Raphaël Butez
email: butez@ceremade.dauphine.fr

Large deviations principle for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distribution

This note provides a large deviations principle for a class of biorthogonal ensembles. We extend the results of Eichelsbacher, Sommerauer and Stotlz to more general type of interactions. Our result covers the case of the singular values of lower triangular random matrices with independent entries introduced by Cheliotis. In particular, we obtain as a consequence a variational formulation for the Dykema-Haagerup as it is the limit law for the singular values of lower triangular matrices with i.i.d. complex Gaussian entries.

Introduction and results

The aim of this note is to extend the work of Eichelbascher, Sommerauer and Stolz [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF] in order to prove a large deviations principle for a wide class of biorthgonal ensembles which include the matrix models introduced by Cheliotis in [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF]. The authors of [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF] proved a large deviations principle for wide variety of models, such as the biorthogonal Laguerre ensembles or the matrix model of Lueck, Sommers and Zirnbauer [START_REF] Lueck | Energy correlations for a random matrix model of disordered bosons[END_REF] for disordered bosons. Those models deal with particle systems in R or C with a density involving a double interaction term of type i<j |x i -x j ||x θ i -x θ j | with θ ∈ N * . Biorthogonal ensembles were introduced by Muttalib in physics in [Mut95] and by Borodin in mathematics in [START_REF] Borodin | Biorthogonal ensembles[END_REF]. The recent article [START_REF] Bloom | Modified logarithmic potential theory and applications[END_REF] develops potential theory for the model we study. Large deviations for particles systems with general repulsion have been studied in [START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF] and we show that their results apply to this kind of problems.

In the article [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF], Cheliotis presented a lower-triangular random matrix model for which the distribution of the singular values can be computed and form a class of biorthogonal ensembles. Later, in [START_REF] Forrester | Muttalib-borodin ensembles in random matrix theory-realisations and correlation functions[END_REF], Forrester and Wang found another matrix model for these ensembles. Large deviations principles for the empirical measures of biorthogonal ensembles enter the general framework of [START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF], but it is not clear that this model fits their technical hypotheses.

Triangular matrices are the elementary object that appear in many factorization algorithms, such at the Cholesky or the LU decomposition, so one could wonder if, starting from a random matrix, we can compute the distribution of the coefficients of it's Cholesky decomposition. Bartlett answered that question in [START_REF] Stevenson | On the theory of statistical regression[END_REF] and proved that the entries of the Cholesky decomposition of a Wishart random matrix are independent Gaussian variables off diagonal and chi random variables on the diagonal. This result is known as the Bartlett decomposition of a Wishart matrix. Cheliotis studied the reverse problem: given a simple model of random triangular matrices T n , what can we say about the eigenvalues of the eigenvalues of T n T * n ? Fix a positive integer n ∈ N, and two parameters b > 0 and θ ≥ 0, we consider the random lower triangular matrix T n = (X i,j ) 1≤i,j≤n with independent random coefficients X i,j distributed according to:

X i,j ∼    N C (0, 1) if i > j, 1 πΓ(c j ) e -|z| 2 |z| 2(c j -1) d C (z) if i = j.
where c j = θ(j -1) + b and d C is the Lebesgue measure on the complex plane. Note that when θ equals 0 and b equals 1, the non-zero entries are i.i.d. complex Gaussians.

In the article [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF], Cheliotis was able to compute the distribution of the ordered eigenvalues of the matrices

S n = T n T * n . He proved that, if we write λ 1 ≥ • • • ≥ λ n the eigenvalues of S n , the distribution of the random vector Λ n = (λ 1 , . . . , λ n )
is absolutely continuous with respect to the Lebesgue measure on R n with density:

1 n j=1 j! θ -n(n-1)/2 n k=1 Γ(c k ) e -n i=1 x i n j=1 x b-1 j i<j (x i -x j )(x θ i -x θ j )1 x 1 >•••>xn>0 (1)
when θ > 0. When θ = 0, the density of the distribution of Λ n is:

1 n j=1 j! 1 n k=1 Γ(c k ) e -n i=1 x i n j=1 x b-1 j i<j (x i -x j )(log x i -log x j )1 x 1 >•••>xn>0 . (2)
We notice that for good choices of θ and b, we can recover many classical ensembles, such as the Laguerre ensembles.

