words, if and only if (p, q) is one of the points dened by γ = n on

(n + 1) parabolas R, R 1 , • • • , R n .
Here are the examples for n = 2, n = 3 and n = 4:

Example 3.2.1. The function F (z) is a polynomial of degree 2 if, and only if,

             p = 0, q = 2 -2κ; or, p = 2, q = 4 -2κ; or, p = 4 -κ, q = 6 -3κ;
Example 3.2.2. The function F (z) is a polynomial of degree 3 if, and only if,

                               p = 0, q = 3 - 9 2 κ; or, p = 2, q = 5 - 9 2 κ; or, p = 4 -κ, q = 7 - 11 2 κ; or, p = 6 -3κ, q = 9 - 15 2 κ.
Example 3.2.3. The function F (z) is a polynomial of degree 4 if, and only if,

                                 p = 0, q = 4 -8κ; or, p = 2, q = 6 -8κ; or, p = 4 -κ, q = 8 -9κ; or, p = 6 -3κ, q = 10 -11κ; or, p = 8 -6κ, q = 12 -14κ.

The

p = q = -2 -κ case
In this section, we introduce a result in the p = q = -2 -κ case. Let us turn to the following theorem.

Theorem 3.3.1. For p = q = -2 -κ, the whole-plane SLE κ moduli one-point function (2.3.16) is given by

E f (z) zf (z) κ+2 (3.3.1) = 1 (1 -z z) κ+2 2 hypergeom - κ 2 -1, - 3κ 4 - 3 2 , κ 4 + 1 2 , z z + z 1 -z + z 1 -z hypergeom - κ 2 -1, - 3κ 4 - 3 2 , κ 4 + 1 2 , z z + (1 -z z) hypergeom - κ 2 , - 3κ 4 - 1 2 , κ 4 + 3 2 , z z .
Proof. Let us give some remarks for the proof of this theorem. We rst note that: for any κ > 0, if p = q = -2 -κ then all the extra diagonals, except for the main diagonal, of the coecient matrix of G(z, z), dened by its corresponding power series, are identical. This fact implies that for all the extra diagonals, their respective sums have a common factor which is a symmetric function (i.e., P (z, z) = P (z z)).

Obviously, the sum of the main diagonal is a symmetric function.

Moreover, relying on the specic results which we obtained for κ = 2 and κ = 4, we realized the existence of the factor 1/(1 -z z) (κ+2)/2 given in (3.3.1). Thus we now look specically for solutions to the PDE (2.3.17), together with the boundary condition G(0, 0) = 1, of the form

G(z, z) = Q 1 (z z) + z 1-z + z 1-z Q 2 (z z) (1 -z z) κ+2 2 . (3.3.2)
Substituting (3.3.2) into the PDE (2.3.17) with p = q = -2 -κ, we get the following system

         (u 2 -1)Q 1 (u) + 2u(1 -u)Q 2 (u) -κ+2 2 u + 5(κ+2) 2 Q 1 (u) +(κ + 2)(u + 2)Q 2 (u) = 0, 4(u -u 2 )Q 1 (u) + 4u(u -1)Q 2 (u) + (κ + 2)(4u + 2)Q 1 (u) -(κ + 2)(5u + 1)Q 2 (u) = 0,
where u := z z. Solving the above system by MAPLE, we nd the particular solution

(Q 1 , Q 2 ) such that the function (3.3.2) equals 1 at (0, 0), given by Q 1 (u) = hypergeom - κ 2 -1, - 3κ 4 - 3 2 , κ 4 + 1 2 , u , Q 2 (u) = hypergeom - κ 2 -1, - 3κ 4 - 3 2 , κ 4 + 1 2 , u + (1 -u) hypergeom - κ 2 , - 3κ 4 - 1 2 , κ 4 + 3 2 , u .
Theorem (3.3.1) gives for some specic values of κ:

• In the κ = 2/3 case, E f (z) zf (z) 8/3 = 1 (1 -z z) 4/3 Q 1 (z z) + z 1 -z + z 1 -z Q 2 (z z) , where Q 1 (z z), Q 2 (z z) are dened as Q 1 (z z) = 2(z z) 2 + 20z z + 5 5 , Q 2 (z z) = (z z) 2 + 16z z + 10 5 . • In the κ = 2 case, E f (z) zf (z) 4 = 1 (1 -z z) 2 Q 1 (z z) + z 1 -z + z 1 -z Q 2 (z z) , where Q 1 (z z), Q 2 (z z) are dened as Q 1 (z z) = 3(z z) 2 + 6z z + 1, Q 2 (z z) = 2(z z) 2 + 6z z + 2. • In the κ = 4 case, E f (z) zf (z) 6 = 1 (1 -z z) 3 Q 1 (z z) + z 1 -z + z 1 -z Q 2 (z z) , where Q 1 (z z), Q 2 (z z) are dened as Q 1 (z z) = 15(z z) 3 + 63(z z) 2 + 45z z + 5 5 , Q 2 (z z) =
10(z z) [START_REF] Bieberbach | Über die Koezienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln[END_REF] + 54(z z) 2 + 54z z + 10 5 . [START_REF] Duplantier | Logarithmic Coecients and Generalized Multifractality of Whole-Plane SLE[END_REF] + 450(z z) [START_REF] De Branges | A proof of the Bieberbach conjecture[END_REF] + 1300(z z) [START_REF] Bieberbach | Über die Koezienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln[END_REF] + 1300(z z) 2 + 450z z + 42 21 .

• In the κ = 6 case, E f (z) zf (z) 8 = 1 (1 -z z) 4 Q 1 (z z) + z 1 -z + z 1 -z Q 2 (z z) , where Q 1 (z z), Q 2 (z z) are dened as Q 1 (z z) = 3(z z) 4 + 20(z z) 3 + 30(z z) 2 + 12z z + 1, Q 2 (z z) = 2(z z) 4 + 16(z z) 3 + 30(z z) 2 + 16z z + 2. • In the κ = 8 case, E f (z) zf (z) 10 = 1 (1 -z z) 5 Q 1 (z z) + z 1 -z + z 1 -z Q 2 (z z) , where Q 1 (z z), Q 2 (z z) are dened as Q 1 (z z) = 63(z z) 5 + 585(z z) 4 + 1430(z z) 3 + 1170(z z) 2 + 315z z + 21 21 , Q 2 (z z) = 42(z z)

Integral means spectrum 3.4.1 Introduction

This section is devoted to summarize the integral means spectrum study of Ref. [5]. In particular, in Section 6 of Ref. [5], we generalized to our work the average integral means spectrum analysis of Refs. [2] and [6] (see also [11, 12, 13]) concerning the whole-plane SLE κ . The original work by BeliaevSmirnov [2] dealt with the exterior version, whereas Ref. [6] and our work concern the interior case. We thus look for the singular behavior of the integral,

r∂D E |f (z)| p |f (z)| q |dz|, (3.4.1)
for r → 1 -, where f stands for the interior whole-plane SLE map at time zero. The generalized average integral means spectrum β(p, q) corresponding to this general-

ized moment integral is the exponent such that r∂D E |f (z)| p |f (z)| q |dz| (r→1 -) (1 -r) -β(p,q) , (3.4.2)
in the sense of the equivalence of the logarithms of both terms.

It is interesting to remark that the map f ,

ζ ∈ C \ D → f (ζ) := 1/f (1/ζ),
is just the exterior whole-plane map from C \ D to the slit plane considered by Beliaev and Smirnov in Ref. [2]. We identically have for 0 < r < 1:

r -1 ∂D E | f (ζ)| p |dζ| = r 2p-2 r∂D E |f (z)| p |f (z)| 2p |dz|. (3.4.3)
We thus see that the standard integral mean of order (p, q = 0) for the exterior whole-plane map studied in Ref. [2] coincides (up to an irrelevant power of r) with the (p, q) integral mean (3.4.1) for q = 2p, for the interior whole-plane map.

Remark 3.4.1. Exterior-Interior Duality. More generally, we obviously have

r -1 ∂D E | f (ζ)| p | f (ζ)| q |dζ| = r 2p-2 r∂D E |f (z)| p |f (z)| 2p-q |dz|, (3.4.4)
so that the (p, q ) exterior integral means spectrum coincides with the (p, q) interior integral means spectrum for q + q = 2p. In particular, the (p, 0) interior derivative moments studied in Ref. [6] correspond to the (p, 2p) mixed moments of the Beliaev Smirnov exterior map.

INTEGRAL MEANS SPECTRUM

In the original BeliaevSmirnov case, the integral means spectrum successively involves three functions [2]:

β tip (p, κ) := -p -1 + 1 4 4 + κ -(4 + κ) 2 -8κp , (3.4.5)
for p ≤ p 0 (κ) := -1 -3κ 8 ;

(3.4.6)

β 0 (p, κ) := -p + 4 + κ 4κ 4 + κ -(4 + κ) 2 -8κp , (3.4.7)
for p 0 (κ) ≤ p ≤ p 0 (κ);

β lin (p, κ) := p - (4 + κ) 2 16κ , (3.4.8)
for p ≥ p 0 (κ) := 3(4 + κ) 2 32κ .

(3.4.9)

As shown in Refs. [6, 11, 12, 13] in the interior case, because of the unboundedness of the interior whole-plane SLE κ map, there exists a phase transition at p = p * (κ), with The singularity analysis given in Ref. [6] led us to introduce the σ-dependent

p * (κ) := 1 16κ (4 + κ) 2 -4 -2 2(4 + κ) 2 + 4 = 1 32κ 2(4 + κ) 2 + 4 -6 2(4 + κ) 2 + 4 + 2 .
function β σ + (p, κ) = (1 -2σ)p - 1 2 1 + 1 -2σκp .
(3.4.12)

For σ = -1, it recovers the integral means spectrum (3.4.11) above for the interior whole-plane SLE, while for σ = +1 it introduces a new spectrum, For general real values of σ,

β (+1) + (p, κ) = -p - 1 2 1 + 1 -2κp ,
σ := q/p -1, (3.4.14)
we can rewrite (3.4.12) as a function of (p, q, κ),

β σ + (p, κ) = β 1 (p, q; κ) := 3p -2q - 1 2 - 1 2 1 + 2κ(p -q). (3.4.15)
The multifractal spectrum (3.4.15) is dened only for 2κ(q -p) ≤ 1, hence for points in the (p, q) plane below the oblique line (see Fig. 3.3):

∆ 1 := (p, q) ∈ R 2 , q = p + 1/2κ . (3.4.16)
Note also that both the tip spectrum (3.4.5) and the bulk spectrum (3.4.7) are dened only to the left of a vertical line in the (p, q) plane, as given by (see Fig. 3.3)

∆ 0 := p = (4 + κ) 2 8κ
, q ∈ R .

(3.4.17)

We claim that the generalized spectrum generated by the integral means (3.4.1) in the general (p, q) case will involve the standard multifractal spectra (3.4.5), (3.4.7), (3.4.8), that are independent of q, and also the new (p, q)-dependent multifractal spectrum (3.4.15). The phase transitions between these spectra will occur along lines drawn in the real (p, q) plane.

Here we will simply describe the corresponding partition of the (p, q) plane into the respective domains of validity of the four spectra above. We thus need to determine the boundary curves where pairs (possibly triplets) of these spectra coincide, which are signaling the onset of the respective transitions.

Phase transition lines 3.4.2.1 The Red Parabola

The parabola R of Theorems 2.2.1 and 2.3.4, which we will hereafter call (and draw in) red (see Fig. 3.3), is given by the simultaneous conditions,

A(p, q, γ) = 0, C(p, γ) = 0, (3.4.18)
which recovers the parametric form (2.2.1)

p = p R (γ) := 2 + κ 2 γ - κ 2 γ 2 , q = q R (γ) := 3 + κ 2 γ -κγ 2 , γ ∈ R. (3.4.19)
We then showed that the BS bulk spectrum β 0 (p) and the novel spectrum β 1 (p, q; κ) coincide along the nite sector of parabola R located between its tangency points T 0 and T 1 with ∆ 0 and ∆ The bulk spectrum β 0 (p) and the generalized spectrum β 1 (p, q) coincide along the arc of red parabola between its tangency points T 0 and T 1 with ∆ 0 and ∆ 1 (thick red line). They also coincide along the innite left branch of the green parabola, up to its tangency point T 2 to ∆ 1 (thick green line). The β 0 (p) spectrum and the linear one β lin (p) coincide along D 0 , whereas β 1 (p, q) and β lin (p) coincide along D 1 .

The Green Parabola

A second parabola in the (p, q) plane, hereafter called (and drawn in) green (see Fig. 3.3) and denoted by G, is such that the multifractal spectra β 0 (p) and β 1 (p, q; κ) coincide on part of it. By using the duality property of the spectrum function [5, 6],

we set the simultaneous seed conditions,

A(p, q, γ ) = 0, C(p, γ ) = 0, γ + γ = 2/κ + 1/2, ( 3
p = p G (γ ) := (4 + κ) 2 8κ - κ 2 γ 2 , q = q G (γ ) := (4 + κ) 2 8κ + γ -κγ 2 , γ ∈ R. (3.4.21)
We showed [5] that the multifractal spectra β 0 (p) and β 1 (p, q; κ) coincide along the innite left branch of parabola G below its tangency point T 2 with ∆ 1 , i.e., corresponding to the domain where γ ∈ [1/κ, +∞) (Fig. 3.3).

Quadruple point

The intersection of the red and green parabolas (3.

P 0 : p 0 = p 0 (κ) = 3(4 + κ) 2 32κ , q 0 = (4 + κ)(8 + κ) 16κ , (3.4.22) P 1 : p 1 = (8 + κ)(8 + 3κ) 32κ , q 0 = (4 + κ)(8 + κ) 16κ . (3.4.23)
Note that these points have same ordinate, while the abscissa of the left-most one, P 0 , is p 0 (κ) (3.4.9), where the integral means spectrum transits from the BS bulk form (3.4.7) to its linear form (3.4.8).

Through this intersection point P 0 further pass two important straight lines in the (p, q) plane. D 1 := (p, q) : q -p = q 0 -p 0 = 16 -κ 2 32κ .

(3.4.25)

A key property of D 1 is the following. The dierence,

β 1 (p, q; κ) -β lin (p, κ) = 1 κ κ 4 -1 + 2κ(p -q) 2 , (3.4.26)
is always positive, and vanishes only on line D 1 , where

∀(p, q) ∈ D 1 , β 1 (p, q; κ) = β lin (p, κ) = p - (4 + κ) 2 16κ .
(3.4.27)

The Blue Quartic

A third locus, the blue quartic Q, will also play an important role, that is where the BS tip-spectrum, β tip (p; κ) (3.4.5), coincides with the novel spectrum, β 1 (p, q; κ) (3.4.15). In the (p, q) plane, the tip condition (3.4.6) [2] descibes the domain to the left of the straight line D 0 (Fig. 3.4), dened by As described in Ref. [5], the parametric form for the blue quartic Q is given by

D 0 := {(p, q) : p = p 0 (κ) = -1 -3κ/8} .
p = p Q (γ) := κ 16 + 1 + κ 4 γ - κ 2 γ 2 - 1 8 ∆ 1 2 (γ), q = q Q (γ) := p Q (γ) + γ - κ 2 γ 2 , γ ∈ R, (3.4.29)
where

∆(γ) := 4κ 2 γ 2 -2κ(4 + κ)γ + 1 4 (8 + κ) 2 + 4κ.
The intersection of the blue quartic Q (3.4.29) with the red parabola R (3.4. 19) is located at

Q 1 : p 0 = -1 - 3κ 8 , q = - 1 2 (3 + κ); γ = - 1 2 ,
(3.4.30) followed by a second intersection at the origin, p = q = 0, for γ = 2/κ.

