Bertrand Duplantier

Yong Han

AND Chi Nguyen

INSTITUT Michel Zinsmeister

Chi Ho

Minh

- Nology

Oanh Vo

WARD 25, BINH Thanh St

H O District

Minh Chi

Vietnam City

UNIVERSIT Denis Poisson

COMPLEX GENERALIZED INTEGRAL MEANS SPECTRUM OF DRIFTED WHOLE-PLANE SLE & LLE

We present new exact results for the complex generalized integral means spectrum (in the sense of [DHLZ18]) for two kinds of whole-plane Loewner evolutions driven by a Lévy process:

(1) The case of a Lévy process with continuous trajectories, which corresponds to Schramm-Loewner evolution SLE κ with a drift term in the Brownian driving function. There is no known result for its standard integral means spectrum, and we show that a natural path to access it goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity.

).

INTRODUCTION

"Il apparut que, entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe."[It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain] (Paul [START_REF] Painlevé | Analyse des travaux scientifiques[END_REF] [START_REF] Painlevé | Analyse des travaux scientifiques[END_REF].

More than two decades ago, Oded Schramm [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF] introduced his celebrated theory of random growth processes SLE κ . As an example, in the so-called chordal case in the halfplane H, it consists of the one-parameter family of Loewner processes driven on the real line ∂H by √ κB t , where κ is a nonnegative number and B t is standard one-dimensional Brownian motion. This is the unique family of random processes satisfying a certain Markov property with continuous driving function, that is symmetric with respect to the imaginary axis. This theory may be generalized along two directions:

(1) One can drop symmetry with respect to the imaginary axis: one then considers SLE κ with a drift term, e.g., the chordal Loewner process driven by a random function of the form

λ(t) = √ κB t + at,
where B t is as before standard one-dimensional Brownian motion and a ∈ R.

(2) One can drop the continuity assumption while keeping symmetry: the process so obtained is Loewner evolution driven by a Lévy process, called LLE (for Lévy-Loewner evolution).

Notice that the first class of continuous drifted processes coincides with the whole class of LLE processes with continuous trajectories. For κ = 0, the Loewner process generated by λ(t) = at becomes deterministic. Several such deterministic chordal Loewner processes were investigated in [KNK04, [START_REF] Marshall | The Loewner differential equation and slit mappings[END_REF][START_REF] Lind | A sharp condition for the Loewner equation to generate slits[END_REF][START_REF] Lind | Collisions and spirals of Loewner traces[END_REF].

In this paper, we shall consider both extended classes in the whole-plane case. In order to understand the multifractal spectra of these processes, such as their integral means spectra (ims), and in the spirit of references [DNNZ15, DHLZ18, Lou12, LY13, LY14, LY19], we shall first investigate the cases for which the expected complex moments, E |f (z) p | z f (z) q , p, q ∈ C, may be computed explicitly, to become part of integrable probability. Here f stands for the time 0 whole-plane map from D to the slit plane in the corresponding Loewner process. Note that complex values of (p, q) are considered here in the case of whole-plane SLE with drift. In agreement with the citation by P. Painlevé above, the suggested passage by the complex plane will help us discover the precise form of the associated integral means spectrum in the case of SLE κ with drift, via its complex and generalized versions [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. We shall make use of the so-called Liouville quantum gravity, in the spirit of [START_REF] Duplantier | Conformally invariant fractals and potential theory[END_REF][START_REF] Duplantier | Harmonic Measure and Winding of Conformally Invariant Curves[END_REF][START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF][START_REF] Duplantier | Liouville Quantum Gravity as a Mating of Trees[END_REF]. For the generalized spectrum of LLE processes, we shall concentrate on (p = 2, q ∈ R) cases. Precise definitions are given in the next sections.

1.1. Interior whole-plane SLE. SLE is a particular case of a growth process called the Loewner process, of which several variants exist, known as chordal, radial, dipolar, or whole-plane [START_REF] Gregory | Conformally invariant processes in the plane[END_REF][START_REF] Beliaev | Conformal Maps and Geometry[END_REF]. In this work we will consider the interior whole-plane case, which is determined by a driving function λ : [0, +∞) → ∂D := {z ∈ C : |z| = 1} obtained as follows. Let us start by defining γ : [0, +∞) → C to be a continuous function such that lim t→+∞ |γ(t)| = +∞ and γ(t) = 0, ∀t ≥ 0. Then, for each t > 0, the slit domain Ω t = C\γ([t, ∞)) is a simply connected domain containing 0. By the Riemann Mapping Theorem, there exists a unique conformal map f t : D → Ω t such that f t (0) = 0 and f t (0) > 0. By the Caratheodory convergence theorem, f t converges to f 0 , the Riemann mapping of Ω 0 , as t → 0. We may assume without loss of generality that f 0 (0) = 1 and, by re-parametrizing the curve if necessary, choose the normalization f t (0) = e t . Loewner's theorem asserts that there exists a continuous function λ taking values in the unit circle such that ∂ ∂t f t (z) = z ∂ ∂z f t (z) λ(t) + z λ(t)z , lim t→+∞ f t (e -t z) = z, ∀z ∈ D.

(1.1)

The Loewner method can be reversed: given a continuous function λ : [0, +∞) → ∂D, the partial differential equation (1.1) has a unique solution f t (z), which is a conformal map from D onto a domain Ω t , and the corresponding family (Ω t ) t is increasing in t.

Nevertheless the domains Ω t need not be slit domains as in the example above. Whole-plane SLE κ is the process driven by

λ(t) = e i √ κBt ,
where κ ∈ [0, +∞) and B t is standard one-dimensional Brownian motion. Note that when κ = 0, f t (z) = e t z (1-z) 2 is the solution to (1.1), so that f 0 is the Koebe function. Thus, as κ → 0 + , whole-plane SLE κ may be seen as a stochastic perturbation of the Koebe map.

In this work, we generalize SLE by adding a drift term to Brownian motion, with a driving function defined as λ(t) := e i( √ κBt+at) , a ∈ R.

(1.

2)

The process driven by λ(t) then appears for small κ as a stochastic perturbation of the (κ = 0, a = 0) case of the logarithmic spiral.

1.2. Complex generalized integral means spectrum. Let f be a conformal map from D to C with f (0) = 0, f (0) = 1. The generalized integral means spectrum of f was originally defined in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF] as follows: for any pair of real numbers (p, q), define the integral moments, for r ∈ [0, 1), M f (p, q) := 2π 0 r q |f (re iθ )| p |f (re iθ )| q dθ, r ∈ [0, 1).

(1.

3)

The generalized integral means spectrum is then defined as β f (p, q) := lim sup r→1 - log M f (p, q)/ log (1r) -1 .

If the limit exists, then

M f (p, q) • ∼ (1 -r) -β f (p,q) , (1.4)
where the notation '

• ∼' between two quantities stands for the equivalence of the logarithms of these quantities [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF].

One recovers for q = 0 the standard integral means spectrum, β f (p) := β f (p, q = 0), which is related by various Legendre transformations to the so-called multifractal spectra [Man74, HP83, FP85, HJK + 86a, HJK + 86b], like those governing the moments of the harmonic measure or the continuum of its local singularities [START_REF] Makarov | Fine structure of harmonic measure[END_REF][START_REF] Garnett | of New Mathematical Monographs[END_REF].

For a random simply connected domain as arising from a whole-plane Loewner process with a random driving function like SLE, the question whether the equivalence (1.4) holds almost surely is notoriously difficult. Earlier works dealt with the 'expected spectrum' for Brownian motion [START_REF] Lawler | Intersection exponents for planar Brownian motion[END_REF][START_REF] Duplantier | Two-Dimensional Copolymers and Exact Conformal Multifractality[END_REF], self-avoiding walk [START_REF] Duplantier | Two-Dimensional Copolymers and Exact Conformal Multifractality[END_REF], percolation [START_REF] Duplantier | Harmonic Measure Exponents for Two-Dimensional Percolation[END_REF][START_REF] Adams | Harmonic Measure for Percolation and Ising Clusters Including Rare Events[END_REF], and SLE [Dup00, Dup03, Has02, Dup04, Dup06, BRGW05, RBGW07, BS09, DNNZ15, BDZ17] as well as with the expected generalized spectrum of whole-plane SLE [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. The almost sure case was solved only recently for the standard spectrum of chordal SLE by Gwynne, Miller and Sun [START_REF] Gwynne | Almost sure multifractal spectrum of Schramm-Loewner evolution[END_REF] by using the so-called imaginary geometry of Miller and Sheffield. (See also the earlier works [START_REF] Viklund | Almost sure multifractal spectrum for the tip of an SLE curve[END_REF] for the SLE a.s. tip spectrum and [START_REF] Alberts | A dimension spectrum for SLE boundary collisions[END_REF] for the SLE a.s. boundary spectrum.)

The case of complex moments corresponds to the mixed multifractal spectrum of the harmonic measure and logarithmic rotations of the conformal map [START_REF] Binder | Rotational Spectrum of Planar Domains[END_REF]. It was studied in expectation in Refs. [START_REF] Duplantier | Harmonic Measure and Winding of Conformally Invariant Curves[END_REF][START_REF] Duplantier | Harmonic measure and winding of random conformal paths: A Coulomb gas perspective[END_REF][START_REF] Belikov | Statistics of harmonic measure and winding of critical curves from conformal field theory[END_REF] for the chordal and radial SLE cases. We shall consider here the whole-plane spectrum defined in expectation for complex moments,

r∂D E |f (z) p | z f (z) q |dz| • ∼ 1 1 -r β(p,q)
, p, q ∈ C.

(1.5)

It is then natural to introduce the one-point function

G(z) := E |f (z) p | z f (z) q , p, q ∈ C.
(1.6)

The setting chosen in (1.5) and (1.6) allows for complex values p, q ∈ C, which we shall need to study the drift case. In the more general case of Lévy processes, we shall see that their defining properties are exactly those needed to obtain a PDE satisfied by

, ζ ∈ C \ D → f (ζ) := 1/f (1/ζ),
is just the exterior whole-plane map from C \ D to the slit plane considered in Ref. [START_REF] Beliaev | Harmonic measure and SLE[END_REF] by Beliaev and Smirnov and in Ref. [START_REF] Beliaev | Integral means spectrum of wholeplane SLE[END_REF]. We identically have for 0 < r < 1 and p ∈ R,

r -1 ∂D E | f (ζ)| p |dζ| = r 2p-2 r∂D E |f (z)| p |f (z)| 2p |dz|. (1.7)
We thus see that the standard integral mean of order (p, q = 0) for the exterior whole-plane map studied in [START_REF] Beliaev | Harmonic measure and SLE[END_REF][START_REF] Beliaev | Integral means spectrum of wholeplane SLE[END_REF] coincides (up to an irrelevant power of r) with the (p, q ) integral mean for q = 2p, for the interior whole-plane map.

Remark 1.1. Interior-Exterior Duality. By conformal inversion, we have for any p, q ∈ C,

r -1 ∂D E f (ζ) p f (ζ) q |dζ| = r 2 p-2 r∂D E f (z) p f (z) 2p-q |dz|,
(1.8) so that the (p, q) exterior integral means spectrum coincides with the (p, q ) interior integral means spectrum for qp = pq. In particular, the (p ∈ R, q = 0) interior derivative moments studied in Ref. [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF] correspond to the (p, q = 2p) mixed moments of the exterior map.

1.4. Generalized spectrum for the logarithmic spiral. In this section we give an example of a generalized integral means spectrum, which is deterministic and corresponds to the κ = 0 case of the drifted SLE κ . It is nothing but the logarithmic spiral with parameter a ∈ R (Fig. 1), i.e., the curve parametrized by

γ(t) = exp [(1 + ia)t] , t ∈ R.
(1.9) FIGURE 1. Logarithmic spiral γ(t) (1.9) for a = +5, restricted to t ≥ 0.

1.4.1. Loewner process for the logarithmic spiral. Let us define, as before, Ω t := C \ γ[t, +∞) and let f t : D → Ω t be the associated Riemann map, i.e., the conformal map such that f t (0) = 0, f t (0) > 0.

(1.10) By the Koebe distorsion theorem, lim t→-∞ f t (0) = 0 and lim t→+∞ f t (0) = +∞. Then, there exists t 0 such that f t 0 (0) = 1. One also has that f t 0 (e iθ 0 ) = γ(t 0 ) = e (1+ia)t 0 for some θ 0 ∈ [0, 2π). Consider now the function ft defined by ft (z) := e (1+ia)t f t 0 (e -iat z).

We have ft (0) = 0, f t (0) = e t , and ft (e i(θ 0 +at) ) = e (1+ia)(t+t 0 ) = γ(t + t 0 ).

(1.11)

Hence from (1.11), ft : D → Ω t+t 0 is the Loewner process corresponding to the curve

γ(t) := γ(t + t 0 ), t ∈ R,
with the associated driving function λ(t) := e i(θ 0 +at) . Define then the curve, η(t) := e -iθ 0 γ(t), t ∈ R, and the conformal map, h t (z) := e -iθ 0 ft (e iθ 0 z).

One still has h t (0) = 0, h t (0) = e t , and h t (e iat ) = η(t), so that h t is the Loewner map corresponding to η(t) and the associated process is driven by λ(t) := e iat . Notice that the curve η is obtained by a time-translation and a rotation of the logarithmic spiral γ. Thus the integral means spectrum will be the same for the time zero Loewner maps h 0 and f0 .

1.4.2. Complex generalized spectrum for the complete logarithmic spiral. We first focus on the complete spiral, for which we first establish the following theorem.