In the rest of this note, we are interested in the eigenvalues of 1 n S n . The factor 1/n is the proper scaling to observe a convergence of the empirical measure. We will keep the notation λ 1 , . . . , λ n for the eigenvalues of 1 n S n and we define its empirical measure:

µ n = 1 n n i=1 δ λ i .
The special case where θ = 0 and b = 1 corresponds to the case where all the coefficients are X i,j are independent complex random variables with variance 1 is of particular interest. In [START_REF] Dykema | Dt-operators and decomposability of voiculescu's circular operator[END_REF], using free probability theory, Dykema and Haagerup proved that (µ n ) n∈N * converges weakly in probability towards a deterministic measure, known as the Dykema-Haagerup distribution. In [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF], the same result is proved using the moments method and path counting. The Dykema-Haagerup distribution µ DH is compactly supported and absolutely continuous with respect to the Lebesgue measure on R + * with density: where W 0 is the Lambert function. W 0 is analytic in C \ (-∞, -e -1 ] and can be extended to C so that it is continuous on the upper half plane, see figure 1.

f DH (x) = 1 π Im - 1 xW 0 (x) 1 [0,e]
The moments of the Dykema-Haagerup distribution are given by: R

x k dµ DH (x) = k k (k + 1)! .
The Stieljes transform of µ DH is defined for all z ∈ C with Im(z) > and is given by:

S(z) = - 1 zW 0 (1/z) = -1 + e W 0 (-1/z) .
The R-transform of µ DH is defined for all z ∈ C such that |z| < 1 and is given by:

R(z) = - 1 (1 -z) log(1 -z) - 1 z .
In [START_REF] Dykema | Dt-operators and decomposability of voiculescu's circular operator[END_REF], Dykema and Haagerup proved, using free probability tools, that the coefficients of T n are i.i.d. complex Gaussians, the empirical measures (µ n ) n∈N converge weakly in probability towards a deterministic measure, called the Dykema-Haagerup distribution. Independently, Cheliotis proved the same result in [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF] using the moments method and path counting. This corresponds to the choice of b = 1 and θ = 0 in our model.

In this note, the term "weak topology" corresponds to the topology associated to continuous and bounded test functions. The Bounded Lipschitz metric d defined as:

∀µ, ν ∈ M 1 (R + ) d(µ, ν) = sup f f µ -f dν
where the supremum is taken over functions bounded by 1 and 1-Lipschitz metricizes the weak topology and makes M 1 (R + ) a complete space, see [Bog07, Section 8.3]. Definition 1.1 (Logarithmic energy). The logarithmic energy is the functional :

E : M 1 (R + ) -→ R ∪ {∞} µ -→ -log |x -y|dµ(x)dµ(y).
We also define the off-diagonal logarithmic energy

E = : M 1 (R + ) -→ R ∪ {∞} µ -→ = -log |x -y|dµ(x)dµ(y).
where we integrate on the complement of the diagonal of (R + ) 2 .

We define the confining potential associated to the eigenvalue distribution (1) and (2). Let V : R → R given by:

V (x) = x.
As the eigenvalues of 1 n S n are the eigenvalues of S n divided by n, we can compute the distribution of the unordered eigenvalues (λ 1 , . . . , λ n ). This distribution is absolutely continuous with respect to the Lebesgue measure on R n with density:

1 Z n e -n n i=1 x i n j=1 x b-1 j i<j |x i -x j ||x θ i -x θ j | when θ > 0 1 Z n e -n n i=1 x i n j=1 x b-1 j i<j |x i -x j || log x i -log x j | when θ = 0
where Z n is a normalizing constant, depending on the model. Those two distributions are of the form:

1 Z n e -n n i=1 V (x i ) n j=1 x b-1 j i<j |x i -x j ||g(x i ) -g(x j )| (3)
if we chose g = g θ where:

g θ (x) = x θ if θ > 0 log x else.
This density can be written in the form:

1 Z n exp -n 2 1 2 E = (µ n ) + 1 2 E = (g * µ n ) + V (x)dµ n (x) n j=1 x b-1 j (4)
where Z n is a normalizing constant and where g * µ is the push-forward of the measure µ by the function g. The term n j=1 x b-1 j will play no role in the large deviations and the same results are valid without this term. We keep this term so that the connection with the model of random matrices introduced by Cheliotis is straightforward. To recover every Laguerre ensemble, one can consider "b = bn", which would correspond to change the function V . Theorem 1.2 (Large deviations principle for µ n ). Let g be a C 1 function on R + * , such that its derivative is positive. Let V be a continuous function on R + such that there exist a constant β > b such that we have:

lim x→∞ V (x) β log |x| > 1 and lim x→∞ V (x) β log |g(x)| > 1.
Let us define I : M 1 (R + ) → R ∪ {∞} given by:

I(µ) = 1 2 E(µ) + 1 2 E(g * µ) + V (x)dµ(x)
The random sequence (µ n ) n∈N satisfies a large deviations principle with speed n 2 in M 1 (R) for the weak topology with good rate function Ĩ = I -inf I. This means that for any Borel set A ∈ M 1 (R + ) we have:

-inf IntA Ĩ ≤ lim 1 n 2 log P(µ n ∈ A) ≤ lim sup 1 n 2 log P(µ n ∈ A) ≤ -inf CloA Ĩ
In addition, the rate function I -inf I is lower semi-continuous and strictly convex on the set of the measures on which it is finite.