Q 0 : p 0 = -1 - 3κ 8 , q 0 := -2 - 7κ 8 ; γ = γ = 1 + 2 κ . (3.4.31)
The tip spectrum and the generalized one coincide in both γ-intervals [1/κ, 1 + 2/κ] and [1 + 2/κ, +∞), which together parameterize the branch of the quartic located below its contact with ∆ 1 (see Fig. 3.4). Because of the tip relevance condition (3.4.6), only the interval [1+2/κ, +∞) describing the lower innite branch of the quartic located to the left of Q 0 will matter for the integral means spectrum. The only possible scenario which thus emerges to construct the average generalized integral means spectrum by a continuous matching of the 4 dierent spectra along the phase transition lines described above, is the partition of the (p, q) plane in 4 dierent regions as indicated in Fig. 3.5:

• a part (I) to the left of D 0 and located above the blue quartic up to point Q 0 , where the average integral means spectrum is β tip (p); • an upper part (II) bounded by lines D 0 , D 0 , and located above the section of the green parabola between points Q 0 and P 0 , where the spectrum is given by β 0 (p); • an innite wedge (III) of apex P 0 located between the upper half-lines D 0 and D 1 , where the spectrum is given by β lin (p); • a lower part (IV) whose boundary is the blue quartic up to point Q 0 , followed by the arc of green parabola between points P 0 and Q 0 , followed by the half-line D 1 above P 0 where the spectrum is β 1 (p, q). The two wings T 1 P 0 and P 0 T 0 of the red parabola (Fig. 3.3), where we know from Theorem 2.3.4 that the average spectrum is given by β 0 (p) = β 1 (p, q), can thus be seen as the respective extensions of region IV into II and of region II into IV. This is summarized by the following theorem (Proposition 6.1 in Ref. [5]). Theorem 3.4.2. The separatrix curves for the generalized integral means spectrum of whole-plane SLE κ are in the (p, q) plane (Fig. 3.5):

• (i) the vertical half-line D 0 above P 0 = (p 0 , q 0 ) (3.4.22), where p 0 = 3(4 + κ) 2 /32κ, q 0 = (4 + κ)(8 + κ)/16κ; • (ii) the unit slope half-line D 1 originating at P 0 , whose equation is q -p = (16 -κ 2 )/32κ with p ≥ p 0 ; • (iii) the section of green parabola, with parametric coordinates p G (γ ), q G (γ )

(3.4.21) for γ ∈ [1/4 + 1/κ, 1 + 2/κ], between P 0 and Q 0 = (p 0 , q 0 ) (3.4.31),
where

p 0 = -1 -3κ/8, q 0 = -2 -7κ/8; • (iv) the vertical half-line D 0 above point Q 0 ; • (v) the branch of the blue quartic from Q 0 to ∞, with parametric coordinates p Q (γ), q Q (γ) (3.4.29) for γ ∈ [1 + 2/κ, +∞).
As mentioned above, the whole-plane SLE κ case studied by Beliaev and Smirnov corresponds to the BS line q = 2p. Owing to Eq. (2.2.2), it intersects the red parabola R only at p = 0. The green parabola G (3.4.21) has for Cartesian equation,

κ 2 (2p -q) 2 - 1 8 (4 + κ) 2 (2p -q) + p + 1 128 (4 + κ) 2 (8 + κ) = 0, (3.4.32) 0 D' 0 D 1 , p q ( ) D P 0 ( ) p ( ) p ( ) p tip lin 0 q p 1 Q 0 Figure 3.5:
Respective domains of validity of integral means spectra β tip (p), β 0 (p), β lin (p), and β 1 (p, q). The thin straight line (coral) q = 2p corresponds to the version of whole-plane SLE studied in Ref. [2]. It does not intersect the lower domain where β 1 holds. which shows that it intersects the BS line at [1] p = p 0 (κ) := -1 128 (4 + κ) 2 (8 + κ), (3.4.33) which is to the left of the tip transition line at p 0 (κ) = -1 -3 8 κ (3.4.6). The quartic

Q (3.4.29) obeys 2p -q - κ 16 2 - c 4 2p -q -1 - κ 8 (2p -q) = κ 2 (p -q) 2p -q - 1 4 - κ 8 2 c = c(κ) := 1 64 (8 + κ) 2 + κ 4 , (3.4.34)
which immediately shows that the BS line q = 2p intersects Q only at the origin and stays above its lower branch.

The BS line therefore does not intersect the segment of green parabola G between P 0 and Q 0 , nor the quartic Q below Q 0 (Fig. 3.5). Thus the novel spectrum β 1 does not a priori appear in the version of whole-plane SLE κ considered in Ref. [2]. The BS line nevertheless intersects G at p 0 (3.4.33) to the left of Q 0 , in a domain lying above the quartic.

We summarize the proof of Theorem 3.4.2 as follows: we rst prove that the generalized integral means spectrum β(p, q) has in the upward wedge sector located between the upper half-lines D 0 and D 1 the linear form β lin (p). Next, we show that the generalized spectrum β(p, q) is given in the whole innite domain of the (p, q) plane located to the left of the innite lower branch of the green parabola G (3.4.21) below point P 0 and to the left of the half-line D 0 above P 0 (Fig. (3.4.36), and intersects D 0 at Q 0 and the red parabola R at point P 3 . The q = 2p continuous straight line in coral , corresponding to the whole-plane SLE version studied in Ref. [2], does not intersect the blue quartic, but intersects the green parabola at a point of abscissa (3.4.33). (In the particular κ = 2 case shown here, the intersection point P 3 of D 3 with R coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.) the generalized integral means spectrum β(p, q) = β tip in the innite wedge sector of apex Q 0 , between the green parabola G end the blue quartic Q, and located to the left of the line D 0 (Fig. 3.6). We further show that β(p, q) = β 1 (p, q) in the semi-innite strip (Fig. 3.6) located under the blue quartic Q up to Q 0 , under the branch of the green parabola G between Q 0 and P 0 , and above the straight line D 2 of equation,

q = 2p -p(κ), p(κ) := (2 + κ)(6 + κ) 8κ . (3.4.35)
To the right, this domain is closed by the vertical segment of line D 0 located above the intersection point of D 0 with D 2 , P 2 = p 0 (κ), q 2 = (4+κ) 2 +8 16κ , and up to point P 0 (Fig. 3.6). Finally, we show that the generalized integral means spectrum β(p, q) = β 1 (p, q) in a nite domain located above the straight line D 3 of equation (Fig. 3.6),

p -q = p(κ), p(κ) := 1 + κ/2.

(3.4.36)

This line intersects the green parabola G at point Q 0 and the red one R at point

P 3 = 1 + 2 κ , 4-κ 2 2κ
. This nite domain is enclosed by the segment of line D 2 between its intersection point with D 3 , 6-3κ 4κ p(κ), 6-7κ 4κ p(κ) , and P 2 , followed by the vertical segment of line D 0 located above P 2 and up to point P 0 , followed by the section of the red parabola R between P 0 and P 3 , followed by the segment of line D 3 between its intersection point with D 2 and P 3 .

Thus, the validity of the various spectra is established in an innite domain of the (p, q)-plane in Fig. 3.6, located above a frontier line obtained by following by continuity the set of lines: D 2 up to its intersection point with D 3 , D 3 up to P 3 , the section of the red parabola R between P 3 and P 0 , and D 1 from P 0 up to innity. This concludes the proof of Theorem 3.4.2. For convenience, we will denote by J this frontier line. The innite branch of R 1 below P 3 (thin coral line), with parametric coordinates p C (γ), q C (γ) (3.4.37) for γ ∈ [1+2/κ, +∞). The innite branch of R below P 3 (thin red line), with parametric coordinates p R (γ), q R (γ) (3.4.19) for γ ∈ [1 + 2/κ, +∞). (In the particular κ = 2 case shown here, point P 3 coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.)

Checks for points below the frontier line J

The parabola R 1 of Theorems 3.1.1 and 3.1.2, which we will hereafter call (and draw in) coral (Fig. 3.7), is given by the simultaneous conditions, A(p, q, γ) = 0, C(p, γ -1) = 0, which recovers the parametric form (3.1.2) , (2+κ) (18-7κ) 16κ , is between that of D 2 with the red parabola R, (6+κ)(10+3κ) 32κ

p = p C (γ) := 2 + 3 2 κ γ - κ 2 γ 2 -κ -2, q = q C (γ) := 3 + 3 2 κ γ -κγ 2 -κ -2, γ ∈ R.
, (6+κ) 2 16κ
, and that of D 2 with D 3 , 6-3κ 4κ p(κ), 6-7κ 4κ p(κ) (Fig. 3.7). Using Theorem 3.1.2, we now calculate the generalized spectrum β(p, q; κ) for some points lying on the innite branch of the coral parabola R 1 below P 3 (Fig. 3.7), with parametric coordinates p C (γ), q C (γ) (3.4.37) for γ ∈ [1 + 2/κ, +∞), in the κ = 2 and κ = 3 cases.

• For κ = 2 and γ = 3, we obtain (p, q) = (p C (γ), q C (γ)) = (2, -4) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 2 r 6 + 9 r 4 + 9 r 2 + 1 -4 r 5 + 12 r 3 + 4 r cos θ

(1 -r) [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] (1 + r) [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] .

Then, [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] , which implies β(2, -4; 2) = 11 = β 1 (2, -4; 2).

1 2π 2π 0 G(re iθ , re -iθ )dθ = r 2 + 1 r 8 + 20 r 6 + 58 r 4 + 20 r 2 + 1 (1 + r) 11 (1 -r) 11 (r→1 -) 1 (1 -r)
• For κ = 2 and γ = 4, we obtain (p, q) = (p C (γ), q C (γ)) = (0, -12) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 3 P 0 (r) + P 1 (r) cos θ

(1 -r) [START_REF] Robertson | On the theory of univalent functions[END_REF] (1 + r) [START_REF] Robertson | On the theory of univalent functions[END_REF] , where P 0 (r) = r 10 + 25 r 8 + 100 r 6 + 100 r 4 + 25 r 2 + 1, P 1 (r) = -6 r 9 + 60 r 7 + 120 r 5 + 60 r 3 + 6 r .

Then, 1 2π 2π 0 G(re iθ , re -iθ )dθ = r 16 + 52 r 14 + 568 r 12 + 2144 r 10 + 3290 r 8 + 2144 r 6 + 568 r 4 + 52 r 2 + 1 (1 -r) [START_REF] Robertson | On the theory of univalent functions[END_REF] (1 + r) 20 [START_REF] Robertson | On the theory of univalent functions[END_REF] , which implies β(0, -12; 2) = 20 = β 1 (0, -12; 2).

(r→1 -) 1 (1 -r)
• For κ = 2 and γ = 5, we obtain (p, q) = (p C (γ), q C (γ)) = (-4, -24) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 4 P 0 (r) + P 1 (r) cos θ

(1 -r) 31 (1 + r) 31 , where P 0 (r) = r 14 + 49 r 12 + 441 r 10 + 1225 r 8 + 1225 r 6 + 441 r 4 + 49 r 2 + 1, P 1 (r) = -8 r 13 + 168 r 11 + 840 r 9 + 1400 r 7 + 840 r 5 + 168 r 3 + 8 r .

Then, 31 , which implies β(-4, -24; 2) = 31 = β 1 (-4, -24; 2), where P (r) = r 20 + 96 r 18 + 2029 r 16 + 15616 r 14 + 51694 r 12 + 77344 r [START_REF] Lebedev | On the coecients of certain classes of univalent functions[END_REF] + 51694 r 8 + 15616 r 6 + 2029 r 4 + 96 r 2 + 1.

1 2π 2π 0 G(re iθ , re -iθ )dθ = (r 2 + 1) P (r) (1 -r) 31 (1 + r) 31 (r→1 -) 1 (1 -r)
• For κ = 3 and γ = 2, we obtain (p, q) = (p C (γ), q C (γ)) = (2, -2) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 5 r 4 + 18 r 2 + 5 -14 r r 2 + 1 cos θ 5(1 -r) [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] (1 + r) [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] .

Then,

2π

2π 0 G(re iθ , re -iθ )dθ = r 2 + 1 5 r 4 + 32 r 2 + 5 5(1 + r) [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] (1 -r) 7 (r→1 -) 1 (1 -r) [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] , which implies β(2, -2; 3) = 7 = β 1 (2, -2; 3).

• For κ = 3 and γ = 3, we obtain (p, q) = (p C (γ), q C (γ)) = ( 1 Next, we use Corollary 2.3.5 to calculate the spectrum β(p, q; κ) for some points lying on the innite branch of the red parabola R below P 3 (Fig. 3.7), with parametric coordinates p R (γ), q R (γ) (3.4.19) for γ ∈ [1 + 2/κ, +∞), in the κ = 2 and κ = 3 cases.

• For κ = 2 and γ = 3, we obtain (p, q) = (p R (γ), q R (γ)) = (0, -6) and G(re iθ , re -iθ ) = (r 2 + 1 -2r cos θ)

3

(1 -r) [START_REF] Binh | Around Milin's conjecture and SLE maps[END_REF] (1 + r) [START_REF] Binh | Around Milin's conjecture and SLE maps[END_REF] .

Then, [START_REF] Binh | Around Milin's conjecture and SLE maps[END_REF] , which implies β(0, -6; 2) = 9 = β 1 (0, -6; 2).

1 2π 2π 0 G(re iθ , re -iθ )dθ = (r 2 + 1) (r 4 + 8 r 2 + 1) (1 -r) 9 (1 + r) 9 (r→1 -) 1 (1 -r)
• For κ = 2 and γ = 5, we obtain (p, q) = (p R (γ), q R (γ)) = (-10, -30) and G(re iθ , re -iθ ) = (r 2 + 1 -2r cos θ)

5

(1 -r) 25 (1 + r) 25 .

Then, 25 , which implies β(-10, -30; 2) = 25 = β 1 (-10, -30; 2).

1 2π 2π 0 G(re iθ , re -iθ )dθ = (r 2 + 1) (r 8 + 24 r 6 + 76 r 4 + 24 r 2 + 1) (1 -r) 25 (1 + r) 25 (r→1 -) 1 (1 -r)
• For κ = 3 and γ = 2, we obtain (p, q) = (p R (γ), q R (γ)) = (1, -3) and G(re iθ , re -iθ ) = (r 2 + 1 -2r cos θ)

2

(1 -r) [START_REF] Duplantier | The Coecient Problem and Multifractality of Whole-Plane SLE and LLE[END_REF] (1 + r) [START_REF] Duplantier | The Coecient Problem and Multifractality of Whole-Plane SLE and LLE[END_REF] .

Then,

1 2π 2π 0 G(re iθ , re -iθ )dθ = r 4 + 4 r 2 + 1 (1 -r) 6 (1 + r) 6 (r→1 -) 1 (1 -r) 6 , which implies β(1, -3; 3) = 6 = β 1 (1, -3; 3). • For κ = 3 and γ = 5, we obtain (p, q) = (p R (γ), q R (γ)) = (-20, -105/2) and G(re iθ , re -iθ ) = (r 2 + 1 -2r cos θ) 5 (1 -r) 75/2 (1 + r) 75/2 .
Then,

1 2π 2π 0 G(re iθ , re -iθ )dθ = (r 2 + 1) (r 8 + 24 r 6 + 76 r 4 + 24 r 2 + 1) (1 -r) 75/2 (1 + r) 75/2 (r→1 -) 1 (1 -r) 75/2 ,
which implies β(-20, -105/2; 3) = 75/2 = β 1 (-20, -105/2; 3).

As above, one always gets the spectrum β(p, q) = β 1 (p, q) for the given points (p, q) below the frontier line J . This is consistent with the proposed partition of the (p, q) plane (Fig. 3.5).