Theorem 1.1. The complex generalized integral means spectrum of the complete logarithmic spiral γ(t) = e (1+ia)t , t ∈ R, is given, for p, q ∈ C, by β(p, q; a) = sup {0, β 1 (p, q; 0, a), β 2 (p, q; 0, a)} , (1.12)

where

β 1 (p, q; 0, a) := 2 p -q 1 -ia + p -1, β 2 (p, q; 0, a) := -2 p -q 1 -ia + p -1.
(1.13)

Proof. Let us define on the unit disk D, the Moebius map ξ : z → ξ(z) := i 1-z 1+z , and consider the function Φ defined on D as,

Φ(z) := exp 2 1 -ia log ξ(z) = i 1 -z 1 + z 2(1+ia) 1+a 2 , z ∈ D.
Define also the strip domain S π := {x + iy : x ∈ R, 0 < y < π}. We know that z → ξ(z) conformally maps D onto upper half-plane H, while z → log(z) conformally maps H onto the strip S π . Lastly, z → exp 2 1-ia z conformally maps the strip domain S π onto C \ γ, with a cut along the whole logarithmic spiral γ := {γ(t) = e (1+ia)t , t ∈ R}. Consequently, Φ is a conformal map from the unit disk D to the complement of the whole logarithmic spiral γ, with Φ(1) = 0, Φ(-1) = ∞.

It enjoys the useful property,

Φ (z) = 2 1 -ia log ξ(z) Φ(z) = - 4 1 -ia Φ(z) 1 -z 2 .
(1.14)

Owing to (1.14),the complex mixed moments of Φ read

Φ (z) p Φ(z) q = - 4 1 -ia 1 1 -z 2 p 1 Φ(z) q-p , (1.15) so that Φ (z) p Φ(z) q = - 4 1 -ia p × |Φ(z) p-q | |(1 -z 2 ) p | . (1.16)
We have explicitly

(1 -z 2 ) p = |1 -z| p |1 + z| p e -p arg(1-z 2 ) , (1.17)
and

Φ(z) p-q = exp 2(p -q) 1 -ia log ξ(z) . (1.18) Setting b = b(p, q) := 2(p-q) 1-ia , we have [b log ξ(z)] = b log |ξ(z)| -b log ξ(z)
, and since log ξ(z) ∈ S π , its imaginary part stays bounded. We thus have the following (logarithmic) equivalence near the two possible singular points z = ±1,

Φ(z) p-q • ∼ |ξ(z)| b(p,q) = 1 -z 1 + z 2(p-q) 1-ia . (1.19)
Using (1.16), (1.17), and (1.19), we finally arrive at

Φ (z) p Φ(z) q • ∼ 1 -z 1 + z 2(p-q) 1-ia |1 -z| -p |1 + z| -p .
(1.20)

Behaviour near infinity and near the origin. For z = re iθ near z = -1 (point at ∞ on the spiral), |1+z| 2 = r 2 +2r cos θ +1 behaves like (1-r) 2 +(π -θ) 2 . Similarly, near z = +1 (point 0 on the spiral), |1 -z| 2 behaves like (1r) 2 + θ 2 . The integral of (1.20) along the circle |z| = r for r → 1 -is thus dominated near z = -1 by the contribution of the angular neighbourhood of θ = π, while near z = +1 it is symmetrically dominated by that of the angular neighbourhood of θ = 0. From the explicit form of the integrand (1.20), we readily obtain the overall asymptotic behaviour as r → 1 -of the integral means,

r∂D Φ (z) p Φ(z) q |dz| • ∼ (1 -r) -β Φ (p,q;a) , r → 1 -, (1.21)
where the integral means spectrum β Φ is given by the largest exponent,

β Φ (p, q; a) := β 1 (p, q; 0, a) ∨ β 2 (p, q; 0, a) ∨ 0, (1.22)
with the two dual spectra defined as,

β 1 (p, q; 0, a) := 2 p -q 1 -ia + p -1, (1.23) β 2 (p, q; 0, a) := 2 q -p 1 -ia + p -1.
(1.24)

Remark 1.2. Singularity localization. Exponent β 1 is associated with the singularity near z = -1 on D in (1.20), i.e., at infinity on the spiral, while β 2 corresponds to that near z = +1, i.e., near the tip at origin 0, around which the spiral indefinitely winds.

Remark 1.3. Conformal invariance by inversion and duality. The full logarithmic spiral is conformally invariant under the complex inversion, z → 1/z, since 1/γ(t) = γ(-t), and t ∈ R. This inversion exchanges the roles of origin and infinity, and maps the interior of D to its exterior. The complex generalized integral means spectrum then obeys the duality property (1.1). Spectra (1.23) and (1.24) are indeed dual of each other under the corresponding exchange qp → pq, resulting in the expected invariance under duality of the integral means spectrum β Φ (1.22) for the complete logarithmic spiral.

1.4.3. Complex generalized spectrum of the half spiral. Consider now h 0 (z), the conformal map corresponding to the whole-plane Loewner process driven by e iat , stopped at time t = 0, the image of which, γ(t) = e (1+ia)t , t ≥ 0, we may call the half spiral (Fig. 1). The complex generalized integral means spectrum of the half logarithmic spiral is given by the following theorem. (See Fig. 2.)

Theorem 1.2. The complex generalized integral means spectrum of h 0 , where h t is the whole-plane Loewner process driven by λ(t) = e iat , and whose trace is the half logarithmic spiral γ(t) = e (1+ia)t , t ≥ 0, is given, for p, q ∈ C, by

β(p, q; κ = 0, a) = sup -p -1, 0, β 1 (p, q; 0, a) = 2 p -q 1 -ia + p -1 . (1.25)
From this, one immediately deduces the following corollary, which yields the real generalized integral means spectrum of the half spiral.

Corollary 1.1. The real generalized integral means spectrum of h 0 , where h t is the wholeplane Loewner process driven by λ(t) = e iat , and whose trace is the half logarithmic spiral γ(t) = e (1+ia)t , t ≥ 0, is given, for p, q ∈ R, by β(p, q; κ = 0, a) = sup -p -1, 0, β 1 (p, q; 0, a) = 2 pq 1 + a 2 + p -1 .

(1.26)

This result for the real case, p, q ∈ R, is illustrated in Fig. 2.

Proof. • Behaviour near infinity. For t ≥ 0, the half spiral and whole spiral are identical, thus have the same spectrum near infinity. So we use the conformal map Φ to calculate the integral means spectrum near ∞, i.e., by considering the mixed moments (1.20) for z → -1 only, as well as the corresponding contribution to integral (1.21). Because of Remark 1.2, the associated spectrum is β 1 (1.23).

• Behaviour near the tip. Let φ(z)

:= z (1-z) 2 , z ∈ D, with φ(-1) = -1 4 , φ(0) = 0, φ(1) = ∞,
:= C \ γ[0, ∞), with g(0) = 0, g(-1 4 ) = γ(0) = 1. Then h 0 = g • φ.
Notice that both g and g are bounded near φ(-1) = -1 4 , hence also h 0 near z = -1. Let us define r∂D ε := {z : |z| = r, |1 + z| < ε}, for some fixed ε such that 1r < ε < 1, as the neighbourhood along the circle r∂D of the pre-image z = -1 by h 0 of the half spiral tip γ(0) = 1. In this domain, we have the logarithmic equivalence, as r → 1 -,

r∂Dε h 0 (z) p h 0 (z) q |dz| • ∼ r∂Dε |φ (z) p ||dz|, r → 1 -. p q 0 -p-1 2 1+a 2 (p-q)+ p-1 FIGURE 2.
The three phases of the generalized integral means spectrum of the logarithmic spiral, with β tip (p; κ = 0) = -p -1, β 0 (p; κ = 0) = 0, β 1 (p, q; κ = 0, a) = 2 1+a 2 (pq) + p -1 (Corollary 1.1).

We thus obtain that the integral means spectrum near the tip of the half spiral is the same as the ims near the tip of the half line, which is simply,

β tip (p; κ = 0) := -p -1.
(1.27)

• Bulk behaviour. Away from ∞ and the tip, the half spiral is rectifiable, and its bulk integral means spectrum is trivial, β 0 (p; κ = 0) = 0. This ends the proof of Theorem 1.2.

COMPLEX GENERALIZED SPECTRUM OF DRIFTED WHOLE-PLANE SLE

2.1. Introduction. In this section, we will predict the exact form of the generalized integral means spectrum β 1 (p, q; κ, a) associated with the whole-plane SLE κ with drift a. As we shall see, its most symmetric and simplest form is obtained for the complex generalized spectrum where the exponents are complex variables p, q ∈ C. We shall use a non-fully rigorous method inherited from theoretical physics. More specifically, we use two-dimensional quantum gravity where the Euclidean Lebesgue measure is replaced by the Liouville quantum measure. This allows us to compute multifractal exponents in Liouville quantum gravity (LQG) for p ∈ C in the q = 0 case, and for any a ∈ R. The conversion to the complex multifractal spectrum in the Euclidean plane is then obtained by using the celebrated Knizhnik-Polyakov-Zamolodchikov (KPZ) relation [KPZ88, DK89, Dav88, DS09, DS11a, RV11, DS11b, DMS21]. The final step to get the complex generalized spectrum for q = 0 is then obtained via the introduction of the packing spectrum, s 1 (p, q; κ, a) := β 1 (p, q; κ, a)p + 1, (2.1) together with the fact that it is a function of variable pq only.

p tip 1 p q ( ) ,
( ) p 0 ( ) p lin q p Q 0 FIGURE 3.
Phase transition lines for the generalized integral means spectrum of whole-plane SLE κ with no drift a = 0. The standard ims of the q = 2p exterior version crosses phases I, II, III only, while the q = 0 standard ims of the interior version crosses phases I, II, IV (from Ref.

[DHLZ18]).

2.2. Driftless case. Let us denote by β(p, q; κ, a) the generalized integral means spectrum of the whole-plane SLE κ with drift coefficient a. Ref. [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF] studied the a = 0 case, for which it is shown that β(p, q; κ, a = 0) has four possible forms, of which three are independent of q,

β tip (p; κ) := -p -1 + 1 4 4 + κ -(4 + κ) 2 -8κp , (2.2) β 0 (p; κ) := -p + 4 + κ 4κ 4 + κ -(4 + κ) 2 -8κp , (2.3)
β lin (p; κ) := p - (4 + κ) 2 16κ , (2.4
)

β 1 (p, q; κ, a = 0) := p + 2(p -q) - 1 2 - 1 2 1 + 2κ(p -q).
(2.5)

The separatrices between the different phases are located as follows [DHLZ18, Theorem 1.7] (See Fig. 3.) For p ≤ -1 -3κ 8 there is a (quartic) curve ending at point Q 0 : p 0 = -1 -3κ 8 , q 0 = -2 -7κ 8 , that separates the half-plane into two parts, β being equal to β tip above that curve and to β 1 below it. In the strip -1 -3κ 8 ≤ p ≤ 3(4+κ) 2 32κ , there is a section of parabola joining Q 0 to point P 0 = (p 0 , q 0 ), with

p 0 = 3(4 + κ) 2 32κ , q 0 = (4 + κ)(8 + κ) 16κ , (2.6)
that separates the strip into two parts, an upper one where β = β 0 and a lower one where β = β 1 . Finally the half-plane p ≥ p 0 is similarly split by the half-line with unit slope starting at P 0 into an upper part where β = β lin , while β = β 1 in the lower part. It should be noticed that the generalized spectrum β is not everywhere the maximum of the four spectra listed above [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. The existence of these phase transition lines was established in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF] within a connected semi-infinite domain of the (p, q) plane, as indicated in Fig. 4. This domain of

0 D 1 P 0 P 2 P 3 D 2 D' 0 p q D Q 0 FIGURE 4
. Domains of validity of the proofs in the driftless case a = 0, κ = 8/3 (from Ref. [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]). validity sweeps the plane from its upper-left part up to a piecewise boundary first made, for increasing values of p, of the dotted green parabola up to its intersection with the straight line D 2 of equation pq = 1 + κ 2 . It then follows this line up to its intersection

P 3 = 1 + 2 κ , 4-κ 2 2κ
with the red parabola. From there, the boundary is made of the section of red parabola up to point P 0 (2.6), followed by the straight line D 1 of equation qp = 16-κ 2 32κ . These restrictions to the domain of proof are due to technicalities involved in the proofs [DNNZ15, BDZ17, DHLZ18], and the spectrum is supposed to be still given by β 1 in the whole connected domain located to the right of the piecewise boundary just described. Recent work by Xuan Hieu Ho extends the domain of validity to the whole interior of the red parabola [START_REF] Hieu | Generalized integral means spectrum of SLE[END_REF].

Drift case.

Claim 2.1. For p, q ∈ C, and a = 0, the complex spectrum of whole-plane SLE can be obtained by combining Liouville quantum gravity and Coulomb gas methods. It is

β 1 (p, q; κ, a = 0) = s 1 (p -q; κ, a = 0) + p -1 (2.7) s 1 (p -q; κ, a = 0) = s 1 (τ ) := 2τ + 1 2 - 1 2 √ 1 + 2κτ , (2.8
)

1 + 2κτ := 1 2 {1 + 2κ (p -q) + |1 + 2κ(p -q)|} .
(2.9)

Claim 2.2. For a = 0, the complex spectrum of whole-plane SLE with drift is given by an extension of the above proofs, as

β 1 (p, q; κ, a) = s 1 (p -q; κ, a) + p -1 (2.10) s 1 (p -q; κ, a) = s 1 (τ ) := 2τ + 1 2 - 1 2 √ 1 + 2κτ , (2.11) 1 + 2κτ := 1 2 (1 + ia) 2 + 2κ(p -q) + (1 + ia) 2 + 2κ(p -q) .
(2.12)

Remark 2.1. In the limit κ → 0, the integral means spectrum (1.23) of the half-spiral is recovered from (2.10) (2.11), by observing that the expansion to order O(κ) of the r.h.s. of (2.12) indeed yields τ = p-q 1-ia . As we shall see in Section 3.4, this complex spectrum yields the correct answer along an integrable complex parabola in the complex space (p, q) ∈ C 2 .