Chapter 4

McMullen's asymptotic variance and SLE 2

Starting motivation

Let us consider a general analytic one-parameter family (φ t ), t ∈ U , a neighborhood of t = 0, of conformal maps with φ 0 = id and φ t (0) = 0, ∀t ∈ U . Then From which follows that In [15], McMullen asked under which condition on the family of (φ t ) it is true that

φ t (z) = z 0 e log φ t (u) du, z ∈ D,
V (z) = ∂ ∂t φ t (z) t=0 = z 0 ∂ ∂t log φ t (u)
d 2 dt 2 H. dim(φ t (∂D)) t=0 = lim r→1 - 1 4π| log(1 -r)| |z|=r |b(z)| 2 |dz| = σ 2 (b) 2 , ( 4
d 2 dt 2 M. dim(φ t (∂D)) t=0 = lim r→1 - 1 4π| log(1 -r)| |z|=r |b(z)| 2 |dz|. (4.1.4)
In particular, they proved that , p ∈ R.

lim p→0 2β(p, φ) p 2 = lim r→1 - 1 4π| log(1 -r)| 2π 0 |b(re iθ )| 2 dθ,
The starting motivation of this chapter was to prove an analog of (4.1.5), in expectation, for the interior whole-plane SLE 2 map at time 0. In particular, we will prove the following theorem.

Theorem 4.1.1. Let f := f 0 be the interior whole-plane SLE κ map at time zero and β(p, f ) be the average integral means spectrum of f dened as Applying the same method as in chapter 2, we use a martingale technique to obtain an equation satised by F (z, t) (4.2.2). For s ≤ t, dene M s := E(log f t (z)|F s ), where F s is the σ-algebra generated by {B u , u ≤ s}. (M s ) s≥0 is by construction a martingale. By the SLE Markov property we have

β(p, f ) = lim sup r-→1 - log 2π 0 E[|f (re iθ )| p ]dθ | log(1 -r)| , p ∈ R; then, for κ = 2, lim p→0 2 β(p, f ) p 2 = lim r→1 - 1 4π| log(1 -r)| 2π 0 E(| log f (re iθ )| 2 )dθ.
M s := E log f t (z)|F s = E log f s (z) + log f τ (z s )|F s = log f s (z) + E log[e τ f τ (z s )]|F s -τ = log f s (z) + F (z s , τ ) -τ,
where z s := fs (z)/λ(s), and τ := t -s.

From Eqs. (2.1.9) and (2.1.11) in section 2.1, we directly arrive at the vanishing condition of the coecient of the ds-drift term of the Itô derivative of M s ,

2 1 - 1 (1 -z s ) 2 + z s z s + 1 z s -1 - κ 2 ∂ z -∂ τ - κ 2 z 2 s ∂ 2 z F (z s , τ ) = 0, (4.2.4)
by the (local) martingale property.

The existence of the limit (4.2.3) further implies that of

lim τ →+∞ ∂ τ F (z, τ ) = 0.
Letting τ → +∞, we obtain the expected ODE for F (z) (4.2.1),

P(∂)[F (z)] := 2 1 - 1 (1 -z) 2 + z z + 1 z -1 - κ 2 ∂ z F (z) - κ 2 z 2 ∂ 2 z F (z) = 0. (4.2.5)
Solving the ODE (4.2.5) by MAPLE, together with the condition F (0) = 0, we get explicit expressions for F (z) for some values of κ. In particular,

• In the κ = 1 case,

F (z) = 7 30 z 2 - 28 15 z + 4 5 log(1 -z).
(4.2.6)

• In the κ = 2 case, F (z) = - 4 3 z + 2 3 log(1 -z). ( 4
F (z) = E f (z) f (z)
. Solving this ODE, we obtain

F (z) = 4 κ (z -1) 4 κ z 2 κ +1 z 2 κ (z -2) (z -1) 4 κ +2 dz, with the condition F (0) = 2E(a 2 ) = -8 κ+2 .

SLE logarithm two-point function 4.3.1 Nonhomogeneous Beliaev-Smirnov type equations

In this section, we study the whole-plane SLE logarithm two-point function for

z 1 , z 2 ∈ D, G(z 1 , z2 ) := E log f (z 1 )log f (z 2 ) . (4.3.1)
As before, we dene a time-dependent, auxiliary two-point function,

G(z 1 , z2 , t) := E log[e t f t (z 1 )]log[e t f t (z 2 )] , (4.3.2)
where ft is the conjugate, reversed radial SLE process in D, as introduced in Denition 2.1.3.

Because of Lemma (2.1.4), the two-point function (4.3.1) is the limit

lim t→+∞ G(z 1 , z2 , t) = G(z 1 , z2 ). (4.3.3)
Let us now consider the two-point martingale (M s ) t≥s≥0 , dened by

M s := E log f t (z 1 )log f t (z 2 )|F s ,
and introduce the shorthand notation, X t (z) := log f t (z).

M s = X s (z 1 )X s (z 2 ) + τ 2 -τ X s (z 1 ) + X s (z 2 ) (4.3.5) + X s (z 1 ) F (z 2s , τ ) + X s (z 2 ) F (z 1s , τ ) -τ F (z 1s , τ ) + F (z 2s , τ ) + G(z 1s , z2s , τ ), τ := t -s,
where, as dened in (2.3.5), z 1s := fs (z 1 )/λ(s); z2s := fs (z 2 )/λ(s) = fs (z 2 )λ(s), and F (z, t) := F (z, t) with F (z, t) dened in (4.2.2). As before, the partial dierential equation satised by G(z 1s , z2s , τ ) is obtained by expressing the fact that the ds-drift term of the Itô dierential of Eq. (4.3.5),

dM s = (X s (z 1 ) -τ ) dX s (z 2 ) + ds + d F + (X s (z 2 ) -τ ) dX s (z 1 ) + ds + d F + (dX s (z 1 ) + ds) F + (dX s (z 2 ) + ds) F + d G, (4.3.6) vanishes.
The dierentials of X s , X s are as in Eq. (2.1.9),

dX s (z 1 ) = 1 - 2 (1 -z 1s ) 2 ds, dX s (z 2 ) = 1 - 2 (1 -z2s ) 2 ds.
Note that the coecient of the ds-drift term in dX s (z 1 ) + ds + d F is the left hand side of the ODE (4.2.4), hence vanishes. The coecient of the ds-drift term in dX s (z 2 ) + ds + d F aslo vanishes, since it is the left hand side of the complex conjugate equation of the ODE (4.2.4).

In addition, the Itô dierential of G(z 1s , z2s , τ ) is given as in (2.3.11),

d G = -i √ κ z 1s ∂ 1 -z2s ∂2 G dB s + z 1s + 1 z 1s -1 z 1s ∂ 1 G ds + z2s + 1 z2s -1 z2s ∂2 G ds -∂ τ G ds - κ 2 (z 1s ∂ 1 -z2s ∂2 ) 2 G ds,
where we used the shorthand notations ∂ 1 := ∂ z 1 , ∂2 := ∂ z2 and the obvious formal identity

(z 1 ∂ 1 ) 2 + (z 2 ∂2 ) 2 -2z 1 ∂ 1 z2 ∂2 = (z 1 ∂ 1 -z2 ∂2 ) 2 .
We therefore directly arrive at the vanishing condition of the overall drift term coecient in dM s ,

z 1s z 1s + 1 z 1s -1 ∂ 1 + z2s z2s + 1 z2s -1 ∂2 -∂ τ - κ 2 (z 1s ∂ 1 -z2s ∂2 ) 2 G (4.3.7) + 2 1 - 1 (1 -z 1s ) 2 F (z 2s , τ ) + 2 1 - 2 (1 -z2s ) 2 F (z 1s , τ ) = 0.
The existence of the limit (4.3.3) further implies that of

lim τ →+∞ ∂ τ G(z 1 , z2 , τ ) = 0.
Letting τ → +∞, we get the expected PDE for G(z 1 , z2 ) (4.3.1),

P(D)[G(z 1 , z2 )] = - κ 2 (z 1 ∂ 1 -z2 ∂2 ) 2 G - 1 + z 1 1 -z 1 z 1 ∂ 1 G - 1 + z2 1 -z2 z2 ∂2 G (4.3.8) + 2 1 - 1 (1 -z 1 ) 2 F (z 2 ) + 2 1 - 2 (1 -z2 ) 2 F (z 1 ) = 0. Recall that F (z 1 ) = E(log f (z 1 )), F (z 2 ) = E(log f (z 2 )) = E(log f (z 2 ) ).

Moduli logarithm one-poin function

Note that one can take the z 1 = z 2 = z case in Denition (4.

P(D)[G(z, z)] = - κ 2 (z∂ -z ∂) 2 G - 1 + z 1 -z z∂G - 1 + z 1 - z z ∂G (4.3.10) + 2 1 - 1 (1 -z) 2 F (z) + 2 1 - 2 (1 -z) 2 F (z) = 0.
Let us now look for the expression for the moduli logarithm one-point function G(z, z) (4.3.9) in the κ = 2 case.

The κ = 2 case

For κ = 2, owing to (4.2.7), the PDE (4.3.10) becomes

-(z∂ -z ∂) 2 G - 1 + z 1 -z z∂G - 1 + z 1 - z z ∂G + 2 1 - 1 (1 -z) 2 - 4 3 z + 2 3 log(1 -z) + 2 1 - 2 (1 -z) 2 - 4 3 z + 2 3 log(1 -z) = 0. (4.3.11)
One can write G(z, z) in the following form,

G(z, z) = F (z)F (z) + R(z, z), (4.3.12)
where

F (z)F (z) = 4 9 2z -log(1 -z) 2z -log(1 -z) . (4.3.13)
As introduced in Section 4.1, the second step of the proof is to accurately compute the coecients of G(z, z) dened by its corresponding power series. From the PDE (4.3.11), we construct a system of recursive equations for these coecients and then get the coecient matrix of G(z, z) by using MAPLE. Owing to (4.3.12) and (4.3.13), we also nd the coecient matrix of R(z, z). In particular, we introduce here a part of the coecient matrix of G(z, z),

                                 0 0 0 0 0 0 0 0 0 0 0 • • • 0 -2 3
                                 ; a part of the coecient matrix of R(z, z),                                  0 0 0 0 0 0 0 0 0 0 0 • • • 0 -4 3 0 0 0 0 0 0 0 0 0 • • • 0 -4 3
                                 ;
together with a part of the coecient matrix of (9/4) • R(z, z),

                                 0 0 0 0 0 0 0 0 0 0 0 • • • 0 -3 0 0 0 0 0 0 0 0 0 • • • 0 -3 7 2 -1 0 0 0 0 0 0 0 0 • • • 0 -1 5 3 -1 2 0 0 0 0 0 0 0 • • • 0 0 -1 2 1 -3 10 0 0 0 0 0 0 • • • 0 0 0 -3 10 17 25 -1 5 0 0 0 0 0 • • • 0 0 0 0 -1 5 1 2 -1 7 0 0 0 0 • • • 0 0 0 0 0 -1 7 19 49 -3 28 0 0 0 • • • 0 0 0 0 0 0 -3 28 5 16
-

1 12 0 0 • • • 0 0 0 0 0 0 0 -1 12 7 27 -1 15 0 • • • 0 0 0 0 0 0 0 0 -1 15 11 50 -3 55 • • • 0 0 0 0 0 0 0 0 0 -3 55 23 121 • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .                                 
.

It is remarkable to note that the coecient matrix of (9/4)•R(z, z) is a symmetric tridiagonal matrix. From the above coecients, we determine the general rules for its three diagonals,

a 1,1 = 9, a n,n = 12 n 2 + 1 n ∀n ≥ 2, a n,n+1 = a n+1,n = - 6 n(n + 1) ∀n ≥ 1,
a n,m = 0 otherwise, whereby we get the explicit expression for (9/4) • R(z, z) as follows: Using the two identities

9 4 • R(z, z) = -4z z + 6(z + z) z z z z 0 log(1 -x)dx -12 z z 0 log(1 -x) x dx -log(1 -z z).
z z 0 ln(1 -x)dx = (z z -1) ln(1 -z z) -z z and z z 0 ln(1 -x) x dx = -dilog(1 -z z) = - ∞ n=1 (z z) n n 2 ,
where the dilog function is dened as

dilog(x) = x 1 ln(t) 1 -t dt = ∞ n=1 (1 -x) n n 2 ,
we can rewrite (4.3.15) as

G(z, z) = - 8 9 z log(1 -z) - 8 9 z log(1 -z) + 4 9 log(1 -z) log(1 -z) - 8 3 (z + z) + 24(z + z)(z z -1) -4z z 9z z log(1 -z z) + 16 3 ∞ n=1 (z z) n n 2 .
(4.3.17 Because of (4.3.17), we have

1 2π 2π 0 E(| log f (re iθ )| 2 )dθ = 16 9 r 2 + 4 9 ∞ n=1 r 2n n 2 - 4 9 log(1 -r 2 ) + 16 3 ∞ n=1 r 2n n 2 = 16 9 r 2 - 4 9 log(1 -r) + log(1 + r) + 52 9 ∞ n=1 r 2n n 2 .
It follows that

2π 0 E(| log f (re iθ )| 2 )dθ -4π log (1 -r) = 2 9 + 16 9 r 2 -4 9 log(1 + r) + 52 9 ∞ n=1 r 2n n 2 -2 log (1 -r) -→ 2 9
as r → 1 -, which encompasses Theorem 4.1.1.

Chapter 5

Grunsky matrices and SLE κ

Grunsky matrices

Let f be a holomorphic and injective (univalent) in the unit disk D, normalized so that f (0) = 0 and f (0) = 1 (i.e. f ∈ S). The corresponding Grunsky coecients b n,m = b m,n of f are dened by the power series

log f (z) -f (ξ) z -ξ = ∞ n,m=0 b n,m z n ξ m , (5.1.1)
which is convergent for |z| < 1, |ξ| < 1. Note that b 0,n = b n,0 is given by log(f (z)/z) = ∞ n=1 b 0,n z n , b 0,0 = 0. We recall here an important inequality related to the Grunsky coecients (b n,m ), which is called Grunsky inequality [18]. Theorem 5.1.1. (Grunsky Inequality) Let f be a holomorphic and injective function in the unit disk and b n,m be the Grunsky coecients of f . Then for any complex vector (λ 1 , λ 2 , ..., λ k , ...), the following inequality holds:

∞ n=1 ∞ m=1 b n,m λ n λ m ≤ ∞ n=1 |λ n | 2
n .

(5.1.2)

Grunsky [7] showed that a holomorphic function f is injective in the unit disk if and only if the Grunsky coecients of f satisfy the Grunsky inequality (5.1.2). In other words, the Grunsky inequality give necessary and sucient conditions that a holomorphic function be injective in the unit disk. From (5.1.2), it will be convenient to work with the normalized Grunsky coecients [18],

c n,m = c m,n = √ nm b n,m , for n, m = 1, 2, ...

Then, the normalized Grunsky matrix (c n,m ) satises ∞ n=1 ∞ m=1 c n,m λ n λ m ≤ ∞ n=1 |λ n | 2 ,
(5.1.3) for any complex vector (λ 1 , λ 2 , ..., λ k , ...).

It is remarkable to note that the Grunsky matrix (c n,m ) of f is unitary if and only if the complement of the image of f has Lebesgue measure zero, i.e., |C \ f (D)| = 0. So, roughly speaking, the image f (D) is a slit region in the complex plane. For instance, it is unitary for SLE κ≤4 .

Besides, the important special case that λ k = 0 for k > m can be expressed in terms of linear algebra. In particular, if f ∈ S then the Grunsky matrix C = (c k,l ) k,l=1,...,m of order m (m = 1, 2, ...) can be written in the form [18]

C = U     d 1 0 . . . 0 d m     U, U unitary (5.1.4)
and U its transpose with |d k | ≤ 1 (k = 1, ..., m).