In the real moment case, (p, q) ∈ R 2 , the generalized integral means spectrum β 1 (p, q; κ, a) associated with whole-plane SLE κ with drift a is given by the explicit formulae:

β 1 (p, q; κ, a) = p + 2τ - 1 2 - 1 2 √ 1 + 2κτ , (2.13) 1 + 2κτ := 1 2 1 -a 2 + 2κ(p -q) + [1 -a 2 + 2κ(p -q)] 2 + 4a 2 . (2.14)
Corollary 2.1. Eq. (2.14) can be inverted into:

p -q = τ 1 + a 2 1 + 2κτ . (2.15)
Therefore the phase transition lines in the (p, pq) plane for a = 0 are obtained from those for a = 0 by the non-linear transform,

p → p, p -q = τ → p -q = τ 1 + a 2 1 + 2κτ . (2.16)
In the work [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF], the location of the various phase transition lines in the case of whole-plane SLE without drift was established with the help of several master curves: a so-called 'red parabola' where the one-point function G (1.6) is integrable, a so-called 'green parabola' where the spectrum changes from β 0 to β 1 , and a 'blue quartic' where it changes from β tip to β 1 , as well as several straight lines, like D 0 where the spectrum changes from β tip to β 0 , D 0 where it changes from β 0 to β lin , and D 1 where it changes from β lin to β 1 (Fig. 3). These curves are also instrumental in delimiting the domains of validity of the proofs (Fig. 4). Applying the non-linear transform (2.16) in the (p, qp) plane to these curves yields the corresponding curves in the case of whole-plane SLE with drift. They are illustrated in Figs. 5 and6.

q-p P 0 D 0 D 1 D 0 Q 0 β tip (p) β 0 (p) β lin (p) β 1 (p, q; a) I II III IV FIGURE 7
. Phase transition lines for the generalized integral means spectrum of simple whole-plane SLE κ with drift (here κ = 2, a = 1). The first bisector with q = 2p (orange continuous line) corresponds to the standard integral means spectrum (ims) for the exterior case which crosses only phases I, II and III, whereas the second bisector with q = 0 (orange dotted line) yields the standard ims for the interior case, which does enter phase IV with the β 1 spectrum.

Phase diagram.

Various cases, relative to the values of parameters κ and a, and drawn thanks to the non-linear mapping (2.16), are depicted in Figs. 7, 8, 9, and 10. In these figures, it is especially interesting to focus on the standard integral means spectra in the p, qp plane, obtained for the whole-plane exterior version, along the line q = 2p, hence qp = p (first bisector, golden continuous line), and for the whole-plane interior version along the line q = 0, hence qp = -p (second bisector, golden dotted line). Point P 0 (2.6) in the drift-less case yields a value of t 0 := p 0q 0 = κ 2 -16 32κ , so that 1 + 2κt 0 = κ 2 /2. The position of the translated point P 0 = (p 0 , q0 ) in the presence of drift a is given by Eq. (2.16) as

q0 = p 0 + 16 -κ 2 32κ 1 + 16a 2 κ 2 .
(2.17)

p q-p P 0 D 0 D 1 D 0 Q 0 β tip (p) β 0 (p) β lin (p) β 1 (p, q; a) I II III IV FIGURE 8
. Phase transition lines for the generalized integral means spectrum of simple whole-plane SLE κ with drift (here κ = 2, a = 2). The standard ims in the q = 2p exterior case crosses all four phases in the order I, II, IV and III, while the q = 0 standard ims in the interior case crosses phases I, II and IV. To determine whether the first bisector enters region IV as in Fig. 8, so that the exterior standard whole-plane spectrum has a β 1 component, or avoids it as in Fig. 7, we need to know the sign of q0 -2p 0 . If positive, the first bisector passes below P 0 so that it successively traverses regions I, II, IV and III as in Fig. 8 The standard ims in the q = 2p exterior case crosses phases I, II, III only, while the q = 0 standard ims in the interior case crosses all four phases in the order I, II, III and IV.

bisector, and the β 1 spectrum does not appear in the standard ims of the exterior wholeplane SLE with drift.

To determine whether the second bisector enters region III and crosses all four phases as in Fig. 10, so that the interior standard whole-plane spectrum has a linear component β lin , or whether it avoids the linear phase III as in Fig. 9, we need to know the position of P 0 with respect to that bisector, hence the sign of q0 . If negative, the second bisector passes above P 0 , so that it successively traverses regions I, II, III and IV as in Fig. 10

s 1 (p) = s 1 (p; κ) := 2p + 1 2 - 1 2 1 + 2κp. (2.21)
When seen as a function of p, it has for inverse in terms of s = s 1 ,

p = s 2 + κ 8 U -1 κ (s),
where we defined

U -1 κ (x) := 1 2κ κ -4 + (4 -κ) 2 + 16κx , U κ (x) := 1 4 x (κx + 4 -κ) , V κ (x) := U κ 1 2 x + 1 - κ 4 = 1 16κ κ 2 x 2 -(4 -κ) 2 .
Here U κ is the KPZ function of Liouville quantum gravity adapted to SLE κ , while V κ is an associated function that relates boundary scaling dimensions to bulk ones [START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF][START_REF] Duplantier | Conformal Random Geometry[END_REF].

Here we generalize methods introduced in [Dup00, DB02] and expounded in [START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF][START_REF] Duplantier | Conformal Random Geometry[END_REF], and use notations similar to those of [START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF], Section 8. For simplicity, we first implicitly assume SLE paths to be simple, i.e., with κ ≤ 4, since the quantum gravity composition rules differ for the simple and non-simple phases of SLE [START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF][START_REF] Duplantier | Conformal Random Geometry[END_REF]. Nevertheless, the results obtained also hold for κ > 4. One has the set of identities,

p = x 1 (s) -x 1 , (2.22) x 1 (s) := 2V κ U -1 κ (s) + U -1 κ (x 1 ) ,
(2.23)

x 1 := x 1 (0) = 1 8κ (6 -κ)(2 -κ), (2.24) x1 := 6 -κ 2κ , U -1 κ (x 1 ) = 2 κ .
(2.25)

The scaling exponent x 1 (s) geometrically corresponds to a configuration where the SLE tip is locally avoiding a bunch of s independent Brownian paths. The tip here should be understood as the so-called SLE 'second tip' at the origin [START_REF] Beliaev | Integral means spectrum of wholeplane SLE[END_REF], after inversion of unbounded (interior) whole-plane SLE [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF], as in Beliaev and Smirnov's bounded (exterior) version of whole-plane SLE [START_REF] Beliaev | Harmonic measure and SLE[END_REF][START_REF] Beliaev | Integral means spectrum of wholeplane SLE[END_REF].

In the LQG approach, s independent Brownian paths avoiding an SLE path near its tip are conformally equivalent to a certain number k(s) of mutually-avoiding SLEs in a star configuration, given by

k(s) = 1 + U -1 κ (s) U -1 κ (x 1 ) , (2.26) such that x 1 (s) = 2V κ (2k(s)/κ). When p ∈ C,
x1 (s, t) := x 1 (s) - κ 2 t 2 k 2 (s) .
(2.27)

The average logarithmic spiral rotation rate a near the tip is then obtained by Legendre transformation as [START_REF] Duplantier | Harmonic Measure and Winding of Conformally Invariant Curves[END_REF][START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF],

a = ∂ ∂ t x1 (s, t).
(2.28)

On the other hand, the real part of p, t := p, is now given by the generalization of (2.22),

p = t = x1 (s, t) -x 1 , (2.29)
whereas the packing spectrum for complex p, s = s 1 (p; κ) = β 1 (p; κ)p + 1, is still given by (2.21), but now in terms of the reduced variable τ ,

τ := x 1 (s) -x 1 , (2.30)
s = s 1 (τ ) = 2τ + 1 2 - 1 2 √ 1 + 2κτ . ( 2
1 2κ k 2 (s) = x 1 (s) + b, b = (4 -κ) 2 8κ .
(2.32)

We thus find for (2.27) the simple formula,

x1 (s, t) = x 1 (s) - 1 4 t2 x 1 (s) + b , (2.33)
from which (2.29) gives,

t = x1 (s, t) -x 1 = x 1 (s) -x 1 - 1 4 t2 x 1 (s) -x 1 + c = τ - 1 4 t2 τ + c , c := b + x 1 = 1 2κ . (2.34) Eq. (2.34) is then inverted into τ = 1 2 t -c ± (t + c) 2 + t2 , (2.35)
which can be recast as

1 + 2κτ = 1 2 (1 + 2κt) ± 1 2 (1 + 2κt) 2 + 4κ 2 t2 .
(2.36)

For t = 0, we have τ = t, which selects the (+)-branch in (2.36), and recalling that t = p, t = p, we obtain

1 + 2κτ = 1 2 (1 + 2κ p) + 1 2 (1 + 2κ p) 2 + 4κ 2 p 2 , (2.37)
which is the announced complex formula (2.14) for p ∈ C, q = 0. When q = 0, we invoke the general validity of the observation made in Ref. [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF] that the generalized packing spectrum, s 1 (p, q; κ, a = 0) = β 1 (p, q; κ, a = 0)p + 1, solely depends on the reduced variable pq, hence s 1 (p, q; κ, 0) = s 1 (pq, 0; κ, 0).

2.4.3. Derivation of Claim 2.2. When a = 0, we modify the above aproach as follows. In the absence of Brownian paths, s = 0, (2.33) becomes, since

x 1 (0) = x 1 , x1 (0, t) = x 1 (0) - 1 4 t2 x 1 (0) + b = x 1 - κ 2 t2 .
(2.38)

The spiral rotation rate a then corresponds via (2.28) to a parameter t0 such that,

a = -κ t0 , x1 (0, t0 ) = x 1 - a 2 2κ .
(2.39)

Re-centering around the spiralling rate a, we define, instead of (2.33),

x 1 (s, t) := x 1 (s) - 1 4 ( t -t0 ) 2 x 1 (s) + b , (2.40)
and substitute to (2.29), (2.34)

t = x 1 (s, t) -x1 (0, t0 ) = x 1 (s) -x 1 + a 2 2κ - 1 4 ( t -t0 ) 2 x 1 (s) -x 1 + c = τ + a 2 2κ - 1 4 ( t -t0 ) 2 τ + c , c = 1 2κ , t0 = - a κ .
(2.41) Thus, instead of (2.35) we find

τ = 1 2   t -c - a 2 2κ ± t + c - a 2 2κ 2 + ( t -t0 ) 2   .
(2.42)

By again selecting the (+)-branch, and recalling that t = p, t = p, this can finally be written as

1 + 2κτ = 1 2 (1 -a 2 + 2κt) + 1 2 (1 -a 2 + 2κt) 2 + 4(a + κ t) 2 , = 1 2 (1 + ia) 2 + 2κp + (1 + ia) 2 + 2κp . (2.43)
This is the announced result (2.12) for p ∈ C, q = 0. Again, for q = 0, we invoke the fact [DHLZ18] that the generalized packing spectrum, s 1 (p, q; κ, a) = β 1 (p, q; κ, a)p + 1, solely depends on the reduced variable pq.

INTEGRABLE PROBABILITY FOR DRIFTED WHOLE-PLANE SLE

In order to anticipate the next section, let us put the computations in a more general setting.

3.1. Some background on Lévy processes. Definition 3.1. A Lévy process is a stochastic process (L t ) t≥0 such that

(1) L 0 = 0 (a.s);

(2) For any discrete ordered set {t i , i ∈ I n := {0, • • • , n}}, such that t 0 = 0 and 0 ≤ t i < t i+1 , ∀i ∈ I n-1 , the successive increments ,L t i+1 -L t i , i ∈ I n-1 , are all mutually independent; (3) For any 0 ≤ s ≤ t, L t -L s has the same law as L t-s . (4) L t is continuous in probability, lim t→0 P(|L t -L 0 | > ε) = 0, ∀ε > 0, which rules out fixed discontinuities of the path t → L t .

Notice that Brownian motion is a special Lévy process, and a general difference with Brownian motion is that random jumps are allowed. The characteristic function of a Lévy process L t has the form

E[e iξLt ] = e -tη(ξ) ,
(3.1) where η, called the Lévy symbol, is a continuous complex function of ξ ∈ R, satisfying η(0) = 0 and η(-ξ) = η(ξ). If η(-ξ) = η(ξ), L t is a symmetric Lévy process. For Brownian motion, the Lévy symbol is η(ξ) = ξ 2 2 . More generally, the function

η(ξ) = |ξ| α 2 , α ∈ (0, 2],
is the Lévy symbol of the so-called α-stable process.

3.2. Derivation of the PDE. The inner whole-plane Loewner process driven by the realvalued function L t is defined as the solution of the ODE in C [Law05],

∂ t g t (z) = g t (z) gt(z)+λ(t) gt(z)-λ(t) lim t→+∞ e t g t (z) = z, ∀z ∈ C, (3.2)
where λ(t) = e iLt , and where g t is (a priori) a mapping from C to D. Its inverse function f t := g -1 t obeys the PDE (1.1),

∂ t f t (z) = zf t (z) λ(t)+z λ(t)-z lim t→+∞ f t (e -t z) = z, ∀z ∈ D, (3.3)
where f t is now a mapping from D to the domain Ω t = C \ K t , where the connected set K t is the hull of the Loewner process. In this section, we will assume L t to be a Lévy process. The (complex) average integral means spectrum of the conformal map f = f 0 , where f t is defined by (3.3), describes the singular behavior of the expectation,

E[|f (z) p |] = E f (z) p 2 f (z) p 2 , p ∈ C. (3.4)
Similarly to the method used in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF], we shall consider the Lévy-Loewner evolution (LLE) two-point function for z 1 , z 2 ∈ D, defined as,

G(z 1 , z 2 ) := 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   , p, q ∈ C. (3.5)
The moment (3.4) is the value G(z, z) at coinciding points z 1 = z 2 = z, for the case q = q = 0. Following essentially the same approach as was introduced in [BS09, BDZ17, DHLZ18], we aim at finding a partial differential equation satisfied by G.

Since f t obeys a PDE instead of an ODE, the use of Itô calculus is problematic. A way to overcome this difficulty [START_REF] Beliaev | Harmonic measure and SLE[END_REF] is to consider the ODE (3.2) for negative times, and then compare the reverse function g -t to the inverse g -1 t . The details are as follows. For any fixed s, define the auxiliary function g (s) t such that: g (s) t (z) = e -t z for t > s, while for t ≤ s, g (s) t is the solution to the differential equation (3.2) with the initial (continuity) condition g (s)

s (z) = e -s z,    ∂ t g (s) t (z) = g (s) t (z) g (s) t (z)+λ(t) g (s) t (z)-λ(t) g (s)
s (z) = e -s z.

(3.6) Lemma 3.1. With g t and g

(s)

t defined as above, we have, for any t ∈ R, lim s→+∞ g (s)

t (z) = g t (z).
This lemma is just the interior version of the following result by Lawler [START_REF] Gregory | Conformally invariant processes in the plane[END_REF] for the exterior whole-plane case.

Lemma 3.2. [Law05, Prop. 4.21] Let gt (z) be the solution of the differential equation,

   ∂ t gt (z) = gt (z) λ(t)+gt(z) λ(t)-gt(z) lim t→-∞ e t gt (z) = z, ∀z ∈ C\{0}. (3.7)
For any fixed s, define g(s) t (z) as:

g(s) t (z) = e -t z if t ≤ -s; for t ≥ -s, g (s)
t (z) is the solution of the above differential equation with initial value g(s)

-s (z) = e s z. Then lim s→+∞ g(s)

t (z) = gt (z).

In order to prove that Lemma 3.1 follows from 3.2, one applies complex inversion and time reversal so as to define, g(s) t (z) := 1/g (s)

-t (1/z) and λ(t) := 1/λ(-t), where g (s)

t (z) is defined by (3.6). Then g(s) t satisfies Lemma 3.2 and for s → +∞, it converges to the limit gt obeying (3.7). It then finally suffices to check that g t (z) := 1/g -t (1/z) satisfies (3.2).

We then define a reversed radial LLE, as the solution to the ODE in the unit disk D,

∂ t ft (z) = ft (z) ft (z) + λ(t) ft (z) -λ(t) , f0 (z) = z, ∀z ∈ D. ( 3
= f 0 (z).

(3.9)

Proof. For any fixed s > 0, let g (s) t be as above. Then we have g

(s) t (z) (law) = ft g (s)
0 (z) , because both obey (3.6), and they coincide at t = 0 because of the initial condition in (3.8). We then have, e t ft (z)

(law) = e t g (s) t (g (s) 0 ) -1 (z) . Letting t = s, we get e t ft (z) (law) = (g (t) 0 ) -1 (z),
and if we let t → +∞, by Lemma 3.1 we have lim t→+∞ e t ft (z)

(law) = g -1 0 (z) = f 0 (z).
Let us define the auxiliary, time-dependent, radial variant of the LLE two-point function

G(z 1 , z2 ) (3.5), 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 Y t (z 2 ) , (3.10)
where ft is the reversed radial Loewner process (3.8), together with the shorthand notations,

X t (z) := f t (z) p 2 ft (z) q 2 , Y t (z) := X t (z) = f t (z) p 2 ft (z) q 2
. By (3.9), the two-point function G(z 1 , z2 ) (3.5) is the limit

lim t→+∞ e (p-q)t G(z 1 , z2 , t) = G(z 1 , z2 ). (3.11)
As explained in [START_REF] Beliaev | Harmonic measure and SLE[END_REF], the idea is then to construct a martingale M s related to G. The vanishing of the drift term in its Itô derivative then yields a partial differential equation obeyed by G.

For s ≤ t, define the two-point martingale (M s ) t≥s≥0 with

M s := E[X t (z 1 )Y t (z 2 )|F s ],
where the random variable is integrable for fixed z 1 and z 2 , and where F s is the σ-algebra generated by the Lévy process filtration {L u , u ≤ s}. By the Markov property of the Lévy process, we know that for any s ≤ t,

ft (z)

(law) = λ(s) ft-s ( fs (z)/λ(s)).

(3.12) Therefore, M s = X s (z 1 )Y s (z 2 ) G(z 1,s , z2,s ; τ ), τ := ts, ∀τ > 0, (3.13) where z 1,s := fs (z 1 ) λ(s) , z2,s := fs (z 2 ) λ(s) = fs (z 2 )λ(s).

In order to prepare for Itô calculus, we have [DHLZ18, Section 4, Eqs. (47-49)]

dX s (z 1 ) =X s (z 1 ) p 2 - q 2 - p (1 -z 1,s ) 2 + q 1 -z 1,s ds, (3.14) dY s (z 2 ) =Y s (z 2 ) p 2 - q 2 - p (1 -z2,s ) 2 + q 1 -z2,s ds, (3.15) ∂z 1,s ∂s λ(s) =z 1s z 1,s + 1 z 1,s -1 , ∂ z2,s ∂s λ(s) = z2,s z2,s + 1 z2,s -1 . (3.16)
Let us write M s as a formal function of two variables,

H(s, L s ) := M s = X s (z 1 )Y s (z 2 ) G(z 1,s , z2,s , t -s).
It is a (local) martingale for all s ≤ t, thus by Itô calculus its total s-derivative vanishes,

ΛH(s, L s ) + ∂ s H(s, L s ) = 0,
where Λ is the generator of the Lévy process L s .

We have from Eqs. (3.14), (3.15), (3.16),

∂ s H = H p 2 - q 2 - p (1 -z 1,s ) 2 + q 1 -z 1,s +H p 2 - q 2 - p (1 -z2,s ) 2 + q 1 -z2,s -X s (z 1 )Y s (z 2 )∂ τ G(z 1,s , z2,s , t -s) +X s (z 1 )Y s (z 2 )∂ z 1 G(z 1,s , z2,s , t -s)z 1s z 1,s + 1 z 1,s -1 +X s (z 1 )Y s (z 2 )∂ z2 G(z 1,s , z2,s , t -s)z 2,s z2,s + 1 z2,s -1 ,
where τ := ts. Since neither X s (z) nor its complex conjugate Y s (z) vanish in D, we deduce that

-Λ G(z 1,s , z2,s , t -s) = G(z 1,s , z2,s , t -s) p 2 - q 2 - p (1 -z 1,s ) 2 + q 1 -z 1,s + G(z 1,s , z2,s , t -s) p 2 - q 2 - p (1 -z2,s ) 2 + q 1 -z2,s -∂ τ G(z 1,s , z2,s , t -s) + ∂ z 1 G(z 1,s , z2,s , t -s)z 1s z 1,s + 1 z 1,s -1 + ∂ z2 G(z 1,s , z2,s , t -s)z 2,s z2,s + 1 z2,s -1 . (3.17)
Notice that by (3.11), it holds that, as t → +∞,

(p -q) exp [ (p -q)t] G(z 1 , z2 , t) + exp [ (p -q)t] ∂ t G(z 1 , z2 , t) → 0, so that lim t→+∞ exp [ (p -q)t] ∂ t G(z 1 , z2 , t) = -(p -q) G(z 1 , z 2 ). (3.18)
Multiplying both sides of (3.17) by exp [ (pq)(ts)], and letting t → +∞, we get

-ΛG(z 1 , z2 ) = G(z 1 , z2 ) p 2 - q 2 - p (1 -z 1 ) 2 + q 1 -z 1 + G(z 1 , z2 ) p 2 - q 2 - p (1 -z2 ) 2 + q 1 -z2 + (p -q)G (z 1 , z2 ) + ∂ z 1 G(z 1 , z2 )z 1 z 1 + 1 z 1 -1 + ∂ z2 G(z 1 , z2 )z 2 z2 + 1 z2 -1 . (3.19)
We finally get that G(z 1 , z2 ) satisfies P(D) G(z 1 , z2 ) = 0, where

P(D) := Λ + z 1 z 1 + 1 z 1 -1 ∂ z 1 + z2 z2 + 1 z2 -1 ∂ z2 + p -q + p - q - p (1 -z 1 ) 2 + q 1 -z 1 - p (1 -z2 ) 2 + q 1 -z2 . (3.20)
Recall the definition of Λ acting on a C ∞ (R 2 ) function u,

Λu(x) = lim t↓0 1 t (E x [u(L t )] -u(x)) .
For k, l ∈ Z, we have for z = re iθ ,

Λ(z k zl ) = r k+l Λ(e iθ(k-l) ) = r k+l lim t↓0 1 t E θ [e i(k-l)Lt ] -e i(k-l)θ = r k+l lim t↓0 1 t (e -tη(k-l) -1) e i(k-l)θ = -η(k -l)z k zl , (3.21)
where η is the Lévy symbol of L t .

3.3. Drifted Brownian motion. In this section, we consider the special Lévy process L t = at + √ κB t , where a ∈ R, κ ≥ 0 and B t is standard one-dimensional Brownian motion. These processes are the most general Lévy processes with a.s. continuous trajectories. By definition of the Lévy symbol

E[e iξLt ] = E[e iξ(at+ √ κBt) ] = e iaξt-t κ 2 ξ 2 = e -tη(ξ) . So η(ξ) = κ 2 ξ 2 -iaξ.
By (3.21), we have

Λ(z k zl ) = -η(k -l)z k zl = - κ 2 (k -l) 2 + ia(k -l) z k zl ,
so that the Lévy generator in the Brownian drift case is explicitly

Λ = - κ 2 (z∂ z -z∂ z ) 2 + ia(z∂ z -z∂ z ).
The operator in (3.20) thus becomes

P(D) = - κ 2 (z 1 ∂ z 1 -z2 ∂ z2 ) 2 + z 1 z 1 + 1 z 1 -1 + ia ∂ z 1 +z 2 z2 + 1 z2 -1 -ia ∂ z2 + p -q + p - q - p (1 -z 1 ) 2 + q 1 -z 1 - p (1 -z2 ) 2 + q 1 -z2 . (3.22)
3.3.1. Algebraic solutions. We want to find some solutions to the PDE

P(D)G(z 1 , z2 ) = 0, G(0, 0) = 1, (3.23)
and follow the method of Ref. [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF], by looking for solutions of the form,

G(z 1 , z2 ) = (1 -z 1 ) α (1 -z2 ) ᾱP (z 1 z2 ), P (0) = 1. (3.24)
The action of the partial differential operator P(D) (3.22) readily gives

P(D)[(1 -z 1 ) α (1 -z2 ) ᾱP (z 1 z2 )] = z 1 z2 (1 -z 1 ) α-1 (1 -z2 ) ᾱ-1 (κα ᾱP (z 1 z2 ) + 2(z 1 z2 -1)P (z 1 z2 )) + [P(∂)(1 -z 1 ) α ](1 -z2 ) ᾱP (z 1 z2 ) + [P( ∂)(1 -z2 ) ᾱ](1 -z 1 ) α P (z 1 z2 ),
where

P(∂) := - κ 2 (z 1 ∂ z 1 ) 2 + z 1 + 1 z 1 -1 + ia z 1 ∂ z 1 + p -q + q 1 -z 1 - p (1 -z 1 ) 2 , (3.25) P( ∂) := - κ 2 (z 2 ∂ z2 ) 2 + z2 + 1 z2 -1 -ia z2 ∂ z2 + p -q + q 1 -z2 - p (1 -z2 ) 2 .
Notice that as complex conjugates,

∀z ∈ D, P(∂)(1 -z) α = 0 ⇔ P( ∂)(1 -z) ᾱ = 0.

So if we have,

P(∂)(1 -z 1 ) α = 0, (3.26)
then equation (3.23) reduces for (3.24) to κα ᾱP (z 1 z2 ) + 2(z 1 z2 -1)P (z 1 z2 ) = 0, P (0) = 1

⇔ P (z 1 z2 ) = (1 -z 1 z2 ) -κ 2 α ᾱ.
(3.27)

Let us now look for α such that Eq. (3.26) is satisfied. A direct computation readily gives [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF],

P(∂)(1 -z) α = (1 -z) α A + (1 -z) α-1 B + (1 -z) α-2 C, where A := - κ 2 α 2 + (1 + ia)α + p -q, (3.28) B := κα 2 - κ 2 + 3 -ia α + q, (3.29) C := - κ 2 α 2 + 2 + κ 2 α -p. ( 3

.30)

Notice that A + B + C = 0. For any α ∈ C, the choice of p, q such that B = 0 and C = 0, yields a solution to (3.26), hence together with (3.27) a solution (3.24) to (3.23).

We thus get the identity for drifted SLE,

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   = (1 -z 1 ) α (1 -z2 ) ᾱ(1 -z 1 z2 ) -κ 2 α ᾱ, (3.31)
where the quadratic equations B = 0, C = 0 yield p and q in terms of α ∈ C under the parametric form,

p = - κ 2 α 2 + 2 + κ 2 α, α ∈ C, (3.32) q = -κα 2 + κ 2 + 3 -ia α. (3.33)
These results generalize those found for real p, q and a = 0 in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. The complex p, q case, still for a = 0, has been thoroughly studied in Ref. [START_REF] Hieu | On multifractality, Schwarzian derivative and asymptotic variance of whole-plane SLE[END_REF]. These equations generalize in the complex p, q case, hence in four-dimensional space, the so-called red parabola of the real (p, q)-plane described in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. As a consequence, we have proven the following Theorem 3.1. Let f (z) = f 0 (z) where f t is the drifted whole-plane Loewner process driven by λ(t) = e i(at+ √ κBt) , a ∈ R. For (p, q) ∈ C 2 , let the complex 'red parabola' R be defined as the two-dimensional manifold,

p = - κ 2 α 2 + 2 + κ 2 α, q -p = - κ 2 α 2 + (1 -ia)α, α ∈ C. (3.34)
For (p, q) ∈ R, z 1 , z 2 ∈ D, we identically have

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   = (1 -z 1 ) α (1 -z2 ) ᾱ(1 -z 1 z2 ) -κ 2 α ᾱ.
In particular, for

z 1 = z 2 = z, G(z, z) = E |z q | f (z) p f (z) q = (1 -z) α (1 -z) ᾱ(1 -z z) -κ 2 α ᾱ.
(3.35)

Hence, in the case of the complex red parabola (3.32), (3.33), we find that the complex generalized bulk spectrum is simply given by β(p, q; κ, a) = 1 2 κ|α| 2 .

(3.36)

Remark 3.1. Tip spectrum. When 2 α + 1 ≤ 0, the presence in (3.35) of the singular factor |(1z) α | 2 , besides that of the bulk singular one, brings in an extra singular contribution to the integral means near z = 1. This yields the new complex generalized tip spectrum [BS09, BDZ17, DHLZ18] along the red parabola (3.34),

β(p, q; κ, a) = 1 2 κ|α| 2 -2 α -1, 2 α ≤ -1. (3.37)
Let us now turn to the case of real points along the complex red parabola R (3.34).