Recall that a complex square matrix U = (u i,j ) is unitary if its conjugate transpose U * = (u * i,j ), where u * i,j = u j,i , is also its inverse (i.e. UU * = U * U = I).

Grunsky matrices for SLE κ processes

Let us now return to the whole-plane SLE κ . We consider (5.1.1) in expectation,

E log f (z) -f (ξ) z -ξ = ∞ n,m=0 b n,m z n ξ m , (5.2.1)
where f is the interior whole-plane SLE κ map at time zero. By dierentiating two sides of (5.2.1) with respect to z and ξ successively, we

get E f (z)f (ξ) (f (z) -f (ξ)) 2 - 1 (z -ξ) 2 = n,m≥1 nm b n,m z n-1 ξ m-1 (z = ξ), (5.2.2) or, equivalently, E (z -ξ) 2 f (z)f (ξ) (f (z) -f (ξ)) 2 = 1 + (z -ξ) 2 n,m≥1 nm b n,m z n-1 ξ m-1 .
(5.2.3)

Owing to (5.2.3), in order to study the Grunsky coecient matrix ( √ nm b n,m ) of f (n, m = 1, 2, ...), we aim at determining the two-point mixed moment below,

E (z -ξ) 2 f (z)f (ξ) (f (z) -f (ξ)) 2 , (5.2.4)
More generally, we will study the natural generalization of (5.2.4) dened as

G(z, ξ) := E (z -ξ) q f p (z)f p (ξ) (f (z) -f (ξ)) q , (p, q) ∈ R 2 , (5.2.5)
where the (p = 1, q = 2) case is the most important one. Our method is to use a martingale technique to obtain a PDE obeyed by G(z, ξ), and then compute coecients of G(z, ξ), with the support of MAPLE, for any p, q ∈ R and κ > 0.

Beliaev-Smirnov type PDE for G(z, ξ)

As introduced above, our aim is to derive a PDE satised by the SLE two-point function G(z, ξ) (5.2.5), which, by construction, stays nite at the origin and such that G(0, 0) = 1.

Let us introduce the shorthand notation,

X t (z, ξ) := f p t (z) f p t (ξ) ft (z) -ft (ξ) q , (5.2.6)
where, as before, ft is the conjugate, reversed radial SLE process in D, as introduced in Denition 2.1.3. Applying the same method as in chapter 2, we consider the auxiliary, time-dependent two-point function

G(z, ξ, t) := E((z -ξ) q X t (z, ξ)), (5.2.7)
such that lim t→+∞ e (2p-q)t G(z, ξ, t) = G(z, ξ).

(

Let us consider the two-point martingale (M s ) t≥s≥0 , dened by

M s = E(X t (z, ξ)|F s ).
By the Markov property of SLE we get

M s = X s (z, ξ) G(z s , ξ s , τ ), τ := t -s, (5.2.9)
where z s := fs (z)/λ(s); ξ s := fs (ξ)/λ(s).

Their Itô dierentials, dz s and dξ s , are as in (2.1.11),

dz s = z s z s + 1 z s -1 - κ 2 ds -i √ κ z s dB s , dξ s = ξ s ξ s + 1 ξ s -1 - κ 2 ds -i √ κ ξ s dB s .
(5.2.10)

As before, the partial dierential equation satised by G(z s , ξ s , τ ) is obtained by expressing the fact that the ds-drift term of the Itô dierential of Eq. (5.2.9),

dM s = GdX s + X s d G, ( 5
dX s (z) = X s (z, ξ)F (z s , ξ s )ds, F (z, ξ) := 2p 1 - 1 (1 -z) 2 - 1 (1 -ξ) 2 -q zξ -z -ξ -1 (1 -z)(1 -ξ)
.

(5.2.12)

We thus get the simple expression

dM s = X s (z, ξ) F (z s , ξ s ) G ds + d G , ( 5
d G(z s , ξ s , τ ) = ∂ z G dz s + ∂ ξ G dξ s -∂ τ G ds (5.2.14) - κ 2 z 2 s ∂ 2 z G ds - κ 2 ξ 2 s ∂ 2 ξ G ds -κz s ξ s ∂ z ∂ ξ G ds,
We observe that the only coupling between the z s , ξ s variables arises in the last term of (5.2.14), the other terms simply resulting from the independent contributions of the z s and ξ s parts.

Using again the Itô dierentials (5.2.10), we can rewrite (5.2.14) as

d G = -i √ κ (z s ∂ z + ξ s ∂ ξ ) G dB s (5.2.15) + z s + 1 z s -1 z s ∂ z G ds + ξ s + 1 ξ s -1 ξ s ∂ ξ G ds -∂ τ G ds - κ 2 (z s ∂ z + ξ s ∂ ξ ) 2 G ds,
where we used the obvious formal identity

(z∂ z ) 2 + (ξ ∂ ξ ) 2 + 2z ∂ z ξ ∂ ξ = (z ∂ z + ξ ∂ ξ ) 2 .
(5.2.16)

We therefore directly arrive at the vanishing condition of the overall drift term coecient in dM s ,

F (z s , ξ s ) + z s z s + 1 z s -1 ∂ z + ξ s ξ s + 1 ξ s -1 ∂ ξ -∂ τ - κ 2 (z s ∂ z + ξ s ∂ ξ ) 2 G = 0.
(5.2.17)

The existence of the limit (5.2.8) further implies that of

lim τ →∞ e (2p-q)τ ∂ τ G(z, ξ, τ ) = -(2p -q)G(z, ξ).
Multiplying the PDE (5.2.17) satised by G by exp((2p -q)τ ) and letting τ → +∞,

we get - κ 2 (z∂ z + ξ∂ ξ ) 2 - 1 + z 1 -z z∂ z - 1 + ξ 1 -ξ ξ∂ ξ + F (z, ξ) + 2p -q G = 0, (5.2.18)
and its fully explicit expression is

P(D)[G(z, ξ)] := - κ 2 (z∂ z + ξ∂ ξ ) 2 G - 1 + z 1 -z z∂ z G - 1 + ξ 1 -ξ ξ∂ ξ G (5.2.19) + - 2p (1 -z) 2 - 2p (1 -ξ) 2 -q zξ -z -ξ -1 (1 -z)(1 -ξ)
+ 4p -q G = 0.

Some observations for coecient matrix of G(z, ξ)

We write G(z, ξ) (5.2.5) in the form of a power series,

G(z, ξ) := ∞ n,m=0 d n,m z n ξ m , (z, ξ) ∈ D × D.
(5.3.1) Substituting (5.3.1) into the PDE (5.2.19) and identifying coecients, we obtain a system of recursive equations for the coecients d n,m . From that system of recursive equations, we can accurately compute the coecients d n,m for any values of p, q and κ. However, in the (p = 1, q = 2) case, we have not yet found any exact formula for G(z, ξ) for specic values of κ. Thus, we only introduce here the square matrix of order 10, (d n,m ) n,m=0,1,...,9 , for κ = 2.

(p = 1, q = 2, κ = 2)                            1 0 -1 3 -2 9 - 2
                          
On the other hand, we found some special cases that their respective coecient matrices are "nite " in the sense that they have a nite numbers of nonzero entries.

We can therefore obtain closed-form expressions for G(z, ξ) in such cases. Let us now turn to the observations (predictions ) below for more details.

Observation 5.3.1. For any nonnegative integer l, if letting

κ = 2 2l + 1
, p = 0 , q = -l(3l + 5) + 2 2(2l + 1) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 3), and we obtain

G(z, ξ) = 3l+2 m=0 |d 3l+2-m,m |(1 -z) 3l+2-m (1 -ξ) m , (5.3.2) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+2 .
In the specic matrices below, the coecients d 3l+2-m,m in (5.3.2) are in bold.

(l = 0)

    1 -1 1 3 -1 1 3 0 1 3 0 0     , G(z, ξ) = 1 3 (1 -z) 2 + 1 3 (1 -z)(1 -ξ) + 1 3 (1 -ξ) 2 , G(z) = (1 -z) 2 . (l = 1)             1 -5 2 3 -2 5
0 0 0 0 0             , G(z, ξ) = 3 28 (1 -z) 5 + 5 28 (1 -z) 4 (1 -ξ) + 3 14 (1 -z) 3 (1 -ξ) 2 + 3 14 (1 -z) 2 (1 -ξ) 3 + 5 28 (1 -z)(1 -ξ) 4 + 3 28 (1 -ξ) 5 , G(z) = (1 -z) 5 . (l = 2)                      1 -4 8 -10
0 0 0 0 0 0 0 0                      , G(z, ξ) = 5 143 (1 -z) 8 + 12 143 (1 -z) 7 (1 -ξ) + 56 429 (1 -z) 6 (1 -ξ) 2 + 70 429 (1 -z) 5 (1 -ξ) 3 + 25 143 (1 -z) 4 (1 -ξ) 4 + 70 429 (1 -z) 3 (1 -ξ) 5 + 56 429 (1 -z) 2 (1 -ξ) 6 + 12 143 (1 -z)(1 -ξ) 7 + 5 143 (1 -ξ) 8 , G(z) = (1 -z) 8 . (l = 3)                              1
0 0 0 0 0 0 0 0 0 0                              , G(z, ξ) = 7 612 (1 -z) 11 + 11 306 (1 -z) 10 (1 -ξ) + 275 3978 (1 -z) 9 (1 -ξ) 2 + 275 2652 (1 -z) 8 (1 -ξ) 3 + 175 1326 (1 -z) 7 (1 -ξ) 4 + 98 663 (1 -z) 6 (1 -ξ) 5 + 98 663 (1 -z) 5 (1 -ξ) 6 + 175 1326 (1 -z) 4 (1 -ξ) 7 275 2652 (1 -z) 3 (1 -ξ) 8 + 275 3978 (1 -z) 2 (1 -ξ) 9 11 306 (1 -z)(1 -ξ) 10 + 7 612
(1 -ξ) [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] , G(z) = (1 -z) [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] .

(l = 4)

                                      1 -7 273
                                      , G(z, ξ) = 18 4807 (1 -z) 14 + 70 4807 (1 -z) 13 (1 -ξ) + 2730 81719 (1 -z) 12 (1 -ξ) 2 + 9555 163438 (1 -z) 11 (1 -ξ) 3 + 637 7429 (1 -z) 10 (1 -ξ) 4 + 819 7429 (1 -z) 9 (1 -ξ) 5 + 945 7429 (1 -z) 8 (1 -ξ) 6 + 990 7429 (1 -z) 7 (1 -ξ) 7 + 945 7429 (1 -z) 6 (1 -ξ) 8 + 819 7429 (1 -z) 5 (1 -ξ) 9 + 637 7429 (1 -z) 4 (1 -ξ) 10 + 9555 163438 (1 -z) 3 (1 -ξ) 11 + 2730 81719 (1 -z) 2 (1 -ξ) 12 70 4807 (1 -z)(1 -ξ) 13 + 18 4807 (1 -ξ) 14 , G(z) = (1 -z) 14 .
Besides, in this case, note that λ = 0 is not a root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l+2 . So, λ = 0 is not an eigenvalue of the matrix. Using MAPLE, we get approximate values of eigenvalues λ. For instance:

• For l = 0, P (λ) = -λ 3 + (4/3) λ 2 + (7/9) λ -1/27, λ = -0.473688871436316; 0.0443581584495084; 1.76266404632014 .

• For l = 1, , p = q = l(3l + 5) + 2 2(2l + 1) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (2l + 2), and we obtain

P (λ) = λ 6 -
G(z, ξ) = (1 -z) l+1 (1 -ξ) l+1 l m=0 |d 2l+1-m,l+1+m |(1 -z) l-m (1 -ξ) m , (5.3.3) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+2 .
In the specic matrices below, the coecients d 2l+1-m,l+1+m in (5.3.3) are in bold. 0 0 -1 [START_REF] Löwner | Untersuchungen über schlichte konforme Abildungendes Einheitskreises[END_REF] .

(l = 0) 1 -1 -1 1 , G(z, ξ) = (1 -z)(1 -ξ), G(z) = (1 -z) 2 . (l = 1)       1 -5 2 2 -1 2 -5 2 6 -9 2 1 2 -9 2 3 -1 2 -1 2 1 -1 2 0       , G(z, ξ) = (1 -z) 2 (1 -ξ) 2 1 2 (1 -z) + 1 2 (1 -ξ) , G(z) = (1 -z) 5 . (l = 2)             1 -4 44
            , G(z, ξ) = (1 -z) 3 (1 -ξ) 3 2 7 (1 -z) 2 + 3 7 (1 -z)(1 -ξ) + 2 7 (1 -ξ) 2 , G(z) = (1 -z) 8 . (l = 3)                   1
2 3 -1 6 0 0 0                   , G(z, ξ) = (1 -z) 4 (1 -ξ) 4 1 6 (1 -z) 3 + 1 3 (1 -z) 2 (1 -ξ) + 1 3 (1 -z)(1 -ξ) 2 + 1 6 (1 -ξ) 3 , G(z) = (1 -z) 11 . (l = 4)                        1 -7 238
0 0 0 0                        , G(z, ξ) = (1 -z) 5 (1 -ξ) 5 14 143 (1 -z) 4 + 35 143 (1 -z) 3 (1 -ξ) + 45 143 (1 -z) 2 (1 -ξ) 2 + 35 143 (1 -z)(1 -ξ) 3 + 14 143 (1 -ξ) 4 , G(z) = (1 -z)
(l = 5)                              1 - 17
                             , G(z, ξ) = (1 -z) 6 (1 -ξ) 6 3 52 (1 -z) 5 + 9 52 (1 -z) 4 (1 -ξ) + 7 26 (1 -z) 3 (1 -ξ) 2 + 7 26 (1 -z) 2 (1 -ξ) 3 + 9 52 (1 -z)(1 -ξ) 4 + 3 52 (1 -ξ) 5 , G(z) = (1 -z) 17 .
Besides, in this case, note that λ = 0 is a root of multiplicity l + 1 of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,2l+1 . Using MAPLE, we obtain approximate values of eigenvalues λ. For instance:

• For l = 0,

P (λ) = λ [λ -2], λ = 0; 2 .
• For l = 1, P (λ) = λ 2 λ 2 -10 λ -5 , λ = -0.4772255751; 0 [START_REF] Beliaev | Harmonic Measure and SLE[END_REF] ; 10.47722558 .

• For l = 2,

P (λ) = 1 7 λ 3 7λ 3 -434 λ 2 -1928 λ + 240 , λ = -4
.27669821438978; 0 [START_REF] Bieberbach | Über die Koezienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln[END_REF] ; 0.121182116828131; 66.1555160975617 .

• For l = 3,

P (λ) = 1 3 λ 4 3 λ 4 -1254 λ 3 -43285 λ 2 + 66836 λ + 2156 , λ = -33
.4355724222944; -0.0316114894591856; 0 (4) ; 1.51113646850890; 449.956047443245 .

• For l = 4,

P (λ) = 1 24167 λ 5 24167 λ 5 -71099314 λ 4 -18613476973 λ 3
+ 270251709000 λ 2 + 137725578900 λ -1166886000 , λ = -254.650361638484; -0.500984761369627; 0 [START_REF] Duplantier | Logarithmic Coecients and Generalized Multifractality of Whole-Plane SLE[END_REF] ; 0.00833625918889979; 14.2641387138316; 3182.87887142683 .

• For l = 5, P (λ) = (1/832) λ 6 832 λ 6 -17680000 λ 5 -34999011392 λ 4 + 4352009336400 λ 3 + 24485667910704 λ 2 -3968526359952 λ -8890849737 , λ = -1931.17205159984; -5.54419489136658; -0.00221021169275717; 0 [START_REF] Duplantier | The Coecient Problem and Multifractality of Whole-Plane SLE and LLE[END_REF] ; 0.159813871185902; 122.511434787320; 23064.0472080444 .