Corollary 3.1. Let f (z) = f 0 (z) where f t is the drifted whole-plane Loewner process driven by λ(t) = e i(at+ √ κBt) . If p, q take the following values:

p = p(κ, a) = (4 + κ) 2 8κ 1 + 4a 2 (2 + κ) 2 , q = q(κ, a) = 4 + κ 2κ 1 + 4a 2 (2 + κ) 2 ;
then the generalized integral means spectrum β(p, q) of f is equal to p.

Proof. Let us look for exponents p, q ∈ R, as parameterized by (3.32) and (3.33), with α = α 1 + iα 2 and α 1 , α 2 ∈ R. The condition p = 0 gives

α 2 -κα 1 + 2 + κ 2 = 0,
hence either α 2 = 0 or α 1 = (4 + κ)/2κ. The condition q = 0 yields

2κα 1 α 2 - κ 2 + 3 α 2 + aα 1 = 0.
So if α 2 = 0, we have either α 1 = 0 or a = 0. The first case is trivial, while the second one is the driftless case studied in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]. So, assuming a = 0, we obtain

α 1 = 4 + κ 2κ , α 2 = - a(4 + κ) κ(2 + κ) ,
and

α = 4 + κ 2κ 1 -i 2a 2 + κ ,
which in turn gives

p = p(κ, a) := (4 + κ) 2 8κ 1 + 4a 2 (2 + κ) 2 , (3.38) q = q(κ, a) := 4 + κ 2κ 1 + 4a 2 (2 + κ) 2 .
(3.39)

Notice the further identity κ 2 α ᾱ = p. So for these special real values of p and q we have

E |z| q |f (z)| p |f (z)| q = |(1 -z) α | 2 (1 -|z| 2 ) κ|α| 2 2
.

Notice also that α > 0, so that the singularity at z = 1 does not contribute to the circle integral

|z|=r<1 |(1 -z) α | 2 (1 -|z| 2 ) κ|α| 2 2 |dz| r→1 -(1 -r) -κ|α| 2 2 .
So in the case (3.38) (3.39) the averaged generalized spectrum is simply β(p, q) = p.

3.4.

Check of integral means spectra on the integrable complex 'red parabola'. As in [DHLZ18, Section 5.2.1], we will find that along the 'red parabola' R, a succession of explicit complex integral means spectra reproduces the result β := κ|α| 2 /2 of Theorem 3.1. In addition to formulae (2.7), (2.9), (2.10), (2.12) for the complex generalized spectrum β 1 of (drifted) whole-plane SLE, we shall need the SLE complex bulk spectrum β 0 (p), p ∈ C, and some extensions of both β 0 and β 1 [DHLZ18, Section 5.1].

3.4.1. SLE complex bulk spectrum. The SLE complex bulk spectrum β 0 (p), p ∈ C, can be obtained from the results of [DB02, DB08], and reads [START_REF] Binder | Multifractal properties of harmonic measure and rotation for Schramm-Loewner Evolution[END_REF],

β 0 (p) = s 0 (p) + p -1, (3.40) s 0 (p) = s 0 (t, t), t := p, t := p, (3.41)
where the expression for s 0 (t, t) is

s 0 (t, t) = 1 + b -t + (b -t) 2 + t2 -(2b ) 1 2 b -t + (b -t) 2 + t2 1 2
, (3.42)

s 0 (t, 0) = 1 + 2(b -t) -2 √ b √ b -t, b := (4 + κ) 2 8κ . (3.43)
By introducing the variables,

τ 0 := b -t, (3.44) τ := 1 2 b -t + (b -t) 2 + t2 = 1 2 τ 0 + τ 2 0 + t2 , (3.45)
the function s 0 (t, t) (3.42) can then be recast as a function of the single variable τ , as

s 0 (t, t) = s(τ ) := 1 + 2τ -2 √ b √ τ , (3.46) s 0 (t, 0) = s(τ 0 ) = 1 + 2τ 0 -2 √ b √ τ 0 .
3.4.2. Extensions of complex spectra β 0 and β 1 . As in Refs. [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF][START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF] it is natural to define auxiliary pseudo-integral means spectra, which help in understanding phase transitions that are mediated by overlaps between various analytic expressions of the spectra. They are obtained by restoring the usual sign indeterminacy in front of square root operations [DNNZ15, Section 4.2], [DHLZ18, Section 5.1]. Let us define the auxiliary functions,

β ± 0 (p) := s ± 0 (p) + p -1, p ∈ C, (3.47) s ± 0 (p) = s ± 0 (t, t) = s ± (τ ) := 1 + 2τ ± 2 √ b √ τ , (3.48) τ = 1 2 (b -p) + |b -p| , b = (4 + κ) 2 8κ = 1 2 b -t + (b -t) 2 + t2 , t := p, t := p,
such that the complex bulk integral means spectrum (3.40) is given by the (-)-branch, β 0 ≡ β - 0 . Similarly, we define β ± 1 (p, q; κ, a) := s ± 1 (pq; κ, a) + p -1, p, q ∈ C, (3.49)

s ± 1 (p -q; κ, a) = s ± 1 (τ ) := 2τ + 1 2 ∓ 1 2 √ 1 + 2κτ , (3.50) 1 + 2κτ = 1 2 (1 + ia) 2 + 2κ(p -q) + (1 + ia) 2 + 2κ(p -q) , (3.51)
such that the complex generalized spectrum (2.10) associated with spiral whole-plane SLE is given by the (+)-branch,

β 1 ≡ β + 1 . 3.4.3. Complex spectra β ± 1 along R.
From parameterization (3.34), we first find the identity along the red parabola R,

(1 + ia) 2 + 2κ(p -q) = (1 + ia -κα) 2 .
Using the general identity,

1 2 (z 2 ) + |z 2 | = ( z) 2 , z ∈ C, (3.52)
we find for (3.51),

1 + 2κτ = [ (1 + ia -κα)] 2 = (1 -κ α) 2 , so that s ± 1 (3.50) reads s ± 1 (τ ) = 1 κ (1 -κ α) 2 -1 + 1 2 ∓ 1 2 |1 -κ α|.
We simultaneously have from (3.34) and (3.52),

p = - κ 2 (α 2 ) + 2 + κ 2 α = κ 2 |α 2 | -κ( α) 2 + 2 + κ 2 α. (3.53)
Combining the last two equations gives

s ± 1 (τ ) + p -1 = κ 2 |α 2 | + κ 2 α - 1 2 ∓ 1 2 |1 -κ α|.
Therefore, we get for (3.49) the branch-dependent identity,

β ± 1 (p, q; κ, a) = s ± 1 (τ ) + p -1 = κ 2 |α 2 |, κ α 1. (3.54)
This shows that the result of Theorem 3.1 for spiral whole-plane SLE is recovered for α ≥ 1/κ by the 'physical' branch β + 1 of the generalized complex spectrum, and for α ≤ 1/κ by its 'unphysical' branch β - 1 , in a way entirely similar to the real case studied in [DHLZ18, Section 5.

-p = κ 2 α - 4 + κ 2κ 2 ,
from which we deduce with the help of (3.52),

τ = 1 2 (b -p) + |b -p| = κ 2 α - 4 + κ 2κ 2 .
This in turn gives

s ± 0 (τ ) = 1 + κ α - 4 + κ 2κ 2 ± 2 + κ 2 α - 4 + κ 2κ ,
which, together with (3.53) yields

s ± 0 (τ ) + p -1 = κ 2 |α 2 | -2 + κ 2 α + κ 4 + κ 2κ 2 ± 2 + κ 2 α - 4 + κ 2κ .
Thus we find the branch-dependent identity,

β ± 0 (p; κ) = s ± 0 (τ ) + p -1 = κ 2 |α 2 |, κ α 2 + κ 2 . (3.55)
We thus see that the result of Theorem 3.1 for spiral whole-plane SLE is recovered for α ≤ 2/κ + 1/2 by the 'physical' branch β - 0 of the standard complex spectrum, and for α ≥ 2/κ + 1/2 by its 'unphysical' branch β + 0 , in a way again similar to the real case studied in [DHLZ18, Section 5.2.1]. We thus arrive at Proposition 3.1. Along the red parabola R (3.34), the integral means spectrum of the drifted whole-plane SLE is successively given by

β - 0 (p; κ) = β - 1 (p, q; κ, a) = κ 2 |α 2 |, α ∈ (-∞, 1/κ], β - 0 (p; κ) = β + 1 (p, q; κ, a) = κ 2 |α 2 |, α ∈ [1/κ, 2/κ + 1/2], β + 0 (p; κ) = β + 1 (p, q; κ, a) = κ 2 |α 2 |, α ∈ [2/κ + 1/2, +∞).
When replacing h by this expansion in the equation, we get a recursion formula between the θ n 's for n ∈ Z. More precisely, by writing that the n'th Fourier coefficient of the left side of (4.2) vanishes, we obtain for n ∈ Z,

2ξ(ξ -1)θ n (ξ) -η n + n + (η n + 2q -n -6)ξ θ n + ξ η n + n + q -2 θ n+1 (ξ) + (η n -n + q -2)θ n-1 (ξ) = 0. (4.4)
Note that the assumption that L t is symmetric implies that h(z, z) is symmetric w.r.t. z and z, which translates into,

θ -n (ξ) = ξ n θ n (ξ), (4.5)
from which we may simply recast expansion (4.3) above as

h(z, z) = θ 0 (ξ) + ∞ n=1 θ n (ξ)(z n + zn ) = θ 0 (ξ) + ∞ n=1 2θ n (ξ)r n cos nt. (4.6)
Before continuing, let us recall that we are looking for the integral means, i.e., the angular integrals,

I(r) = 2π 0 G(re it , re -it )dt,
that can be easily expressed in terms of θ j 's as

I(r) 2π = (1 + r 2 )θ 0 (r 2 ) -2r 2 θ 1 (r 2 ), (4.7)
so that we only need to compute θ 0 and θ 1 . For later purposes, let us also mention that θ 0 (0) = 1. We thus focus on the equations for n = 0 and n = 1 (recall that η 0 = 0 and that θ -1 (ξ) = ξθ 1 (ξ)),

(ξ -1)θ 0 (ξ) -(q -3)θ 0 (ξ) + (q -2)θ 1 (ξ) = 0, (4.8) 2ξ(ξ -1)θ 1 (ξ) -[η 1 + 1 + (η 1 + 2q -7)ξ] θ 1 (ξ) + (η 1 + q -1)ξθ 2 (ξ) + (η 1 + q -3)θ 0 (ξ) = 0.
(4.9)

There are two simple cases where we can explicitly compute θ 0 and θ 1 .

(1) The first case is when the coefficient of the θ 0 -term in the second equation vanishes, i.e., when η 1 = 3q, (4.10) which requires q < 3. In this case we may take θ 1 = 0 (and actually θ n = 0 for n ≥ 1) and θ 0 to be the solution to

(ξ -1)θ 0 (ξ) -(q -3)θ 0 (ξ) = 0, θ 0 (0) = 1.
This gives

h(z, z) = θ 0 (ξ) = 1 (1 -ξ) 3-q , ( 4.11)
and

G(z, z) = (1 -z)(1 -z) (1 -z z) 3-q ,
so that β(2, q) = 3q > 0.

(2) The second case is by letting the coefficient of the θ 2 -term vanish in the second equation, i.e., by taking η 1 = 1q, (4.12) which requires q ≤ 1. We then get a system of coupled ODEs for θ 0 and θ 1 , which we must solve with initial data θ 0 (0) = 1, θ 1 (0) finite. We find

   θ 0 (ξ) = (1 + ξ)(1 -ξ) -(4-q) θ 1 (ξ) = -2 2-q (1 -ξ) -(4-q) , (4.13)
from which we deduce that β(2, q) = 4q(> 3).

Eqs. (4.11) and (4.13) generalize results of Ref. [START_REF] Loutsenko | New exact results in spectra of stochastic Loewner evolution[END_REF] to q = 0. As noticed in [START_REF] Loutsenko | Stochastic Loewner evolutions, Fuchsian systems and orthogonal polynomials[END_REF], the preceding method generalizes: more precisely for any n ≥ 1, if we let the coefficient of the θ n-1 -term vanish in the nth equation, i.e., by taking η n = 2q + n, then the solution of the system is θ p = 0 for p ≥ n, while θ 0 , ..θ n-1 are the solutions of the n first equations, with the initial data θ 0 (0) = 1. Another possible generalization is by letting vanish, again in the nth equation, the coefficient of the θ n+1 -term, i.e., by taking η n = 2qn; then the n first equations allow us to compute θ 0 , ..θ n-1 , which is more than needed since we only need to know θ 0 and θ 1 . Having dealt in the last section with the n = 1 case, let us now investigate the n = 2 case. 4.1. The η 2 = 4q case. We take θ n≥2 = 0, and have to solve the following system of differential equations:

(x -1)θ 0 (x) + (3 -q)θ 0 (x) + (q -2)θ 1 (x) = 0 2x(x -1)θ 1 (x) -[1 + η 1 + (η 1 + 2q -7)x] θ 1 (x) + (η 1 -3 + q)θ 0 (x) = 0.
We can assume the existence of a real number δ (to be determined later) and of two functions f 0 , f 1 : [0, 1] → R such that θ 0 and θ 1 have the following form,

θ 0 (x) = (1 -x) -δ f 0 (x), θ 1 (x) = (1 -x) -δ f 1 (x).
The ODE system satisfied by f 0 and f 1 is then

(x -1)f 0 + (3 -q -δ)f 0 + (q -2)f 1 = 0 2x(x -1)f 1 -[1 + η 1 + (η 1 + 2q -7 + 2δ)x] f 1 + (η 1 -3 + q)f 0 = 0. (4.14)
The form of the equation coefficients suggests we define δ as,

δ := δ + q -3, (4.15)
so that

(x -1)f 0 -δ f 0 + (q -2)f 1 = 0 2x(x -1)f 1 -1 + η 1 + (η 1 -1 + 2δ )x f 1 + (η 1 -3 + q)f 0 = 0. (4.16)
From the first equation we extract f 1 as a function of f 0 and f 0 for q = 2,

f 1 (x) = x -1 2 -q f 0 (x) - δ 2 -q f 0 (x), q = 2. (4.17)
Substituting into (4.16), we obtain the following degree two differential equation satisfied by f 0 ,

2x(x -1) 2 2 -q f 0 (x) + 2x(x -1) 1 -δ 2 -q - x -1 2 -q (1 + η 1 + (η 1 -1 + 2δ )x) f 0 (x)+ η 1 + q -3 + δ 2 -q (1 + η 1 + (η 1 -1 + 2δ )x) f 0 (x) = 0. (4.18)
We want to find δ such that this equation reads

x(x -1)f 0 (x) + ((a + b + 1)x -c)f 0 (x) + abf 0 (x) = 0, (4.19)
so that f 0 (x) = 2 F 1 (a, b, c; x), the hypergeometric function. This identification shows that δ must obey the following relation,

E(δ , η 1 , q) := 2δ (δ + η 1 ) + (2 -q)(η 1 + q -3) = 0, (4.20)
so that (4.18) simplifies into

x(x -1)f 0 (x) + 3 -η 1 -4δ 2 x - 1 + η 1 2 f 0 (x) (4.21) -δ 1 -η 1 -2δ 2 f 0 (x) = 0. (4.22)
This yields

a = -δ , b = 1 -η 1 2 -δ , c = 1 + η 1 2 , (4.23)
and we can choose

f 0 (x) = 2 F 1 (a, b; c; x) and θ 0 (x) = (1 -x) -δ 2 F 1 (a, b; c; x).
The other independent solution to the hypergeometric equation is

x 1-c 2 F 1 (1 + a -c, 1 + b -c; 2 -c;
x), which is non-analytic at the origin x = 0, hence is discarded as a candidate for f 0 .

Before continuing, let us consider general symmetric Lévy processes: what are the possible couples (η 1 , η 2 )? We know that we must have η 2 ≥ 0 and the Lévy-Khinchine formula implies that η 1 ≥ η 2 /4. It happens 1 that every couple (η 1 , η 2 ) such that η 1 ≥ η 2 /4 ≥ 0 actually corresponds to some (symmetric) Lévy process. Moreover the case η 1 = η 2 /4 exactly corresponds to an SLE process, while in the case η 2 = 0, L t is a pure jump process with jumps equal to kπ, k odd (notice that the other case is similar with k even, yielding a continuous process on the circle).

For η 2 = 4q, the preceeding constraints on (η 1 , η 2 ) become q ≤ 4 and η 1 ≥ η 2 /4 = 1q/4, with equality for SLE κ , with κ = 2q/2. Let us then define D 4-q := {(q, η 1 ) ∈ R 2 ; q ≤ 4, η 1 ≥ 1q/4}.

(4.24)

Let us return to Eq. (4.20). It can be shown (see below) that η 2 1 -2(2q)(η 1 + q -3) ≥ 0 in D 4-q ; we may thus extract δ from (4.20): Replacing in (4.23) δ by its value in terms of η 1 , q, we get:

δ ± = δ ± + 3 -q δ ± = δ ± (q, η 1 ) := 1 2 -η 1 ± η 2 1 -2(2 -q)(η 1 + q -3) .
a ± = a ± (η 1 , q) := 1 2 η 1 ∓ 1 2 η 2 1 -2(2 -q)(η 1 + q -3) (4.26) b ± = b ± (η 1 , q) := 1 2 ∓ 1 2 η 2 1 -2(2 -q)(η 1 + q -3) (4.27) c = c(η 1 , q) := 1 + η 1 2 . (4.28)
We can now compute f 1 as,

f 1 (x) = x -1 2 -q f 0 (x) + -δ 2 -q f 0 (x) = a 2 -q 2 F 1 (a, b; c; x) + ab(x -1) c(2 -q) 2 F 1 (a + 1, b + 1; c + 1; x). (4.29)
Using expression (4.7) for the integral means, we get We then have ca

1 2π I(r) = (1 -r 2 ) -δ (1 + r 2 )f 0 (r 2 ) -2r 2 f 1 (r 2 ) . ( 4
+ -b + = η 2 1 -2(2 -q)(η 1 + q -3) > 0, so that 0 < 2 F 1 (a, b; c; 1) = Γ(c)Γ(c -a -b) Γ(c -a)Γ(c -b) < ∞. (4.31)
If furthermore, a + b + 1 < c, then 2 F 1 (a + 1, b + 1; c + 1; 1) < ∞, and

∆ := lim r→1 (1 + r 2 )f 0 (r 2 ) -2r 2 f 1 (r 2 ) = 2 1 - a 2 -q 2 F 1 (a, b; c; 1) < ∞.
We have thus proven that if ∆ > 0, the spectrum is

β(2, q) = 3 -q + δ + (η 1 , q) = 3 -q + 1 2 η 2 1 -2(2 -q)(η 1 + q -3) -η 1 .
One can check that the coefficient A := 1a + /(2q) > 0 for q < 3, whereas for q ≥ 3, it vanishes for η 1 = 1q or η 1 = 3q. In D 4-q , A is thus positive since η 1 ≥ 1q/4 is located outside the non-positive interval [1q, 3q]. So we get that the spectrum is equal to δ + on the subset of D 4-q of points for which a + b + 1 < c. This condition is just

Z > 1, with Z = Z(η 1 , q) := η 2 1 -2(2 -q)(η 1 + q -3). (4.32)
Note that Z can be written as

Z = (q -3) 2 + (η 1 + q -2) 2 -1 (4.33) = X 2 + Y 2 -1, X := q -3, Y := η 1 + q -2, (4.34)
so that the condition Z > 1 corresponds to the exterior of the blue ellipse of equation Z = 1, i.e., X 2 + Y 2 = 2 in the (q, η 1 )-plane (Fig. 11). Notice that the set Z = 0 is the co-centered green ellipse of equation X 2 + Y 2 = 1, which only intersects D 4-q at the tangency point ( 12 5 , 2 5 ) with the (η 1 = 1q/4)-line, implying that Z ≥ 0 in D 4-q , as mentioned above.

In the interior of the blue ellipse, we instead have a + b < c < a + b + 1, and we use the Euler transformation,

2 F 1 (a, b; c; x) = (1 -x) c-a-b 2 F 1 (c -a, c -b; c; x), (4.35) so that 2 F 1 (a + 1, b + 1; c + 1; x) = (1 -x) c-a-b-1 2 F 1 (c -a, c -b; c + 1; x). (4.36)
We then get from (4.29)

f 1 (x) = a 2 -q 2 F 1 (a, b; c; x) - ab c 1 (2 -q) (1 -x) c-a-b 2 F 1 (c -a, c -b; c + 1; x). (4.37)
We now have 2 F 1 (a, b; c; 1) < ∞ and 2 F 1 (ca, cb; c + 1; 1) < ∞, so the second term in (4.37) still vanishes as x → 1. This again yields

0 < ∆ = 2 1 - a 2 -q 2 F 1 (a, b; c; 1) < ∞,
i.e., the same result as found outside the blue ellipse. We thus find for the (+)-branch, We then use both (4.35) and (4.36) in (4.29),

1 2π I(r) ∼ 2(1 -r 2 ) -δ + 1 - a + 2 -q Γ(c)Γ(c -a + -b + ) Γ(c -a + )Γ(c -b + ) , r → 1 -, q = 2.
f 1 (x) = (1 -x) c-a-b a 2 -q 2 F 1 (c -a, c -b; c; x) - ab c 1 (2 -q) 2 F 1 (c -a, c -b; c + 1; x) , (4.39)
where now cab < 0, and where

2 F 1 (c -a, c -b; c; 1) = Γ(c)Γ(a + b -c) Γ(a)Γ(b) < ∞, 2 F 1 (c -a, c -b; c + 1; 1) = Γ(c + 1)Γ(a + b -c + 1) Γ(a + 1)Γ(b + 1) < ∞.
From the well-known identity Γ(x + 1) = xΓ(x), we finally get for (4.39)

f 1 (x) = (1 -x) c-a-b c -b 2 -q Γ(c)Γ(a + b -c) Γ(a)Γ(b) . I(r) (4.30) is now equivalent for r 2 = x → 1 to 1 2π I(r) ∼ 2(1 -x) -δ -+c-a-b 1 - c -b 2 -q Γ(c)Γ(a + b -c) Γ(a)Γ(b) .
where we recall that a = a -, b = b -. We now use the duality formulae

c -a -= b + , c -b -= a + , a -+ b --c = c -a + -b + ,
and

δ ± = 3 -q + δ ± , δ --δ + = c -a --b -, so that for r → 1 -, 1 2π I(r) ∼ 2(1 -r 2 ) -δ + 1 - a + 2 -q Γ(c)Γ(c -a + -b + ) Γ(c -a + )Γ(c -b + ) ,
which is exactly the same as the result (4.38) for the (+)-choice in Section 4.1.1.

Remark 4.1. The q = 2 case. Up to now, we have assumed that q = 2. The solution for q = 2, thus η 2 = 2, η 1 ≥ 1/2, can be obtained by continuity as the q → 2 limit of f 0 and f 1 (4.17). Eq. (4.26) gives

a + = 1 2η 1 (η 1 -1)(q -2) + O (q -2) 2 , b + = 1 2 (1 -η 1 ) + O(q -2), c = 1 2 (1 + η 1 ),
so that f 0 = 1, and (4.17) has a finite limit when q → 2,

f 1 (x) = η 1 -1 2η 1 1 - 1 -η 1 1 + η 1 (1 -x) 2 F 1 (1, 1 2 (3 -η 1 ); 1 2 (3 + η 1 ); x) , (4.40)
together with δ + = 0 and δ + = 1. Eq. (4.38) simply becomes 1 2π I(r) ∼ (1-r 2 ) -1 (3 -1/η 1 ), as r → 1 -, so that β(p = 2, q = 2) = 1.

We therefore proved the following Theorem 4.1. For a Lévy process, with symbols η 2 = 4q, η 1 ≥ η 2 /4 for q ≤ 4, the generalized integral means spectrum of the corresponding LLE is

β(2, q) = 3 -q + 1 2 η 2 1 -2(2 -q)(η 1 + q -3) -η 1 . (4.41)
In particular, if q = 2, η 2 = 2, the standard integral means spectrum at p = 2 of the logarithm of the LLE is independent of η 1 ≥ 1/2, and equal to β(2, 2) = 1.

In the q = 0 case, Eq. (4.41) recovers a result of [START_REF] Loutsenko | Stochastic Loewner evolutions, Fuchsian systems and orthogonal polynomials[END_REF].

4.1.3. Algebraic solutions. As a transition to the next section, let us look for purely algebraic solutions of the form, θ j (x) = (1x) -α f j (x), f j (x) = A 0 j + A 1 j (1x), j = 0, 1. where A k j , k = 0, 1 are fixed coefficients and with the understanding that θ j = 0 for j ≥ 2. Recall that the hypergeometric function 2 F 1 is given by the well-known series expansion,

2 F 1 (a, b; c; x) = ∞ n=0 (a) n (b) n (c) n x n n! = 1 + ab c x 1! + a(a + 1)b(b + 1) c(c + 1) x 2 2! + • • • , (4.42) with (a) n = 1 n = 0 a(a + 1) • • • (a + n -1) n ≥ 1,
so that the fact that f 0 (x) = 2 F 1 (a, b; c; x) is at most linear in x implies that either: a = a + = 0; a + = -1; b = b + = 0; or b + = -1. The function f 1 (4.17) is then also linear, assuming for now that q = 2. For later convenience, let us write (4.26) (4.27) as

a + = 1 2 η 1 -1 2 Z 1/2 , b + = 1 2 -1 2 Z 1/2 , (4.43) Z = η 2 1 -2(2 -q)(η 1 + q -3).
FIGURE 11. Domain lines in the (q, η 1 ) plane, for the η 2 = 4q case. The two purple straight lines have for equations, η 1 = 1q and η 1 = 3q.

The green, blue and red ellipses have for respective equations X 2 + Y 2 = 1, 2, 10, with X = q -3, Y = η 1 + q -2, with same center located at (q, η 1 ) = (3, -1).

• The a + = 0 case. The equation Z = η 2 1 gives for q = 2, η 1 = 3q, which recovers the algebraic solutions, Eqs. (4.10) and (4.11). For q = 2, Remark (4.1) and Eq. (4.40) yield for η 1 = 1, f 0 = 1, f 1 = 0, in agreement with (4.11).

• The a + = -1 case. The equation Z = (η 1 + 2) 2 yields (4q)(η 1 + q -1) = 0. Hence we first recover the algebraic case η 1 = 1q, as in Eqs. (4.12) and (4.13). The other case, q = 4, is the vertical boundary line for D 4-q (Fig 11 ), where η 2 = 0, and for which Eq. (4.41) gives δ + = β(p = 2, q = 4) = 0, so that θ j = f j , j = 0, 1. One further finds b + = -c = -1 2 (1 + η 1 ), so that one gets the polynomial solutions, f 0 (x) = 2 F 1 (-1, b + ; c; x) = 1 + x, and from (4.17) f 1 (x) = 1 2 (1 + x).

• The b + = 0 case. From (4.43) we get the condition Z = 1, which in parameterization (4.33) of Z (4.32) is just that defining the blue ellipse as X 2 + Y 2 = 2. Its solution is given by η 1 = 2q ± -q 2 + 6q -7. The condition (q, η 1 ) ∈ D 4-q (4.24) selects the (+)-branch only, and restricts the range of parameter q to q ∈ [ 8 5 , 16 5 ] (see Fig. 11). This yields a first line of algebraic solutions,

     η 1 = 2 -q + 1 + (q -2)(4 -q) β(2, q) = 1 2 5 -q -1 + (q -2)(4 -q) q ∈ [ 8 5 , 16 5 ].
• The b + = -1 case. From (4.43) we get the condition Z = 9, which in parameterization (4.33) is defining a red ellipse as X 2 + Y 2 = 10. Its solution is given by η 1 = 2q ± 1 + 6qq 2 . The condition (q, η 1 ) ∈ D 4-q (4.24) allows for both (±)-branches, but restricts the range of parameter q to [3 -√ 10, 4] for the (+)-branch, and to [3 -√ 10, 0] for the (-)-branch (see Fig. 11). This finally gives algebraic solutions for,

     η 1 = 2 -q + 1 + 6q -q 2 β(2, q) = 1 2 (7 -q -1 + 6q -q 2 ) q ∈ [3 - √ 10, 4],
and

     η 1 = 2 -q -1 + 6q -q 2 β(2, q) = 1 2 (7 -q + 1 + 6q -q 2 ) q ∈ [3 - √ 10, 0].
From (4.42) and for n ≥ 2, one further finds a whole series of algebraic solutions where the f j 's are polynomials of degree n, when either (a) n+1 = 0 or (b) n+1 = 0, i.e., either a + = -n or b + = -n.