• For l = 6, setting N := 12165074387,

P (λ) = (1/N ) λ 7 12165074387 λ 7 -1899673686125146 λ 6 -28501179335024060199 λ 5 + 29133804061664982944264 λ 4 + 1572201809578397651593940 λ 3 -3228651447196418488605360 λ 2
-162246287526486885321600 λ + 96480960960480096000 , λ = -14667.3115363778; -53.2761804199220; -0.0496543405839516; 0 [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] ; 0.000587784249828495; 2.02835995819537 1009.28359609128; 1.69867324827305 • 10 5 .

Observation 5.3.3. For any integer l ≥ 1, if letting

κ = 2 2l + 1 , p = 3l(l + 1) 2(2l + 1) , q = l(3l + 1) -2 2(2l + 1) ,
then, the corresponding coecient matrix (d n,m ) is a square matrix of order (2l + 3), and we obtain

G(z, ξ) = (1 -z) l (1 -ξ) l l+2 m=0 |d 2l+2-m,l+m |(1 -z) l+2-m (1 -ξ) m , (5.3.4)
G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+2 .
In the specic matrices below, the coecients d 2l+2-m,l+m in (5.3.4) are in bold.

(l = 1)

         1 -5 2
-1 5 0 0 0          , G(z, ξ) = (1 -z)(1 -ξ) 1 5 (1 -z) 3 + 3 10 (1 -z) 2 (1 -ξ) + 3 10 (1 -z)(1 -ξ) 2 + 1 5 (1 -ξ) 3 , G(z) = (1 -z) 5 . (l = 2)                1 -4 47
0 0 0 0                , G(z, ξ) = (1 -z) 2 (1 -ξ) 2 5 42 (1 -z) 4 + 5 21 (1 -z) 3 (1 -ξ) + 2 7 (1 -z) 2 (1 -ξ) 2 + 5 21 (1 -z)(1 -ξ) 3 + 5 42 (1 -ξ) 4 , G(z) = (1 -z) 8 . (l = 3)                      1 - 11
0 0 0 0 0                      , G(z, ξ) = (1 -z) 3 (1 -ξ) 3 7 99 (1 -z) 5 + 35 198 (1 -z) 4 (1 -ξ) + 25 99 (1 -z) 3 (1 -ξ) + 25 99 (1 -z) 2 (1 -ξ) 3 + 35 198 (1 -z)(1 -ξ) 4 + 7 99 (1 -ξ) 5 , G(z) = (1 -z) 11 . (l = 4)                           1 -7 243
0 0 0 0 0                           , G(z, ξ) = (1 -z) 4 (1 -ξ) 4 6 143 (1 -z) 6 + 18 143 (1 -z) 5 (1 -ξ) + 30 143 (1 -z) 4 (1 -ξ) 2 + 35 143 (1 -z) 3 (1 -ξ) 3 + 30 143 (1 -z) 2 (1 -ξ) 4 + 18 143 (1 -z) 1 (1 -ξ) 5 + 6 143 (1 -ξ) 6 , G(z) = (1 -z) 14 . (l = 5)                                 1 - 2 13
                                , G(z, ξ) = (1 -z) 5
(1 -ξ) [START_REF] Duplantier | Logarithmic Coecients and Generalized Multifractality of Whole-Plane SLE[END_REF] 11

442 (1 -z) 7 + 77 884 (1 -z) 6 (1 -ξ) + 147 884 (1 -z) 5 (1 -ξ) 2 + 49 221 (1 -z) 4 (1 -ξ) 3 + 49 221 (1 -z) 3 (1 -ξ) 4 + 147 884 (1 -z) 2 (1 -ξ) 5 + 77 884 (1 -z)(1 -ξ) 6 + 11 442 (1 -ξ) 7 , G(z) = (1 -z) 17 .
In addition, in this case, note that λ = 0 is a root of multiplicity l of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,2l+2 . Using MAPLE, we get approximate values of eigenvalues λ. For instance:

• For l = 1, P (λ) = -(1/500) λ [500 λ 4 -4000 λ 3 -7100 λ 2 + 1200 λ + 9], λ = -1.62742614692069; -0.00719495255354571; 0; 0.162287966346161; 9.47233313312807 .

• For l = 2, setting N := 27783,

P (λ) = - 1 N λ 2 27783 λ 5 -1481760 λ 4 -14089950 λ 3 + 13067250 λ 2 + 752500 λ -1250 , λ = -8
.97492212870802, -0.0559421034347205; 0 [START_REF] Beliaev | Harmonic Measure and SLE[END_REF] ; 0.00161586776450401; 0.902321238068622; 61.4602604596429 .

• For l = 3, setting N := 12679867332,

P (λ) = -(1/N ) λ 3 12679867332 λ 6 -4733817137280 λ 5 -281944534895100 λ 4 + 1651970595429000 λ 3 + 691542051440625 λ 2
-13028876437500 λ -5252187500 , λ = -56.9534576360948; -0.409444533297716; -0.000394851578279199; 0 (3) ; 0.0184422272384619; 5.74034083133930; 424.937847295726 .

• For l = 4, setting N := 494190983,

P (λ) = -(1/N ) λ 4 494190983 λ 7 -1328385362304 λ 6 -532354498059384 λ 5 + 21076866188265888 λ 4 + 64421994743778672 λ 3
-10888016701161600 λ 2 -65082380126400 λ + 6613488000 , λ = -386.073540317202; -3.00228847130902; -0.00587683289047773; 0 (4) ; 0.0000999469835051707; 0.166100767964351; 38.9534817870098; 3037.96202311944 .

• For l = 5, setting N := 230787600532736, , p = l(3l + 7) + 2 2(2l + 1) , q = l(3l + 9) + 2 2(2l + 1) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (2l + 1), and we obtain

P (λ) = -(1/N ) λ
G(z, ξ) = (1 -z) l+2 (1 -ξ) l+2 l-2 m=0 |d 2l-m,l+2+m |(1 -z) l-2-m (1 -ξ) m , (5.3.5)
G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+2 .
In the specic matrices below, the coecients d 2l-m,l+2+m in (5.3.5) are in bold.

(l = 2)

         1 -4 6 -4 1 -4 16 -24 16 -4 6 -24 36 -24 6 -4 16 -24 16 -4 1 -4 6 -4 1          , G(z, ξ) = (1 -z) 4 (1 -ξ) 4 , G(z) = (1 -z) 8 . (l = 3)                1 -11 2 25 2 - 15 10 -7 2 1 2
-5 2 5 -5 5 2 -1 2 0                , G(z, ξ) = (1 -z) 5 (1 -ξ) 5 [(1/2)(1 -z) + (1/2)(1 -ξ)] , G(z) = (1 -z) 11 . (l = 4)                      1 -7 234
0 0                      , G(z, ξ) = (1-z) 6 (1-ξ) 6 (3/11)(1 -z) 2 + (5/11)(1 -z)(1 -ξ) + (3/11)(1 -ξ) 2 , G(z) = (1-z) 14 .
(l = 5)

                          1 - 17
                          , G(z, ξ) = (1 -z) 7 (1 -ξ) 7 2 13 (1 -z) 3 + 9 26 (1 -z) 2 (1 -ξ) + 9 26 (1 -z)(1 -ξ) 2 + 2 13 (1 -ξ) 3 , G(z) = (1 -z) 17 . (l = 6)                                 1 -10 228
0 0 0 0                                 , G(z, ξ) = (1 -z) 8 (1 -ξ) 8 3 34 (1 -z) 4 + 21 85 (1 -z) 3 (1 -ξ) + 28 85 (1 -z) 2 (1 -ξ) 2 + 21 85 (1 -z)(1 -ξ) 3 + 3 34 (1 -ξ) 4 , G(z) = (1 -z) 20 .
In addition, in this case, note that λ = 0 is a root of multiplicity l + 2 of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,2l . Using MAPLE, we obtain approximate values of eigenvalues λ. For instance:

• For l = 2, P (λ) = -λ 4 [λ -70] , λ = 0 [START_REF] De Branges | A proof of the Bieberbach conjecture[END_REF] ; 70 .

• For l = 3, P (λ) = -λ 5 λ 2 -462 λ -4851 , λ = -10.27163115; 0 [START_REF] Duplantier | Logarithmic Coecients and Generalized Multifractality of Whole-Plane SLE[END_REF] ; 472.2716312 .

• For l = 4, P (λ) = -λ 6 λ 3 -3198 λ 2 -396279 λ + 821340 , λ = -121.419850584612; 0 [START_REF] Duplantier | The Coecient Problem and Multifractality of Whole-Plane SLE and LLE[END_REF] ; 2.03909734478645; 3317.38075323983 .

• For l = 5, P (λ) = -λ 7 λ 4 -22814 λ 3 -26142413 λ 2 + 874035756 λ + 403213140 λ = -1124.52007490399; -0.455130190587252; 0 [START_REF] Grunsky | Koezienten Bedingungen für schlicht abbidende meromorphe Funktionen[END_REF] ; 32.9552447197475; 23906.0199603748 . , p = 1 , q = 2 -3l(l -1) 2(2l + 1) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 2), and we obtain

G(z, ξ) = (1 -z)(1 -ξ) 3l m=0 |d 3l+1-m,m+1 |(1 -z) 3l-m (1 -ξ) m , (5.3.6) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+2 .
In the specic matrices below, the coecients d 3l+1-m,m+1 in (5.3.6) are in bold. [START_REF] Loutsenko | SLE κ : correlation functions in the coecient problem[END_REF] .

(l = 0) 1 -1 -1 1 , G(z, ξ) = (1 -z)(1 -ξ), G(z) = (1 -z) 2 . (l = 1)          1 -5 2
         , G(z, ξ) = (1 -z)(1 -ξ) 1 5 (1 -z) 3 + 3 10 (1 -z) 2 (1 -ξ) + 3 10 (1 -z)(1 -ξ) 2 + 1 5 (1 -ξ) 3 , G(z) = (1 -z) 5 . (l = 2)                   1 -4 51
0 0 0 0 0 0                   , G(z, ξ) = (1 -z)(1 -ξ) 2 33 (1 -z) 6 + 3 22 (1 -z) 5 (1 -ξ) + 15 77 (1 -z) 4 (1 -ξ) 2 + 50 231 (1 -z) 3 (1 -ξ) 3 + 15 77 (1 -z) 2 (1 -ξ) 4 + 3 22 (1 -z)(1 -ξ) 5 + 2 33 (1 -ξ) 6 , G(z) = (1 -z) 8 . (l = 3)                           1 - 11
-1 52 0 0 0 0 0 0 0 0                           , G(z, ξ) = (1 -z)(1 -ξ) 1 52 (1 -z) 9 + 3 52 (1 -z) 8 (1 -ξ) + 15 143 (1 -z) 7 (1 -ξ) 2 + 21 143 (1 -z) 6 (1 -ξ) 3 + 49 286 (1 -z) 5 (1 -ξ) 4 + 49 286 (1 -z) 4 (1 -ξ) 5 + 21 143 (1 -z) 3 (1 -ξ) 6 + 15 143 (1 -z) 2 (1 -ξ) 7 + 3 52 (1 -z)(1 -ξ) 8 + 1 52 (1 -ξ) 9 , G(z) = (1 -z)
(l = 4)                                    1 - 7

24

-53 13 [START_REF] Loutsenko | On exact multi-fractal spectrum of the wholeplane SLE[END_REF] , G(z) = (1 -z) [START_REF] Löwner | Untersuchungen über schlichte konforme Abildungendes Einheitskreises[END_REF] .

                                   , G(z, ξ) = (1 -z)(1 -ξ) 2 323 (1 -z) 12 + 15 646 (1 -z) 11 (1 -ξ) + 33 646 (1 -z) 10 (1 -ξ) 2 + 55 646 (1 -z) 9 (1 -ξ) 3 + 495 4199 (1 -z) 8 (1 -ξ) 4 + 594 4199 (1 -z) 7 (1 -ξ) 5 + 630 4199 (1 -z) 6 (1 -ξ) 6 + 594 4199 (1 -z) 5 (1 -ξ) 7 + 495 4199 (1 -z) 4 (1 -ξ) 8 + 55 646 (1 -z) 3 (1 -ξ) 9 + 33 646 (1 -z) 2 (1 -ξ) 10 + 15 646 (1 -z)(1 -ξ) 11 + 2 323 (1 -ξ)
In the specic matrices below, the coecients d 3l+1-m,m in (5.3.7) are in bold.

(l = 0) 1 -1 2 -1 2 0 , G(z, ξ) = 1 2 (1 -z) + (1 -ξ) , G(z) = 1 -z. (l = 1)          1 -2 24
0 0 0 0          , G(z, ξ) = 77 494 (1 -z) 4 + 55 247 (1 -z) 3 (1 -ξ) + 60 247 (1 -z) 2 (1 -ξ) 2 + 55 247 (1 -z)(1 -ξ) 3 + 77 494 (1 -ξ) 4 , G(z) = (1 -z) 4 . (l = 2)                   1 - 7
                  , G(z, ξ) = 299 5890 (1 -z) 7 + 322 2945 (1 -z) 6 (1 -ξ) + 462 2945 (1 -z) 5 (1 -ξ) 2 + 539 2945 (1 -z) 4 (1 -ξ) 3 + 539 2945 (1 -z) 3 (1 -ξ) 4 + 462 2945 (1 -z) 2 (1 -ξ) 5 + 322 2945 (1 -z)(1 -ξ) 6 + 299 5890 (1 -ξ) 7 , G(z) = (1 -z) 7 . (l = 3)                           1 -5 63
                          , G(z, ξ) = 28652 1726235 (1 -z) 10 + 16588 345247 (1 -z) 9 (1 -ξ) + 149292 1726235 (1 -z) 8 (1 -ξ) 2 + 211479 1726235 (1 -z) 7 (1 -ξ) 3 + 7293 49321 (1 -z) 6 (1 -ξ) 4 + 38709 246605 (1 -z) 5 (1 -ξ) 5 + 7293 49321 (1 -z) 4 (1 -ξ) 6 + 211479 1726235 (1 -z) 3 (1 -ξ) 7 + 149292 1726235 (1 -z) 2 (1 -ξ) 8 + 16588 345247 (1 -z)(1 -ξ) 9 + 28652 1726235 (1 -ξ) 10 , G(z) = (1 -z) 10 . (l = 4)                                    1 - 13
                                   , G(z, ξ) = 915325 168480536 (1 -z) 13 + 475969 24068648 (1 -z) 12 (1 -ξ) + 516477 12034324 (1 -z) 11 (1 -ξ) 2 + 860795 12034324 (1 -z) 10 (1 -ξ) 3 + 2414425 24068648 (1 -z) 9 (1 -ξ) 4 + 2973555 24068648 (1 -z) 8 (1 -ξ) 5 + 5748873 42120134 (1 -z) 7 (1 -ξ) 6 + 5748873 42120134 (1 -z) 6 (1 -ξ) 7 + 2973555 24068648 (1 -z) (1 -ξ) + 2414425 24068648 (1 -z) (1 -ξ) + 860795 12034324 (1 -z) (1 -ξ) + 516477 12034324 (1 -z) (1 -ξ) + 475969 24068648 (1 -z)(1 -ξ) 12 + 915325 168480536 (1 -ξ) 13 , G(z) = (1 -z) 13 .
Furthermore, in this case, note that λ = 0 is not a root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l+1 . Using MAPLE, we get approximate values of eigenvalues λ. For instance:

• For l = 0, P (λ) = λ 2 -λ -1/4, λ = -0.2071067812; 1.207106781 .

• For l = 1, P

= 6 6l + 1 , p = 1 , q = 2 -3l(3l -5) 2(6l + 1) ,
then, the corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 1), and we obtain

G(z, ξ) = (1 -z)(1 -ξ) 3l-1 m=0 |d 3l-m,m+1 |(1 -z) 3l-1-m (1 -ξ) m , (5.3.8) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+1 .
In the specic matrices below, the coecients d 3l-m,m+1 in (5.3.8) are in bold.

(l = 1) 0 0 [START_REF] Loutsenko | Average harmonic spectrum of the wholeplane SLE[END_REF] . Furthermore, in this case, note that λ = 0 is a simple root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l . Using MAPLE, we get approximate values of eigenvalues λ. For instance:

      1 -2 17
      , G(z, ξ) = (1 -z)(1 -ξ) (4/13)(1 -z) 2 + (5/13)(1 -z)(1 -ξ) + (4/13)(1 -ξ) 2 , G(z) = (1 -z) 4 . (l = 2)                1 - 7
0 0 0 0 0                , G(z, ξ) = (1 -z)(1 -ξ) 17 190 (1 -z) 5 + 17 95 (1 -z) 4 (1 -ξ) + 22 95 (1 -z) 3 (1 -ξ) 2 + 22 95 (1 -z) 2 (1 -ξ) 3 + 17 95 (1 -z)(1 -ξ) 4 + 17 190 (1 -ξ) 5 , G(z) = (1 -z) 7 . (l = 3)                        1 -5 296
                       , G(z, ξ) = (1 -z)(1 -ξ) 34684 1233025 (1 -z) 8 + 95381 1233025 (1 -z) 7 (1 -ξ) + 161161 1233025 (1 -z) 6 (1 -ξ) 2 + 210749 1233025 (1 -z) 5 (1 -ξ) 3 + 9163 49321 (1 -z) 4 (1 -ξ) 4 + 210749 1233025 (1 -z) 3 (1 -ξ) 5 + 161161 1233025 (1 -z) 2 (1 -ξ) 6 + 95381 1233025 (1 -z)(1 -ξ) 7 + 34684 1233025 (1 -ξ) 8 , G(z) = (1 -z) 10 . (l = 4)                                 1 - 13
                                , G(z, ξ) = (1 -z)(1 -ξ) 3116 345247 (1 -z) 11 + 1558 49321 (1 -z) 10 (1 -ξ) + 3230 49321 (1 -z) 9 (1 -ξ) 2 + 5100 49321 (1 -z) 8 (1 -ξ) 3 + 46920 345247 (1 -z) 7 (1 -ξ) 4 + 15249 98642 (1 -z) 6 (1 -ξ) 5 + 15249 98642 (1 -z) 5 (1 -ξ) 6 + 46920 345247 (1 -z) 4 (1 -ξ) 7 + 5100 49321 (1 -z) 3 (1 -ξ) 8 + 3230 49321 (1 -z) 2 (1 -ξ) 9 + 1558 49321 (1 -z)(1 -ξ) 10 + 3116 345247 (1 -ξ) 11 , G(z) = (1 -z)
• For l = 1, P (λ) = (1/2197) λ 2197 λ 3 -10478 λ 2 -8112 λ + 320 , λ = -0.711111180491101; 0; 0.0376328658944030; 5.44270908382747 .

• For l = 2, setting N := 2940367562500, , p = 0 , q = -3 (3l + 4) (l + 1) 2 (6l + 5) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 4), and we obtain

P (λ) = - 1 N λ 2940367562500
G(z, ξ) = 3l+3 m=0 |d 3l+3-m,m |(1 -z) 3l+3-m (1 -ξ) m , (5.3.9) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+3 .
In the specic matrices below, the coecients d 3l+3-m,m in (5.3.9) are in bold.

(l = 0) In addition, in this case, note that λ = 0 is not a root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l+3 . Using MAPLE, we obtain approximate values of eigenvalues λ. For instance:

      1 -3 2
      , G(z, ξ) = 5 22 (1-z) 3 + 3 11 (1-z) 2 (1-ξ)+ 3 11 (1-z)(1-ξ) 2 + 5 22 (1-ξ) 3 , G(z) = (1-z) . (l = 1)                1 -3 75
0 0 0 0 0 0                , G(z, ξ) = 836 11339 (1 -z) 6 + 1596 11339 (1 -z) 5 (1 -ξ) + 2100 11339 (1 -z) 4 (1 -ξ) 2 + 2275 11339 (1 -z) 3 (1 -ξ) 3 + 2100 11339 (1 -z) 2 (1 -ξ) 4 + 1596 11339 (1 -z)(1 -ξ) 5 + 836 11339 (1 -ξ) 6 , G(z) = (1 -z) 6 . (l = 2)                        1 - 9
                       , G(z, ξ) = 2635 109388 (1 -z) 9 + 6975 109388 (1 -z) 8 (1 -ξ) + 2925 27347 (1 -z) 7 (1 -ξ) 2 + 3900 27347 (1 -ξ) 6 (1 -ξ) 3 + 4446 27347 (1 -ξ) 5 (1 -ξ) 4 + 4446 27347 (1 -ξ) 4 (1 -ξ) 5 + 3900 27347 (1 -ξ) 3 (1 -ξ) 6 + 2925 27347 (1 -z) 2 (1 -ξ) 7 + 6975 109388 (1 -z)(1 -ξ) 8 + 2635 109388 (1 -ξ) 9 , G(z) = (1 -z) 9 . (l = 3)                                 1 -6 528 - 1045
• For l = 0, P (λ) = λ Besides, in this case, note that λ = 0 is a simple root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l+2 . Using MAPLE, we get approximate values of eigenvalues λ. For instance:

• For l = 0, F (n, d, z) everywhere. When p = q + 1, the series converges for |z| < 1. F (n, d, z) is then dened for |z| ≥ 1 by analytic continuation. The point z = 1 is a branch point, and the interval (1, +∞) is the branch cut. When q + 1 < p, the series diverges for all z = 0. In this case, the series is interpreted as the asymptotic expansion of F (n, d, z) around z = 0. The positive real axis is the branch cut.

P (λ) = - 1

A.2 Hausdor dimension and Minkowski dimension

Let α > 0. The α-dimensional Hausdor measure of a Borel set E ⊂ C is dened by

Λ α (E) = lim -→0 inf (B k ) k (diamB k ) α ,
where the inmum is taken over the covers (B k ) of E with diam B k ≤ for all k.

The Hausdor dimension is dened by H. dim(E) = inf{α : Λ α (E) = 0}.

Sets of non-integer number dimension are called "fractals". Proof. see [19].

Let E be a bounded set in C and let N ( , E) denote the minimal numbers of disks of diameter that are needed to cover E. Up to bounded multiplies it is the same as the number of squares of grid of mesh size that intersect E. We dene the Proof. see [19].

the Brownian motion, also started at B0 = 0. where Xs := e -s -i √ κ Bs , s ≥ 0, is an independent copy of that process, with X0 = 1.

The operator J (B.0.14) can then be written as J v(t) As mentioned in Theorem 1.2.1, the conjugate whole-plane Schramm Loewner evolution e -i √ κBt f t (e i √ κBt z) shold have the same law as f 0 (z). At order n, we are thus interested in the stochastically rotated coecients:

e i(n-1) √ κBt a n (t) = (X t ) -(n-1) u n (t). if n < 0 then return 0 elif m < 0 then return 0 elif (n, m) = (0, 0) then return 0 elif (n, m) = (1, 1) then return 8 elif (n, m) = (1, 0) then return 0 elif (n, m) = (0, 1) then return 0 elif (n, m) = (2, 0) then return 0 elif (n, m) = (0, 2) then return 0 elif (n, m) = (2, 1) then return (here slit by a single curve γ([t, ∞)) for SLE κ≤4 ). One has f t (0) = 0, ∀t ≥ 0. At t = 0, the driving function λ(0) = 1, so that the image of z = 1 is at the tip γ(0) = f 0 (1) of the curve (Fig. 1 in [6]). . . . . .

2.1

Integral curves R of Theorem 2.2.1, for κ = 2 (blue), κ = 4 (red), and κ = 6 (green). In addition to the origin, the q = 0 intersection point with the p-axis is at p(κ) := (6 + κ)(2 + κ)/8κ, with p(2) = p(6) = 2 [6, 11]

3.1

Integral curves R 1 of Theorem 3.1.1, for κ = 2 (blue), κ = 4 (red), and κ = 6 (green). The p = 0 intersection points with the q-axis are at q 1 (κ) := -2(2 + κ)(1 + κ)/κ, with q 1 (2) = -12, and q 2 (κ) := (2 -κ)/2 (not marked). . . . . . . . . . . . . . . . . . . . . . . . . . The bulk spectrum β 0 (p) and the generalized spectrum β 1 (p, q) coincide along the arc of red parabola between its tangency points T 0 and T 1 with ∆ 0 and ∆ 1 (thick red line). They also coincide along the innite left branch of the green parabola, up to its tangency point T 2 to ∆ 1 (thick green line). The β 0 (p) spectrum and the linear one β lin (p) coincide along D 0 , whereas β 1 (p, q) and β lin (p) coincide along D 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

The blue quartic 3.4.29 for κ = 6. It intersects the green parabola at point Q 0 (3.4.31) and the red parabola at point Q 1 (3.4.30) (not marked), both of abscissa p 0 (κ) = -1 -3κ/8. . . . . . . . . . . . . . .

Introduction 1

 1 Radial, whole-plane SLE κ processes 1.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Simply connected domains . . . . . . . . . . . . . . . . . . . . 1.1.2 Caratheodory convergence theorem . . . . . . . . . . . . . . . 1.1.3 Brownian motion -Itô formula . . . . . . . . . . . . . . . . . . 1.1.4 Radial, whole-plane stochastic Loewner evolution . . . . . . . 1.2 Expectation of f 0 (z) . . . . . . . . . . . . . . . . . . . . . . . . . . .

  (z) := z f (z 2 )/z 2 = ∞ n=0 b 2n+1 z 2n+1 , (0.0.2) with b 1 = 1. Robertson's conjecture states that: ∀n ≥ 0, n k=0 |b 2k+1 | 2 ≤ n + 1.

  b 1 b 2n-1 + b 3 b 2n-3 + • • • + b 2n-1 b 1 ,

n k=0 |b 2k+1 | 2 ≤

 2 (n + 1) exp 1 n + 1 n m=1 m k=1 k|γ k | 2 -1 k .

  are said to be homotopic if there exists Γ :[a, b] × [0, 1] → X continuous such that ∀s ∈ [a, b], Γ(s, 0) = γ 1 (s), Γ(s, 1) = γ 2 (s). Denition 1.1.1. The space X is called simply-connected if it is connected and if every closed arc γ : [a, b] → X is homotopic to a constant arc γ 0 : [a, b] → γ(a).When X is a plane domain we have the following equivalent characterizations of simply connected domains: Theorem 1.1.2. For a connected open subset Ω of C the followings are equivalent:i . Ω is simply connected, ii . C \ Ω is connected, iii .For any closed arc γ whose image lies in Ω and any z /

3 .

 3 (Riemann Mapping Theorem).

( 1 . 3 . 8 ) 1 . 3 .

 13813 EXPECTATION OF LOG F 0 (Z) Z Note that one can rewrite the dierential recursion (1.3.7) as βn = X t [ βn-1 + 2(n -1)β n-1 ].

  ) and(1.3.18) lead us to the following explicit representation of β n (t) (1.3.14):

Theorem 2 . 1 . 1 .

 211 Let f (z) := f 0 (z) be the interior whole-plane SLE κ map at time 0

which encompasses Theorems 2 . 3 .1 and 2 . 1 . 1 .

 23211 We end this chapter by briey returning to the Lebedev-Milin theory. By Theorem 2.3.1, we have for SLE 2 ,

.1. 6 )Figure 3 . 1 :

 631 Figure 3.1: Integral curves R 1 of Theorem 3.1.1, for κ = 2 (blue), κ = 4 (red), and κ = 6 (green). The p = 0 intersection points with the q-axis are at q 1 (κ) := -2(2 + κ)(1 + κ)/κ, with q 1 (2) = -12, and q 2 (κ) := (2 -κ)/2 (not marked).

( 3 . 1 . 2 )

 312 or(3.1.4) or(3.1.5), and for any z ∈ D,

Figure 3 . 2 :

 32 Figure 3.2: Parabolas R (red), R 1 (coral), R 2 (magenta), R 3 (cyan) of the family (R k ) k∈N , dened by Eqs. (3.2.3) or (3.2.4), for (a) κ = 2, (b) κ = 4, and (c) κ = 6.The two lines ∆ 0 := {p = (4 + κ) 2 /8κ} and ∆ 1 := {(p, q) ∈ R 2 , q = p + 1/2κ} are

( 3 . 4 . 10 )

 3410 The integral means spectrum is afterwards given by β(p, κ) := 3p -, for p ≥ p * (κ).

( 3 . 4 . 11 )

 3411 Since p * (κ) < p 0 (κ)(3.4.9), this transition precedes and supersedes the transition from the bulk spectrum (3.4.7) towards the linear behavior(3.4.8).

( 3 .

 3 4.13) the relevance of which for the exterior whole-plane SLE case is analyzed in a joint work of M. Zinsmeister and B. Duplantier with D. Beliaev[1].

Figure 3 . 3 :

 33 Figure 3.3: Red parabola R (3.4.19) and green parabola G (3.4.21) (for κ = 6). From the intersection point P 0 (3.4.22) originate the two (half )-lines D 0 (3.4.24) and D 1 (3.4.25).The bulk spectrum β 0 (p) and the generalized spectrum β 1 (p, q) coincide along the arc of red parabola between its tangency points T 0 and T 1 with ∆ 0 and ∆ 1 (thick red line). They also coincide along the innite left branch of the green parabola, up to its tangency point T 2 to ∆ 1 (thick green line). The β 0 (p) spectrum and the linear one β lin (p) coincide along D 0 , whereas β 1 (p, q) and β lin (p) coincide along D 1 .

  4.19) and(3.4.21) can be found by combining the seed equations (3.4.18) and(3.4.20). We nd either γ = γ = 1/κ + 1/4, or γ = 2/κ + 1/4, γ = -1/4, which lead to the two intersection points,

  Denition 3.4.1. D 0 and D 1 are, respectively, the vertical line and the slope one line passing through point P 0 , of equations D 0 := {(p, q) : p = p 0 }, (3.4.24)

Figure 3 . 4 :

 34 Figure 3.4: The blue quartic 3.4.29 for κ = 6. It intersects the green parabola at point Q 0 (3.4.31) and the red parabola at point Q 1 (3.4.30) (not marked), both of abscissa p 0 (κ) = -1 -3κ/8.

3. 4 . 3

 43 Whole-plane SLE κ generalized spectrum 3.4.3.1 Four-domain structure for the generalized spectrum β(p, q)

Figure 3 . 6 :

 36 Figure 3.6:Various geometrical elements appearing in the proof of Theorem 3.4.2, when establishing the respective domains of validity of integral means spectra β tip (p), β 0 (p), β lin (p), and β 1 (p, q). The dashed coral line D 2 corresponds to Eq.(3.4.35) and intersects D 0 at point P 2 . The dashed cyan line D 3 corresponds to Eq. (3.4.36), and intersects D 0 at Q 0 and the red parabola R at point P 3 . The q = 2p continuous straight line in coral , corresponding to the whole-plane SLE version studied in Ref.[2], does not intersect the blue quartic, but intersects the green parabola at a point of abscissa(3.4.33). (In the particular κ = 2 case shown here, the intersection point P 3 of D 3 with R coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.)