• The a + = -n case. From (4.43), one finds

η 1 = (2 -q)(3 -q) -2n 2 2 -q + 2n . (4.44)
One recovers the two linear cases seen above, n = 0, η 1 = 3q with q ≤ 8/3 and n = 1, η 1 = 1q with q ≤ 0. For n ≥ 2, the Lévy symbol condition η 1 ≥ 1q/4 requires that q ≤ 2 -2n. Eq. (4.44) can be recast as

(2q + 2n)(η 1 + q -3 + 2n) = 2n(n -1).

This corresponds to a branch of a hyperbola H n in the (q, η 1 ) plane, defined in affine coordinates by

X n Y n = 2n(n-1), X n := 2-q+2n, Y n := η 1 +q-3+2n, q ≤ 2-2n, n ≥ 2. (4.45)
• The b + = -n case. From (4.43) one readily finds

Z = (2n + 1) 2 , (4.46)
which in parameterization (4.33) is defining the ellipse E n by the equation

X 2 + Y 2 = 1 + (2n + 1) 2 .
This in turn yields η 1 = 2q ± (2q)(q -4) + (2n + 1) 2 , (4.47) together with

β(2, q) = 3 -q + 1 2 (2n + 1 -η 1 ) (4.48) = 1 2 2n + 5 -q ∓ (2 -q)(q -4) + (2n + 1) 2 .
The condition (q, η 1 ) ∈ D 4-q (4.24) allows for both (±)-branches, but restricts for the (+)-branch the range of parameter q to [3 -(2n + 1) 2 + 1, 4], and for the (-)-branch to [3 -(2n + 1) 2 + 1, 8 5 (1n)]. For n = 0, 1 one recovers the blue and red ellipses of (1) On the η 1 = 1q/4 line:

As we have already seen, this case occurs when η 2 = 4η 1 and the process is an SLE κ with κ = 2η 1 = 2q/2, for which we get from the above, β(2, q) = 1 -1 4 q, q ≥ 12 5 , = 4 -3 2 q, q ≤ 12 5 .

(4.49)

It is known that the SLE generalized spectrum has several phases [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF], among which, the standard 'bulk' spectrum [START_REF] Beliaev | Harmonic measure and SLE[END_REF],

β 0 (p; κ) := -p + (4 + κ) 2 4κ - κ + 4 4κ (4 + κ) 2 -8κp, (4.50)
and the 'unbounded whole-plane' one [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF],

β 1 (p, q; κ) := 3p -2q - 1 2 - 1 2 1 + 2κ(p -q). (4.51) We thus have          β 0 (2; κ = 2 -q/2) = 1 -1 4 q, q ≥ -4, = 16
4-q , q ≤ -4. β 1 (2, q; κ = 2q/2) = 4 -3 2 q, q ≤ 3, = 7 -5 2 q, q ≥ 3. Hence, on the SLE boundary line, a phase transition takes place at q = 12/5 in the spectrum (4.49), in the sense that β(2, q) = β 1 (2, q; κ = 2q/2) if q ≤ 12/5 and β(2, q) = β 0 (2; κ = 2q/2) if q ≥ 12/5. One can check that the phase transition point (p = 2, q = 12/5) is located on the so-called 'green parabola' that delineates the respective domains of validity of β 0 and β 1 for whole-plane SLE κ=4/5 [DHLZ18, Sec. 5.2.2].

(2) On the q = 4 line:

Here, η 2 = 0 and the spectrum is β(2, 4) = 0, ∀η 1 . This case corresponds to a pure jump Lévy process, whereas the equality q = 2p = 4 corresponds to a bounded LLE process [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF]; the above result then agrees with [START_REF] Chen | Schramm-Loewner equations driven by symmetric stable processes[END_REF]. Fig. 11 summarizes the results of this section, showing

• the domain D 4-q , domain of validity of the hypergeometric analysis;

• the domain of definition of the square root involved in the expression of the spectrum β(2, q) (4.41), which is the exterior of the green ellipse (thus containing D 4-q ); • the η 1 = 1q and η 1 = 3q lines, corresponding to degenerate hypergeometric solutions with a + = -1, 0; • the special solutions with b + = -1, 0, for which the hypergeometric f 0 , f 1 are degree 1 polynomials, which respectively correspond to the two red and blue ellipses (intersected with D 4-q ). Note that the phase-transition point (q = 12/5, η 1 = 2/5)

This system is best written under a matricial form, by defining successively,

D k (α) :=   k -(α + q -3) q -2 0 η 1 + q -3 2k -2(η 1 + α + q -3) η 1 + q -1 0 -4 2k -2(α -3)   and C k-1 (α) :=   0 0 0 0 2(k -1) -(η 1 + 2α + 2q -7) η 1 + q -1 0 0 2(k -1) -(2α + q -8)   .
Let us finally define the column vectors,

A k :=   A k 0 A k 1 A k 2   , k ≥ 0, so that recursions (4.57) become for k ≥ 1 D k (α)A k = C k-1 (α)A k-1 , k ≥ 1, (4.58)
together with the initial condition,

D 0 (α)A 0 = 0, (4.59)
and the closure relation

A n+1 = 0, (4.60)
such that A k = 0, ∀k ≥ n + 1. Equation (4.59) shows that for a non-trivial solution to exist, one must have det D 0 (α) = 0, with A 0 an eigenvector of vanishing eigenvalue. The determinant of D 0 is

det D 0 (α) = 2(1 -α)E 0 (α) (4.61) E 0 (α) := 2(α + q -3)(η 1 + α + q -5) -q(η 1 + q -3).
Let us look for the solutions to

E 0 (α) = 0 ⇔ α = α ± 0 := 3 -q + 1 2 2 -η 1 ± Ẑ , (4.62)
Ẑ = Ẑ(η 1 , q) := (η 1 -2) 2 + 2q(η 1 + q -3), (4.63) which yields the set of non-trivial zeroes of det D 0 . Ẑ can also be written as Ẑ = (q -1) 2 + (η 1 + q -2) 2 -1 = X2 + Y 2 -1, X := q -1, Y := η 1 + q -2. (4.64)

Thus Ẑ is non-negative outside the green ellipse Ẑ = 0 (Fig 12 ), and since q ≤ 0 in D -q , expression (4.64) is clearly non-negative there, vanishing only for q = 0, η 1 = 2, so that α ± 0 (4.62) is defined and real in D -q . Observe also that a translation maps Z (4.33) to Ẑ (4.64), Ẑ(η 1 , q) = Z(η 1 -2, q + 2).

(4.65)

The null eigenvectors A 0 (α) of D 0 (α) with vanishing eigenvalue are given, either for α = α ± 0 or for α = 1, by the one-dimensional space

A 0 = A 0 (α) =      A 0 0 ∈ R A 0 1 = 1 q-2 (α + q -3)A 0 0 A 0 2 = -2 α-3 A 0 1 .
(4.66)

As it will appear shortly, the value of the generalized spectrum is given by the root α + 0 in (4.62). To check this, consider the first integrability line, η 1 = 3q, where (4.62) gives, α + 0 = 3q, α - 0 = 2, q ≤ 1 = 2 = 3q, q ≥ 1.

Therefore, the choice of root α + 0 reproduces for q ≤ 1, hence q ≤ 0 in D -q the expected spectrum (4.11) β(2, q) = 3-q. For the second integrability line, η 1 = 1-q, one similarly finds α + 0 = 4q, α - 0 = 3, q ≤ 1 = 3 = 4q, q ≥ 1.

The condition η 1 ≥ 0 requires that q ≤ 1, thus the choice of root α + 0 again gives for q ≤ 0 in D -q the expected spectrum (4.13) β(2, q) = 4q. 4.2.2. Recursion. Assume for the time being that for k ≥ 1, det D k (α) = 0, so that D k (α) is invertible. We immediately get from (4.58) Eq. (4.74) gives for ellipse Ên the equations in Cartesian coordinates (q, η 1 ) ∈ D -q ,      η 1 = 2q + q(2q) + (2n -1) 2 α = α + 0 = β(2, q) = 1 2 (5q + 2kq(2q) + (2n -1) 2 ) q ∈ [1 -1 + (2n -1) 2 , 0], (4.75) and      η 1 = 2qq(2q) + (2n -1) 2 α = α + 0 = β(2, q) = 1 2 (5q + 2k + q(2q) + (2n -1) 2 ) q ∈ [1 -1 + (2n -1) 2 , inf{-8 5 (n -3 2 ), 0}].

A k = M k A 0
(4.76)

In the latter set, the q = -8 5 (n-3 2 ) point for n ≥ 2 corresponds to SLE κ with κ = 4 5 (n-3 2 ) and α = β 2, -8 5 (n -3 2 ) = β 1 2, -8 5 (n -3 2 ); 4 5 (n -3 2 ) in (4.51). The results are summarized in Fig. 12. The blue and red ellipses in Fig. 12 above FIGURE 12. Domain lines in the (q, η 1 ) plane, for the η 2 = -q case. The two purple straight lines have for equations, η 1 = 1q and η 1 = 3q. The green, blue and red ellipses have for respective equations X2 + Y 2 = 1, 2, 10, with X = q -1, Y = η 1 + q -2, with a common center located at (q, η 1 ) = (1, 1). respectively correspond to the first n = 1 and n = 2 cases. Because of (4.65), these ellipses are images by the (q → q -2, η 1 → η 1 + 2) translation of the corresponding same color ellipses of Fig. 11 in Section 4.1, with a common center now located at (q, η 1 ) = (1, 1). Note also that if the generalized spectrum is given in the whole region D -q by α + 0 in Eq. (4.62), then on the SLE κ -line where κ = 2η 1 = η 2 /2 = -q/2 it coincides with the spectrum β 1 (2, q; κ = -q/2) = 5 -3q/2, and there is no phase transition along that line, in contrast to the (η 2 = 4q)-case. The phase transition now takes place on the q = 0 = η 2 line, at its η 1 = 2 contact point with the green ellipse, since the predicted spectrum α + 0 is there equal to 5η 1 for 0 ≤ η 1 ≤ 2 and to 3 for η 1 ≥ 2. This agrees with results (4.13), β(2, 0) = 4 for η 1 = 1, and (4.11), β(2, 0) = 3 for η 1 = 3. Remark 4.3. Alternative condition. For completeness, let us also mention that the alternative condition in (4.72), 2n-(2α+q -8) = 0 with α = α ± 0 , yields 2n+q +η 1 = ± Ẑ, leading to η 1 + q -1 = q(q-2) 4(n+1)n and to the selection of the (+)-branch. From (4.71), one further needs to check that (η 1 + q + 1)A n 1 = (η 1 + q -1)A n 2 upon starting the recursion (4.67) from the null-eigenvector (4.66). Using again Mathematica® shows this equality not to hold, thus leading to no further solution. 4.2.4. General solution to the Fuchsian system. From the perspective of Fuchsian systems [IY08, LY19], the initial equations (4.53) for the vector function, ϑ(ξ) := (θ 0 (ξ), θ 1 (ξ), θ 2 (ξ)) t can be written under the matrix form,

ϑ (ξ) = A ξ ϑ(ξ) + B 1 -ξ ϑ(ξ),
where matrices A and B do not depend on ξ. They are simply given here by

A = 1 2   0 0 0 η 1 + q -3 -(η 1 + 1) 0 0 -4 q -2   , (4.77) B = 1 2   2(3 -q)
2(q -2) 0 η 1 + q -3 -2(η 1 + q -3) η 1 + q -1 0 -4 6   .

(4.78)

The discriminant of B is det(B -α1) = (1α) 1 2 (η 1 + q -3)(2α + q -6) + (α -2)(α + q -3) = 1 2 (1α)E 0 (α) = 1 4 det D 0 (α), so that B has for eigenvalues the zeroes of det D 0 , α ± 0 (4.62) and 1. For ξ → 1 -, the vector functions ϑ(ξ) = (θ 0 (ξ), θ 1 (ξ), θ 2 (ξ)) t , have for asymptotic behavior,

ϑ(ξ) = (1 -ξ) -α + 0 f + (ξ) + (1 -ξ) -α - 0 f -(ξ) + (1 -ξ) -1 f 1 (ξ), (4.79)
where f ± , f 1 are vector functions with Taylor series expansions in ξ. In the generic nonresonant case, where the eigenvalues do not differ by integer numbers, these functions converge at ξ = 1 towards the eigenvectors of B corresponding to their respective eigenvalue powers α ± 0 , 1. In the resonant case, they can be polynomials in ξ, or can involve polynomials inlog(1ξ), which then dominate the limit when ξ → 1 -.

In the resonant case of the E n ellipses of Section 4.2.3, we have α - 0 = α + 0 -(2n -1) with n ≥ 1. These vector functions are then simple polynomials, with f + (ξ) = (f 0 (ξ), f 1 (ξ), f 2 (ξ)) t , and f -(ξ) = f 1 (ξ) = 0. At ξ = 1, f + (ξ) becomes the eigenvector f + (ξ = 1) = A 0 (α + 0 ), as given in (4.66).

Let us conclude with the following Theorem.