Figure 3 . 7 :

 37 Figure 3.7: The coral parabola R 1 (3.4.37) and the red parabola R (3.4.19) have the same intersection point P 3 with the straight line D 3 (3.4.36).The innite branch of R 1 below P 3 (thin coral line), with parametric coordinates p C (γ), q C (γ) (3.4.37) for γ ∈ [1+2/κ, +∞). The innite branch of R below P 3 (thin red line), with parametric coordinates p R (γ), q R (γ)(3.4.19) for γ ∈ [1 + 2/κ, +∞). (In the particular κ = 2 case shown here, point P 3 coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.)

( 3 .

 3 4.37) In Cartesian coordinates, the coral parabola R 1 (3.4.37) has for equation(3.1.4).It is interesting to note that the coral parabola R 1 and the red parabola R have the same intersection pointP 3 = 1+ 2 κ , 4-κ 22κ with the straight line D 3 (3.4.36) (Fig. 3.7). The unique intersection point of the straight line D 2 (3.4.35) with the coral parabola R 1 , 5(2+κ)(6-κ) 32κ

2 (r→1 -) 1 ( 1 7 + 4327554 r 5 +

 21175 , -19/2) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 2 P 0 (r) + P 1 (r) cos θ 15(1 -r) 35/2 (1 + r) 35/2 , where P 0 (r) = 15 r 10 + 315 r 8 + 1190 r 6 + 1190 r 4 + 315 r 2 + 15, P 1 (r) = -78 r 9 + 728 r 7 + 1428 r 5 + 728 r 3 + 78 r . iθ , re -iθ )dθ = r 2 + 1 15 r 12 + 516 r 10 + 3561 r 8 + 7016 r 6 + 3561 r 4 + 516 r 2 + 15) 15 (r -1) 35/2 (r + 1) 35/-r) 35/2 , which implies β(1, -19/2; 3) = 35/2 = β 1 (1, -19/2; 3). • For κ = 3 and γ = 4, we obtain (p, q) = (p C (γ), q C (γ)) = (-3, -23) and G re iθ , re -iθ = r 2 + 1 -2r cos θ 3 P 0 (r) + P 1 (r) cos θ 3315(1 -r) 31 (1 + r) 31 , where P 0 (r) = 3315 r 16 + 175032 r 14 + 1973972 r 12 + 7592200 r 10 + 11723250 r 8 + 7592200 r 6 + 1973972 r 4 + 175032 r 2 + 3315, P 1 (r) = -25194 r 15 + 646646 r 13 + 4327554 r 11 + 10606750 r 9 + 10606750 r 646646 r 3 + 25194 r . iθ , re -iθ )dθ = (r 2 + 1) P (r) 3315(1 -r) 31 (1 + r) 31 (r→1 -) 1 (1 -r) 31 , which implies β(-3, -23; 3) = 31 = β 1 (-3, -23; 3), where P (r) = 3315 r 20 + 277134 r 18 + 5468645 r 16 + 40345964 r 14 + 130356040 r 12 + 193312844 r 10 + 130356040 r 8 + 40345964 r 6 + 5468645 r 4 + 277134 r 2 + 3315.

  φ t (u))e log φ t (u) du.

( 1 -

 1 and b(z) = V (z) = ∂ ∂t log φ t (z) t=0 belongs to the Bloch space B, which is dened as follows:B = b holomorphic in D; sup D |z| 2 )|b (z)| < ∞ .We recall that the McMullen's asymptotic variance of a Bloch function b ∈ B iθ )| 2 dθ.

( 4 .

 4 1.5) which, together with Proposition 3 in Ref.[8], implies (4.1.4), where β(p, φ) is the integral means spectrum of φ (φ (z) = exp b(z)) dened as β(p, φ) = lim sup r-→1 - log 2π 0 |φ (re iθ )| p dθ | log(1 -r)|

( 4 . 1 . 6 )F

 416 On the other hand, as shown in Ref.[6] and mentioned in Section 3.4 of Chapter 3, the average integral means spectrum of the interior whole-plane SLE κ map f is Owing to Lemma (2.1.4), we have lim t→+∞ (z, t) = F (z).

( 4 . 3 . 4 )

 434 By the Markov property of SLE we get

  3.1) above, therebyobtaining the moduli logarithm one-point function,G(z, z) = E | log f (z)| 2 .

( 4 . 3 . 9 )

 439 Because of Eq. (4.3.8), it obeys the corresponding PDE,

( 4 . 3 . 14 ) 1 -

 43141 From Eqs. (4.3.12), (4.3.13) and (4.3.14), we obtain for κ = 2,G(z, z) = (2z -log(1 -z))(2z -log(1 -z))z z).

( 4 .

 4 3.15)

) 4 . 4

 44 The proof of Theorem 4.1.1

•λ 8 2125 λ 5 -352925950 λ 4 -3442903220445 λ 3 +• For l = 7 ,P (λ) = - 1 19 λ 9 19 λ 6 -23305210 λ 5 -1865382914425 λ 4 + 6862029786001500 λ 3 + 837937538085037500 λ 2 - 6 . 5 . 3 . 5 .

 5437654326535 For l = 6, P (λ) = -1 2125 1297186309131804 λ 2 + 12023236369144728 λ -1310828169819168 , λ = -9585.20521130524; -9.15673328716147; 0 (8) ; 0.107771811816883; 372.001966142034; 1.75305052206639 • 10 5 . 2177250042747780000 λ -58081904080200000 , λ = -78723.3474876778; -120.779114558606; -0.0264083818971425; 0 (9) ; 2.57120092294019; 3637.23752000905; 1.30179434428969 • 10 Observation For any nonnegative integer l, if letting

( 1 -

 1

( 1 -

 1 ) = (1 -z)[START_REF] Loutsenko | On exact multi-fractal spectrum of the wholeplane SLE[END_REF]

.

Observation 5 . 3 . 9 .

 539 For any nonnegative integer l, if letting corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 3), and we obtainG(z, ξ) = (1 -z)(1 -ξ) 3l+1 m=0 |d 3l+2-m,m+1 |(1 -z) 3l+1-m (1 -ξ) m , (5.3.10) G(z) := lim ξ→z G(z, ξ) = E (f (z)) 2p-q = (1 -z) 3l+3 .In the specic matrices below, the coecients d 3l+2-m,m+1 in (5.3.10) are in bold. ξ)[START_REF] De Branges | A proof of the Bieberbach conjecture[END_REF] ,G(z) = (1 -z) 6 . , ξ) = (1 -z)(1 -ξ) 55 1334 (1 -z) 7 + 275 2668 (1 -z) 6 (1 -ξ) + 429 2668 (1 -z) 5 (1 -ξ) 2 + 130 667 (1 -z) 4 (1 -ξ) 3 + 130 667 (1 -z) 3 (1 -ξ) 4 + 429 2668 (1 -z) 2 (1 -ξ) 5 ξ) 7 , G(z) = (1 -z) 9 .

4 λ 4 λ 2 - 1 N λ 9138686662951 λ 5 -127474161278094 λ 4 -468590087451369 λ 3 +λ 8 -845346615291859421061229824 λ 7 -23433777081891534413505969216 λ 6 + 123361612548242494613840183520 λ 5 + 110881829698785003854135170800 λ 4 -

 2154387654 12 λ -3 , λ = -0.2320508076; 0; 3.232050808 . • For l = 1, setting N := 9138686662951, P (λ) = 224236344353658 λ 2 + 9070727237208 λ -14106849120 , λ = -3.37458314603931; -0.0389335016002556; 0; 0.00149977501807599; 0.463618278988831; 16.8972476984920 . • For l = 2, setting N = 10028746107459051168522496, 12964010361790242372682779000 λ 3 -141365208240235581888268125 λ 2 + 90490096744360666031250 λ + 1600938223203515625 , λ = -25.5147759584886; -0.876131063858002; -0.0106098153021541; -0.0000172288744522820; 0; 0.000622903526212357; 0.113920834224742; 5.22550233298296; 105.353841818877 . Remark 5.3.1. In all the above observations, it is interesting to note that one always has 4p -2q = p(κ) := (6 + κ)(2 + κ) 8κ , which implies that the point (4p -2q; 0) coincides the intersection point (p(κ), 0) of the red parabola R (3.4.19) with the p-axis (Fig. 2.1). From Theorem 2.2.1, the function G(z) := E ((f (z)) 2p-q ) has the integrable form (1 -z) γ .

Proposition A. 2 . 1 .

 21 The Hausdor dimension H.dim(E) is the unique real d ≥ 0such that Λ α (E) = +∞ if 0 < α < H. dim(E) and Λ α (E) = 0 if α > H. dim(E).

  Minkowski dimensionof E by M. dim(E) = lim sup -→0 log N ( , E) log(1/ ) . Proposition A.2.2. If E is any bounded set in C then H. dim E ≤ lim inf -→0 log N ( , E) log(1/ ) ≤ M. dim E.

  Xs v(s + t)ds. By iteration of the use of the Markov property, Eq (B.0.15) can be rewritten asu n (law) = J • [X (1 + (n -1) J [n-2] )] • • • • • [X (1 + 2 J [1] )]1 (1 + (k + 1) J [k] )]1, (B.0.19)where the integral operators J[k] , k = 1, • • • , n -2, involve successive independent copies, X[k] s k , k = 1, • • • , n -2, ofthe original process X s . We therefore arrive at the following explicit representation of the solution (B.0.15):u n (t) s k ) k ds k ds.(B.0.20)

  Using again the identity in law (B.0.16) in (B.0.20), we arrive at e i(n-1) √ κBt a n (t) it must, no longer depends on t. All factors in (B.0.21) involve successive independent copies of the process X s , and their expectations can now be taken independently. Recall that E[(Xs ) k ] = e -( κ 2 k 2 +k)s . Function Coes_Loga for the coecients of G(z, z) := E (| log f (z)| 2 ) (for SLE 2 ).Coes_Loga := proc(n, m) function computes the coecients of G(z, z) := E(| log f | 2 ) when κ = 2 option remember;

-2 3 elif+ m - 4 -+ 2 •Function

 342 (n, m) = (1, 2) then return -2 3 elif (n, m) = (2, 2) then return 5 3 elif (n, m) = (3, 0) then return 0 elif (n, m) = (0, 3) then return 0 elif (n, m) = (3, 1) then return 4 9 elif (n, m) = (1, 3) then return 4 9 elif (n, m) = (3, 2) then return -10 27 elif (n, m) = (2, 3) then return -10 27 elif n ≥ 4 and m = 2 then returnfactor (n -m) 2 •Coes_Loga(n -2, m -2) (n -m -1) 2 + 2 -n •Coes_Loga(n -2, m -1) + (n -m -2) • (m -n + 3)•Coes_Loga(n -2, m) + 2 • (n -m + 1) 2 + 2 -m •Coes_Loga(n -1, m -2) -4 • (n -m) 2 •Coes_Loga(n -1, m -1) + 2 • (n -m -1) 2 + m •Coes_Loga(n -1, m) + (n -m + 2) • (m -n -3)•Coes_Loga(n, m -2) + 2 • (n -m + 1) 2 + n •Coes_Loga(n, m -1) -8 3•(n-2)•(n-1)•n n ≥ 4 and m = 1 then return factor   simplify   1 n+m+(n-m) 2 •   n + m -4 -(n -m) 2 •Coes_Loga(n -2, m -2) + 2 • (n -m -1) 2 + 2 -n •Coes_Loga(n -2, m -1) + (n -m -2) • (m -n + 3)•Coes_Loga(n -2, m) + 2 • (n -m + 1) 2 + 2 -m •Coes_Loga(n -1, m -2) -4 • (n -m) 2 •Coes_Loga(n -1, m -1) + 2 • (n -m -1) 2 + m •Coes_Loga(n -1, m) + (n -m + 2) • (m -n -3)•Coes_Loga(n, m -2) + 2 • (n -m + 1) 2 + n •Coes_Loga(n, m --m) 2 •   n + m -4 -(n -m) 2 •Coes_Loga(n -2, m -2) + 2 • (n -m -1) 2 + 2 -n •Coes_Loga(n -2, m -1) + (n -m -2) • (m -n + 3)•Coes_Loga(n -2, m) + 2 • (n -m + 1) 2 + 2 -m •Coes_Loga(n -1, m -2) -4 • (n -m) 2 •Coes_Loga(n -1, m -1) + 2 • (n -m -1) 2 + m •Coes_Loga(n -1, m) + (n -m + 2) • (m -n -3)•Coes_Loga(n, m -2) + 2 • (n -m + 1) 2 + n •Coes_Loga(n, m -1) --m) 2 •   n + m -4 -(n -m) 2 •Coes_Loga(n -2, m -2) + 2 • (n -m -1) 2 + 2 -n •Coes_Loga(n -2, m -1) + (n -m -2) • (m -n + 3)•Coes_Loga(n -2, m) + 2 • (n -m + 1) 2 + 2 -m •Coes_Loga(n -1, m -2) -4 • (n -m) 2 •Coes_Loga(n -1, m -1) + 2 • (n -m -1) 2 + m •Coes_Loga(n -1, m) + (n -m + 2) • (m -n -3)•Coes_Loga(n, m -2) + 2 • (n -m + 1) 2 + n •Coes_Loga(n, m -1) + 16 3•(m-2)•(m-1)•m -m) 2 •   n + m -4 -(n -m) 2 •Coes_Loga(n -2, m -2) + 2 • (n -m -1) 2 + 2 -n •Coes_Loga(n -2, m -1) + (n -m -2) • (m -n + 3)•Coes_Loga(n -2, m) + 2 • (n -m + 1) 2 + 2 -m •Coes_Loga(n -1, m -2) -4 • (n -m) 2 •Coes_Loga(n -1, m -1) + 2 • (n -m -1) 2 + m •Coes_Loga(n -1, m) + (n -m + 2) • (m -n -3)•Coes_Loga(n, m -2) + 2 • (n -m + 1) 2 + n •Coes_Loga(n, m -1)Coes_F F for the coecients of E (log f (z)) • E (log f (z)) (for SLE 2 ).

2 G

 2 Coes_F F := proc(n, m) auxiliary function computes the coecients ofF (z)F (z) := E (log f (z)) E (log f (z)) when κ = (z, z) = F (z)F (z) + R(z, z) with G(z, z) = E | log f (z)| 2option remember;if n = 0 then return 0 elif m = 0 then return 0 elif (n, m) = (1, 1) then return 4 elif n = 1 and m ≥ 2 then return 4 3•m elif n ≥ 2 and m = 1 then return 4 3•n elif n ≥ 2 and m ≥ 2 then return

4 9 •

 9 n•m end if; end proc; 1 Loewner map z → f t (z) from D to the slit domain Ω t = C\γ([t, ∞))

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . 2 Parabolas

 32 R (red), R 1 (coral), R 2 (magenta), R 3 (cyan) of the family (R k ) k∈N , dened by Eqs. (3.2.3) or (3.2.4), for (a) κ = 2, (b) κ = 4, and (c) κ = 6. The two lines ∆ 0 := {p = (4 + κ) 2 /8κ} and ∆ 1 := {(p, q) ∈ R 2 , q = p + 1/2κ} are tangent lines to each parabola of this family. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Red parabola R (3.4.19) and green parabola G (3.4.21) (for κ = 6). From the intersection point P 0 (3.4.22) originate the two (half )-lines D 0 (3.4.24) and D 1 (3.4.25).

  Let g(t, x) ∈ C 2 ([0, +∞)×R) (i.e. g is twice continuously dierentiable on [0, +∞)× R). Then Y t = g(t, X t ) X t ) • (dX t ) 2 ,where(dX t ) 2 = (dX t ) • (dX t ) is computed according to the rules dt • dt = dt • dB t = dB t • dt = 0, dB t • dB t = dt.

	Denition 1.1.7. A standard, n-dimentional Brownian motion is a (n-dimentional) continuous-time stochastic process B t = (B (1) t , • • • , B (n) t ) such that each B (i) t is a standard Brownian motion and the B (i) t 's are independent of each other. Theorem 1.1.8. is again an Itô process, and dY t = ∂g ∂t (t, X t )dt + ∂g ∂x (t, X t )dX t + 1 2 ∂ 2 g ∂x 2 (t, Theorem 1.1.9.

(The one -dimensional Itô formula) Let X t be an Itô process given by dX t = udt + vdB t . (The general Itô formula) Let

dX t = udt + vdB t be an n-dimensional Itô process. Let g(t, x) = (g 1 (t, x), • • • , g p (t, x)) be a C 2 map from [0, +∞) × R n into R p .

Then the process

  [START_REF] Loutsenko | On exact multi-fractal spectrum of the wholeplane SLE[END_REF] + 3025κ 11 + 131609κ 10 + 3216919κ 9 + 47640321κ[START_REF] Le | On Minkowski dimension of Jordan curves[END_REF]

	+ 422153664κ 7 + 2049303168κ 6 + 5164417376κ 5 + 6709663264κ 4
	+ 4916208896κ 3 + 2042489088κ 2 + 447056640κ + 39916800

  [START_REF] Robertson | On the theory of univalent functions[END_REF] + 292381824κ 19 + 16336358935κ[START_REF] Ch | Univalent functions[END_REF]

	+ 609663097723κ 17 + 16046685312588κ 16 + 305815223149938κ 15
	+ 4249934755510755κ 14 + 42691956825394491κ 13 + 303775371493846966κ 12
	+ 1500161524181640952κ 11 + 5069881966077351360κ 10
	+ 11650539687097032144κ 9 + 18247361263799099424κ 8
	+ 19713761511688071936κ 7 + 14894859141855481600κ 6
	+ 7892452487790806272κ 5 + 2905579088214538752κ 4
	+ 724920375549081600κ 3 + 116517087890841600κ

  There is an alternative proof of this corollary, which is based on the study of SLE one-point function. We rst restate Corollary 1.3.3 as

	Chapter 2
	Logarithmic coefficients of
	whole-plane SLE κ processes
	2.1 Expectations of logarithmic coecients

1 2n 2 holds in the SLE 2 case. The rigorous proof of this property is given in the next chapter. The starting point was to consider Milin's conjecture. It yields the study of the logarithmic coecients γ n of the interior whole-plane SLE κ map f (z) := f 0 (z). By Corollary 1.3.3, we have already seen a computation of E log f (z) z for SLE 2 .

  The intersection of the blue quartic Q (3.4.29) with the green parabola G

	(3.4.21) is located at

  ) is an analytic family. There exists a neighborhood U of 0 such that if t ∈ U then φ t is a conformal map with quasiconformal extension.In Ref.[8], using a probability argument, N.Le and M. Zinsmeister described a relatively large family of function b ∈ B for which if φ t is dened by (4.1.3) (t being real), then (4.1.2) is true with Hausdor dimension replaced by Minkowski dimension (see also Appendix A),

	4.1. STARTING MOTIVATION
	Conversely, starting from a function b ∈ B, it is known that if we put
	z
	φ t (z) =	e tb(u) du, b ∈ B,	(4.1.3)
	0
	then (φ t

.

1.2)

where H. dim(φ t (∂D)) denotes Hausdor dimension of φ t (∂D) (see Appendix A).

  .2.11) vanishes. The Itô dierential of X s is simply computed from Eqs. (2.1.9) and(2.1.10)

as:

  .2.13) and the vanishing of the ds-drift term in dM s requires that of the drift term in the right-hand side bracket in (5.2.13), since X s (z, ξ) does not vanish in D × D.

The Itô dierential of G(z s , ξ s , τ ) can be obtained from Eqs. (5.2.10) and Itô calculus as

  [START_REF] Duplantier | Logarithmic Coecients and Generalized Multifractality of Whole-Plane SLE[END_REF] 230787600532736 λ 8 -4549285181701292032 λ 7 -12772751297221978270208 λ 6 + 3551620916003369797812224 λ 5 + 80085686812398032552698912 λ 4 -112698292125376973991399424 λ 3 -7166982949539552041048608 λ 2 + 13276095142966573682688 λ

	Observation 5.3.4. For any integer l ≥ 2, if letting
	κ =	2 2l + 1
		+ 346792539273426477 ,
	λ = -2717.79332022905; -22.2130354765117; -0.0626880018584103;
		-0.257634008830183 • 10 -4 ; 0 (5) ; 0.00182649779303946; 1.38680121956311;
		274.361067547899; 22176.3193742056 .

  λ 6 -73447286587500 λ 5

	• For l = 4, N := 497478873035110874035149889519235117915098040306388138248641,
	P (λ) = -	1 N	λ 497478873035110874035149889519235117915098040306388138248641 λ 12
	-532727542350127989718544701200095388951136642800569321024466020 λ
	-λ 10
	+ 17849005796821652589787822322841407132312910655798157087370931340240 λ 9
	+ 344297864288138251319200229961508446258433495691330778278154810608976 λ 8
	-1305809037984406113653123425917082362518950072864121831298194147971584 λ 7
	-857459709928706636281627128531536748933260220065754221494033396979968 λ 6
	+ 82350349827169436218268442272600235900268467554762816633894717952000 λ 5
	+ λ 4 -536778108041875 λ 4 + 613580056937750 λ 3 -973299559128652607584349977483082614411646344627307408652288000 λ 3 + 83531381609725 λ 2 -833056632100 λ -282969148 , -66196938606096751635477616175451066002147406809598197760000 λ 2 λ = -6.71908029571401; -0.131562669586427; -0.000328857911389764; + 187891311363831342261122696760078459289425226752000000 λ
	0; 0.00964640672192552; 1.11775628056074; 30.7025165043502 . + 10640383921128326167258251725563452748800000000 ,
	• For l = 3, N := 52703192856349857947384370700745662529366485595703125,
	P (λ) = λ = -358.467163208234; -19.0072656658816; -0.646157263047298; 1 λ 52703192856349857947384370700745662529366485595703125 λ 9 -0.0111599266641292; -0.0000666533778004677; N -5.55444566928419 • 10 -8 ; 0; 0.00000278259567776007; -8300609792668264459482464091106643611433296496582031250 λ 8 0.00101759218979804; 0.0943002426171891; -445494630869450831025054635527711314147490581862060546875 λ 7 + 4794778996142210900232623988837260417742766309970898437500 λ 3.74532560247919; 86.5431005254185; 1358.60267157073 .
	+ 9756465613003402089003884988756390021440062708106251171875 λ Observation 5.3.8. For any nonnegative integer l, if letting
			-3017384501505283273071518606155848657382720371906916875000 λ -110136365938455321211992245505985628913509930246874312500 λ 3 6 κ = 6l + 5
			+ 339407505728887685551143517950660327903074561069962500 λ 2
			+ 55789600918061271349143165830214064792716089155560 λ
			-231366636087183602796285517158453376756432736 ,
		λ = -49.3913171271021; -1.99073754558809;
			-0.0356150192571145; -0.000160321576336723;
			0; 0.00000404759002436705; 0.00300504302403685;
			0.302505715828807; 10.7402583250749; 197.869342014100 .

  Therefore, the process X t (B.0.8) is,

	in law,
	X s	(law) = X t Xs-t , ∀s ≥ t,	(B.0.16)

2.3. SLE TWO-POINT FUNCTION

Remerciements

Therefore, β(p, f ) has the following development at p = 0: From this, the proof of Theorem 4.1.1 is twofold:

• Using a martingale technique, we derive a PDE satised by E(| log f (z)| 2 ) for κ = 2. • From that PDE together with the support of MAPLE, we determine the coecient matrix of E(| log f (z)| 2 ) whereby we can nd the explicit expression for E(| log f (z)| 2 ) and show that Eq. (4.1.7) is true for κ = 2.

SLE logarithm one-point function

Let us now consider the whole-plane SLE logarithm one-point function,

which, by construction, stay nite at the origin and such that F (0) = 0. Our aim is to derive an ODE satised by F (4.2.1). We consider the auxiliary, time-dependent, radial variant of the function F (z)

above,

where ft is a modied radial SLE map at time t as in Denition 2.1.3.

In addition, in this case, note that λ = 0 is a simple root of the characteristic polynomial P (λ) of the coecient matrix (d n,m ) n,m=0,1,...,3l+1 . Using MAPLE, we obtain approximate values of eigenvalues λ. For instance:

• For l = 0, P (λ) = λ [λ -2] , λ = 0; 2 .

• For l = 1,

λ 500 λ 4 -4000 λ 3 -7100 λ 2 + 1200 λ + 9 , λ = -1.62742614692069; -0.00719495255354571; 0; 0.162287966346161; 9.47233313312807 .

• For l = 2, setting N := 180470690631, , p = 0 , q = -2 + 9l(l + 1) 2(6l + 1) , then, the corresponding coecient matrix (d n,m ) is a square matrix of order (3l + 2), and we obtain

Appendix A

The generalized hypergeometric function, Hausdor dimension and Minkowski dimension

A.1 Generalized hypergeometric function

Let n = [n 1 , n 2 , ...] (list of upper parameters, may be empty), p = nops(n) (the number of elements in list n), d = [d 1 , d 2 , ...] (list of lower parameters, may be empty), q = nops(d) (the number of elements in list d) and complex number z. The hypergeom(n, d, z) calling sequence is the generalized hypergeometric function F (n, d, z). This function is frequently denoted by pF q(n, d, z).

Formally, F (n, d, z) is dened by the series

where the pochhammer symbol is dened for the positive integer n and complex number z as

with pochhammer(z, 0) = 1.

One notes that:

• If some n i is a non-positive integer, the series is nite (that is, F (n, d, z) is a polynomial in z). If some d j is a non-positive integer, the function is undened for all non-zero z, unless there is also a negative upper parameter of smaller absolute value, in which case the previous rule applies.

• For the remainder of this description, assume that no n i or d j is a non-positive integer. When p ≤ q, this series converges for all complex z, and hence denes Appendix B

The proof of Theorem 1.2.1

This section will present the proof of Theorem 1.2.1 in Section 1.2. This is the SLE κ case of Theorem 3.1 in [6], with the Lévy process L t := √ κB t .

Proof. [6] Let us rst recall that

By expanding both sides of Loewner's equation (0.0.1) as power series, and identifying coecients, leads one to the set of equations

where a 1 = 1; the dot means a t-derivative, and λ(t

The rst dierential equation (B.0.2) (together with the uniform bound, ∀t ≥ 0,

where X t is dened as

The recursion (B.0.7) can be rewritten under the simpler form:

(B.0.9)

Recall that u 1 = a 1 = 1, while the next term of this recursion, as already seen in Eq. (B.0.4), is

Similarly, we can write the general solution u n , for n ≥ 2, under the form

with v 2 (s) = 1, and rewrite the dierential equation (B.0.9) as an integral equation:

(B.0.12)

Dene then the multiplicative and integral operators X and J such that

The solutions to (B.0.10), (B.0.11) and (B.0.12) can then be written as the operator product

•(X + (k + 1)J )1. = B t + Bs-t , where Bs is an independent copy of Thus,

The proof of Theorem 1.2.1 is completed.

Appendix C

Matlab Code for the logarithmic coecient problem

In this section, we will present Matlab code to calculate expectation of coefcients' absolute values squared of SLE process. In the rst part, we construct structure of a "term" with B contains values of (α, β) in integrals (1. Coes_Gzz := proc(k, n, m, p, q) function computes the coecients of G(z, z) := E (|f (z)| p /|f (z)/z| q ) option remember;

end if; end proc;

Function Coes_Gzξ for the coecients of G(z, ξ)

Respective domains of validity of integral means spectra β tip (p), β 0 (p), β lin (p), and β 1 (p, q). The thin straight line (coral) q = 2p corresponds to the version of whole-plane SLE studied in Ref. [2]. It does not intersect the lower domain where β 1 holds. . . . . . . . . . . . . . . .

3.6

Various geometrical elements appearing in the proof of Theorem 3.4.2, when establishing the respective domains of validity of integral means spectra β tip (p), β 0 (p), β lin (p), and β 1 (p, q). The dashed coral line D 2 corresponds to Eq. (3.4.35) and intersects D 0 at point P 2 . The dashed cyan line D 3 corresponds to Eq. (3.4.36), and intersects D 0 at Q 0 and the red parabola R at point P 3 . The q = 2p continuous straight line in coral , corresponding to the whole-plane SLE version studied in Ref. [2], does not intersect the blue quartic, but intersects the green parabola at a point of abscissa (3.4.33). (In the particular κ = 2 case shown here, the intersection point P 3 of D 3 with R coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.) . . . . . .

3.7

The coral parabola R 1 (3.4.37) and the red parabola R (3.4.19) have the same intersection point P 3 with the straight line D 3 (3.4.36). The innite branch of R 1 below P 3 (thin coral line), with parametric coordinates p C (γ), q C (γ) (3.4.37) for γ ∈ [1 + 2/κ, +∞). The innite branch of R below P 3 (thin red line), with parametric coordinates p R (γ), q R (γ) (3.4.19) for γ ∈ [1 + 2/κ, +∞). (In the particular κ = 2 case shown here, point P 3 coincides with the intersection point (p(2) = 2, q = 0) of R with the p-axis.) . . . . . . . . . . . . . . . . .

Thanh Binh LE

SUR LE PROBLÈME DE COEFFICIENT ET LA MULTIFRACTALITÉ DE WHOLE-PLANE SLE

Résumé :

Le point de départ de cette thèse est la conjecture de Bieberbach : sa démonstration par De Branges utilise deux ingrédients, à savoir la théorie de Loewner des domaines plans croissants et une inégalité de Milin qui concerne les coefficients logarithmiques. Nous commençons par étudier les coefficients logarithmiques du whole-plane SLE en utilisant une méthode combinatoire, assistée par ordinateur. Nous retrouvons les résultats en utilisant une équation au dérivées partielles analogue à celle obtenue par Beliaev et Smirnov. Nous généralisons ces résultats en définissant le spectre généralisé du whole-plane SLE, que nous calculons par la même méthode, à savoir en dérivant, par le calcul d'Itô, une EDP parabolique satisfaite par les quantités que nous moyennons. Cette famille à deux paramètres d'EDP admet une riche structure algébrique que nous étudions en détail. La dernière partie de la thèse concerne l'opérateur de Grunsky et ses généralisations. Plus expérimentale, nous y mettons à jour, grâce à un logiciel de calcul formel, une structure assez complexe dont nous avons commencé l'exploration.

Mots clés : Whole-plane SLE, moments logarithmiques, équation de Beliaev-Smirnov, spectre généralisé, variance asymptotique de McMullen, coefficients de Grunsky.

ON THE COEFFICIENT PROBLEM AND MULTIFRACTALITY OF WHOLE-PLANE SLE

Abstract :

The starting point of this thesis is Bieberbach's conjecture: its proof, given by De Branges, uses two ingredients, namely Loewner's theory of increasing plane domains and an inequality from Milin about the logarithmic coefficients. We start with a study of the logarithmic coefficients of the whole-plane SLE by using a combinatorial method, assisted by computer. We find the results by using a partial differential equation similar to that obtained by Beliaev and Smirnov. We generalize these results by defining the generalized spectrum of the whole-plane SLE, that we calculate by the same method, namely by deriving, thanks to Itô calculus, a parabolic PDE satisfied by the quantities of which we take the average. This two-parameter family of PDEs admits a rich algebraic structure that we study in detail. The last part of this thesis is about the Grunsky operator and its generalizations. In this part that is more experimental we update, thanks to a computer algebra system, a rather complex structure of which we began the exploration.