Theorem 4.2. The generalized integral means spectrum β(2, q) of a whole-plane Lévy-Loewner process with Lévy symbols η 1 and η 2 = -q is given in the whole D -q domain (4.52) by α + 0 in Eq. (4.62).

  be the Koebe function, conformally mapping the unit disk to the straight cut plane asD → C \ (-∞, -1 4 ].Let g be the conformal map from C \ (-∞, -1 4 ] to the plane cut by the half spiral, Ω 0

FIGURE 5 .

 5 FIGURE 5. Non-linear mapping (2.16) of the red and green parabolae and blue quartic of Ref. [DHLZ18] (here κ = 2, a = 1).

FIGURE 6 .

 6 FIGURE 6. Non-linear mapping (2.16) of the domains of validity of the proofs, as shown in Fig. 4 from [DHLZ18] (here κ = 2, a = 1).

  p

FIGURE 9 .

 9 FIGURE 9. Phase transition lines for the generalized integral means spectrum of non-simple whole-plane SLE κ with drift (here κ = 6, a = 2). The successive phase crossings of the two standard ims bisector lines are analogous to those depicted in Fig. 7.

βFIGURE 10 .

 10 FIGURE 10. Phase transition lines for the generalized integral means spectrum of non-simple whole-plane SLE κ with drift (here κ = 6, a = 8). The standard ims in the q = 2p exterior case crosses phases I, II, III only, while the q = 0 standard ims in the interior case crosses all four phases in the order I, II, III and IV.

  2.1]. 3.4.4. Complex spectra β ± 0 along R. From parameterization (3.34) we first get the identity, b

  .30) 4.1.1. The (+)-branch. Let us first consider the case where a = a + , b = b + .

  (4.38) 4.1.2. The (-)-branch. Let us now consider the other possible choice, a = a -, b = b -.

Fig. 11 .

 11 Let us finally investigate what happens on the boundary of D 4-q .

  α)C k--1 (α) := D -1 k (α)C k-1 (α)D -1 k-1 (α)C k-2 (α) • • • D -1 1 (α)C 0 (α). (4.68)Notice that D k and C k obey a simple shift relation,D k (α) = D 0 (αk), C k (α) = C 0 (αk). (4.69)Result (4.68) can then be rewritten simply as, k + )C 0 (αk + + 1) (4.70):= D -1 0 (αk)C 0 (αk + 1)D -1 0 (αk + 1)C 0 (αk + 2) • • • D -1 0 (α -1)C 0 (α).

4.2. 3 .

 3 Polynomial solutions. Requiring the f j 's to be polynomials of given degree n ≥ 1 is equivalent to requiring that A n+1 = 0 (4.60). From (4.58), we getD n+1 (α)A n+1 = C n A n = 0, i.e., [2n -(η 1 + 2α + 2q -7)]A n 1 + (η 1 + q -1)A n 2 = 0, (4.71) [2n -(2α + q -8)]A n 2 = 0.(4.72)

  [START_REF] Duplantier | Liouville Quantum Gravity and KPZ[END_REF][START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF][START_REF] Duplantier | Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation[END_REF]. Although the LQG method, which originates in theoretical physics, is heuristic and not fully rigorous, it often offers the quickest and most natural path to the derivation of scaling exponents and multifractal spectra. It is also intimately related to the recently developed and rigorous wedge-welding theory in Liouville quantum gravity[START_REF] Sheffield | Conformal weldings of random surfaces: SLE and the quantum gravity zipper[END_REF][START_REF] Duplantier | Liouville Quantum Gravity as a Mating of Trees[END_REF] (See in particular Appendix B in[START_REF] Duplantier | Liouville Quantum Gravity as a Mating of Trees[END_REF] for a mathematically precise description of the KPZ interpretation.)

	2.4.2. Derivation of Claim 2.1. Let us first recall that in the original work on whole-plane
	SLE [DNNZ15], the novel integral means spectrum, β 1 (p; κ) := β 1 (p; q = 0; κ, a = 0),
	derived there for p ∈ R, was related to some Liouville quantum gravity results obtained in [Dup04]. (See [DNNZ15, Section 1.3].) It was found that the related packing spectrum,
	defined as,
	s 1 (p; κ) := β 1 (p; κ) -p + 1,	(2.20)
	is given by
					.
	This happens for
	a 2 κ 2 ≥ ã0 (κ) :=	1 8	8 + κ κ -4	, κ > 4.	(2.19)
	2.4. Derivation of Claims 2.1 and 2.2.
	2.4.1. Discourse on the Method. We are going to use here a Liouville quantum gravity
	(LQG) approach, which historically gave the first derivation of the standard SLE multi-
	fractal spectrum [Dup00], which was later confirmed by a standard mathematical approach

This phenomenon thus occurs only for non-simple SLE κ>4 curves, and for a sufficiently strong drift term a. Otherwise, one is in the configuration of Figs. 7, 8, and 9 for the second bisector, and the β lin spectrum does not appear in the standard ims of the interior whole-plane SLE with drift, which takes the successive forms β tip , β 0 , β 1 .

Remark 2.2. The two conditions on the reduced drift parameter, a/κ ≥ a 0 (κ), in fact obey SLE duality

[START_REF] Duplantier | Conformally invariant fractals and potential theory[END_REF][START_REF] Duplantier | Conformal fractal geometry & boundary quantum gravity[END_REF][START_REF] Duplantier | Conformal Random Geometry[END_REF][START_REF] Zhan | Duality of chordal SLE[END_REF][START_REF] Dubédat | Duality of Schramm-Loewner evolutions[END_REF]

. Defining the dual SLE parameter κ = 16/κ, with κ > 4 and κ < 4, one checks that ã0 (κ ) = ã0 (16/κ) = a 0 (κ). The occurence here of this reduced drift parameter a/κ may seem natural, if one recalls that the quadratic variation of √ κB t + at is κt and its mean at. [BS09, BDZ17, GMS18]. It is based on the celebrated Knizhnik-Polyakov-Zamolodchikov (KPZ) relation

[START_REF] Knizhnik | Fractal Structure of 2D-quantum gravity[END_REF][START_REF] David | Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge[END_REF][START_REF] Distler | Conformal Field Theory and 2D Quantum Gravity[END_REF]

between scaling exponents in the Euclidean plane, and their counterparts under a random LQG measure that gives the scaling limit of the area measure on a random planar map. The KPZ relation is now mathematically proved [

  its imaginary part t := p corresponds to exponentially weighting by exp( t arg C) the mutually-avoiding SLE-Brownian path configurations C, with local winding angle arg C around the tip. One can then show by Coulomb gas arguments [DB02, Dup04, DB08] that the new scaling exponent associated with the tip is

  .8) Lemma 3.3. For f t as defined in (3.3) and ft as defined in (3.8), we have the equivalence in law, lim

	t→+∞	e t	ft (z)

Acknowledgements: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while Bertrand Duplantier participated in the program "Analysis and Geometry of Random Spaces", hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2022 semester. The work by Yong Han is supported by the National Natural Science Foundation of China under Grant No. 12131016. B.D. also wishes to warmly thank Emmanuel Guitter for his help with Mathematica® and the figures, and Thomas C. Halsey for a critical reading of the manuscript.

Remark 3.2. The two 'physical' integral means spectra β - 0 (3.40) and β + 1 (2.10) overlap along the red parabola R (3.34) in the interval α ∈ [1/κ, 2/κ + 1/2], a result which can be directly compared to [DHLZ18, Eqs. (93)-( 95)].

This corresponds to the presence of a two-dimensional "overlap ribbon" on the red parabola, where the complex generalized integral means spectrum takes both the β - 0 and β + 1 forms. In the 4-dimensional (p, q) space, there exists a larger phase-transition manifold, that is defined by the single condition that these two spectra are equal. This threedimensional manifold must intersect the above overlap ribbon on a certain phase-transition line. The study of such phase-transition manifolds is left to a future work.

GENERAL L ÉVY PROCESSES WITH SPECIAL SYMBOLS

In this section, we generalize the results in [START_REF] Duplantier | Logarithmic coefficients and generalized multifractality of whole-plane SLE[END_REF], [START_REF] Duplantier | The coefficient problem and multifractality of whole-plane SLE & LLE[END_REF], [START_REF] Loutsenko | SLE κ :correlation functions in the coefficient problem[END_REF], [START_REF] Loutsenko | Average harmonic spectrum of the whole-plane SLE[END_REF], [START_REF] Loutsenko | New exact results in spectra of stochastic Loewner evolution[END_REF] and [START_REF] Loutsenko | Stochastic Loewner evolutions, Fuchsian systems and orthogonal polynomials[END_REF] to the generalized integral means spectrum: in other words, we investigate the values of (p, q) for which the generalized integral means spectrum for Lévy-Loewner evolution has an exact form.

For this purpose, we assume in this section that G(z, z) (3.5) may be written as

where h(z, z) is doubly analytic in z, z and satisfies the boundary condition h(0, 0) = 1.

By applying (3.20), we get

The coefficient of h = h(z, z) in this equation is the sum of a polynomial in z, z, and of the polar part,

. The latter clearly becomes pole free, i.e., a polynomial in z, z if and only if p = p = 2, which we shall hereafter assume. Under this condition, the above equation becomes

Besides the restriction to p = 2, we shall also assume that q ∈ R and that the Lévy process L t is symmetric. We then get

In order to analyze this equation, we use the Fourier expansion of t → h(re it , re -it ):

is the intersection point of the boundary of D 4-q with the green ellipse (a single tangency point).

4.2. The η 2 = -q case. Here the points (q, η 1 ) must belong to

In this case, the first three equations (4.4) together with (4.5) form a system of coupled ODEs with unknowns θ j ,j = 0, 1, 2,

Let us now consider for n ≥ 0 the following Ansatz,

Consider then in each left-hand side of the three equations [1], [2], [3] in (4.55), the contributions arising for a fixed k from the monomials A k j P k in f j , j = 0, 1, 2 (4.54). Because of the universal presence of factors (ξ -1) or ξ(ξ -1) in front of derivatives P k (ξ) = -kP k-1 (ξ), and of polynomials of degree at most 1 in ξ in front of P k (ξ), only P k and P k+1 monomials will result from P k . We explicitly find for the three lines the resulting contributions,

(4.56)

When summing up over k ∈ {0, • • • , n} to reconstruct the f j 's in (4.55), each monomial P k must get an overall vanishing coefficient in order to satisfy the equations. Therefore, collecting all coefficients of terms P k , we are led to the recursions,

(4.57)

Note that in the k = 0 case, there are no A k-1 1 and A k-1 2 terms, which are thus set by convention equal to 0.

If 2n -(2α + q -8) = 0, then A n 2 = 0, and for a non-vanishing solution to exist, one needs the condition 2n -(η 1 + 2α + 2q -7) = 0 to hold for α = α ± 0 . This gives ± Ẑ = 2n -1, with n ≥ 1. This selects the (+)-branch α = α + 0 , together with Ẑ = (2n -1) 2 , n ≥ 1.

(4.73)

Because of (4.64), one thus finds from (4.73) a set of ellipses Ên in the (q, η 1 ) plane, satisfying the equation

It is interesting to note that it is far from obvious that condition (4.73), while necessary, is also sufficient to obtain that A n 2 = 0 at level n of recursion (4.67), when starting from eigenvector A 0 (α) (4.66) for α = α + 0 . We checked with Mathematica® that this is indeed the case, but despite repeated attempts, a combinatorial-like proof has eluded us.

Remark 4.2. Degeneracy of D k (α). Let us finally consider the degenerate case when

This gives Ẑ = k 2 , and since Ẑ = (2n -1) 2 , we get k = 2n -1. Since the recursion stops at level n+1, we are only interested in the cases where k ≤ n+1, hence n ≤ 2. When n = 1, a direct computation gives the solution at level k = n = 1 with the necessary condition A 1 2 = 0, as A 1 1 = -1 4 (η 1 + q -1)A 0 2 , and

would also provide a solution to the recursion at level k = 1, but not its closure at level n + 1 = 2. Taking into account the boundary condition θ 0 (0) = f 0 (0) = A 0 0 + A 1 0 = 1 yields the explicit solution on ellipse Ê1 ,

When n = 2, we get k = n + 1 = 3. In that case, A 3 = 0 is a trivial solution yielding A k = 0, ∀k ≥ 3, and while adding to it any null vector of D 3 (α + 0 ) would still satisfy the recursion at level 3, it would not close the latter at next levels.

• Case α + 0k = 1. From (4.62) and (4.73), one finds η 1 = 5 + 2(nk) -2q, and one has to consider the intersection in D -q of this straight line with ellipse Ên . One finds two solutions, q ± n,k :

As before, we are only interested in recursion levels 1 ≤ k ≤ n + 1, so that ∆ ≥ 0. One also has ∆ = (nk + 2) 2 + 2(k -2)(2n + 1k), so that for k ≥ 2, there is one admissible root, q - n,k ≤ 0, whereas q + n,k ≥ 0, and for k = 1, q - n,1 = 2, q + n,1 = 2n which are not in D -q . By continuity, at points q - n,k with 2 ≤ k ≤ n + 1 on Ên , the generalized integral means spectrum is still β(2, q) = α + 0 .

Proof. Because of (4.79), the generalized integral means spectrum must be equal to one of the three eigenvalues α + 0 , α - 0 , 1. Eigenvalues α + 0 and α - 0 are equal only when Ẑ = 0, i.e., on the green ellipse which lies outside D -q , except for the point P 0 = (q = 0, η 1 = 2). Therefore in D * -q := D -q \ P 0 , we have α + 0 > α - 0 . One can also check that on D -q , α + 0 > 1, whereas the equality α - 0 = 1 is realized in D -q on the branch of hyperbola of equation (η 1 + q -8)(q -4) = 4 with q ≤ 0. We know that β(2, q) = 3q = α + 0 on the half-line η 1 = 3q, q ≤ 0, as well as β(2, q) = 4q = α + 0 on the half-line η 1 = 1q, q ≤ 0. Because of the Hölder inequality, the generalized integral means spectrum is convex in (p, q) [DHLZ18], hence continuous. By continuity, β(2, q) = α + 0 on these integrability lines cannot jump to α - 0 < α + 0 or 1 < α + 0 in D * -q , hence β(2, q) = α + 0 in the whole domain. At the singular point P 0 , the spectrum is still α + 0 , with a change of its analytic form.