Remark 1.3 (Assumptions on g and V .). The assumptions on g mean that the two interaction terms play the same role of short range repulsion, but at different scales. Our hypothesis on g can be rephrased as "g is locally a

C 1 -diffeomorphism of RR + * .
The assumptions on V are very standard in large deviations for Coulomb gases. They ensure that e -V (x) dx is finite and that that rate function is well defined.

In [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF], Eichelsbacher, Sommerauer and Stolz proved a large deviations principle for the empirical measures µ n when g = g θ and θ is an integer and where V can depend on n. The classical techniques to prove large deviations for the empirical measures of Coulomb gases apply here with no modification.

The novelty of our approach is to extend the result of [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF] to any function g. Our theorem covers the original model of Muttalib from [Mut95] with g(x) = Argsh 2 ( √ x) which was the starting point of the study of biorthogonal ensembles. The matrix model introduced by Cheliotis corresponds to the choice of g = g θ where θ > 0. Choosing g(x) = exp(x) gives the large deviations for the model of [START_REF] Claeys | Random matrices with equispaced external source[END_REF]. The key of this article is the way we deal with the lower bound. Instead of inspiring from the proof of the lower bound originally given by Ben Arous and Guionnet in [START_REF] Ben | Large deviations for Wigner's law and Voiculescu's non-commutative entropy[END_REF], we adapt the proof of Hiai and Petz from [START_REF] Hiai | The semicircle law, free random variables and entropy[END_REF]. We show that the article [CGZ14] covers a wide class of biorthogonal ensembles, which did not seem obvious. The authors of [START_REF] Bloom | Modified logarithmic potential theory and applications[END_REF] consider a very close model as the study holomorphic functions g while the density (3) is integrated with respect to more general measures on C or R. Our techniques rely on the classical probabilistic approach of large deviations while they adopt a Bernstein-Markov approach.

From this result we obtain two important corollaries, which are the motivation for our study: a variational formulation for the Dykema-Haagerup distribution and the almost sure convergence of (µ n ) n∈N towards this measure. We also state a large deviations principle for the top right particle.

Corollary 1.4 (Almost sure convergence towards the minimizer). Let g be a C 1 function on R + * , such that its derivative is positive. Let ν be the unique minimizer of the functional I. Then the random sequence of measures (µ n ) n∈N converges weakly almost surely towards the deterministic measure ν.

Corollary 1.5 (Variational characterization of the Dykema-Haagerup Distribution). The Dykema-Haagerup distribution µ DH is the unique minimizer on M 1 (R + ) of the functional :

I(µ) = 1 2 E(µ) + 1 2 E(log * µ) + xdµ(x)
which is strictly convex.

Theorem 1.6 (Large deviations for the largest particle). Let (x 1 , . . . , x n ) be distributed according to (3) and let x * n = max 1≤i≤n x i . Suppose that the hypotheses of Theorem 1.2 are satisfied and assume that there exist a constant ζ such that:

lim n→∞ 1 n log Z * n-1 Z n = ζ
where Z * n-1 is the normalizing constant of the gas (3) with n -1 particles and confining potential n n-1 V . Let µ eq be the limit measure of (µ n ) n∈N * and let b e q be the right endpoint of its support. The random sequence (x * n ) n∈N * satisfies a large deviations principle in R + with speed n and good rate function:

J(x) =   -1 2 log |x -y| + log |g(x) -g(y)|dµ eq (y) + V (x) -κ if x ≥ b eq ∞ if x < b eq .
where κ is such that J(b eq ) = 0. This theorem will not be proved in this note as the authors of [START_REF] Credner | Large deviations for the largest eigenvalue of disordered bosons and disordered fermionic systems[END_REF] already proved this theorem for the model of [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF]. In the setting of [START_REF] Credner | Large deviations for the largest eigenvalue of disordered bosons and disordered fermionic systems[END_REF], the number of particles at step n is not n but p(n) which makes their result more technical. One could also adapt the proof of the similar theorem from [AGZ10] as the scheme of the proof is the same. First, the product structure of the density (3) allows us to separate the variables and integrate with respect to x 1 < • • • < x n-1 . Then, the large deviations principle for the empirical measure allows us to says that the particles x 1 < • • • < x n-1 generate the same potential as the measure µ e q. Finally, the assumption on the normalizing constants allows us to control the error we do by changing the measure from n particles to n -1 particles.

Remark 1.7 (Large deviations for the top eigenvalue for Cheliotis' matrix model). In the article [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF], Cheliotis gives exact formulas for the normalizing constants Z n when g = g θ and V (x) = x. It is straightforward to check that this model satisfies the assumptions of Theorem 1.6. One can also obtain another proof of the fact that for the Dykema-Haagerup model, λ max converges almost surely towards e.

The rest of the note is devoted to the proofs of the theorems. We start by proving the large deviations principles and then we deduce the variational formula and the almost sure convergence.

Proof of the large deviations principle.

The proof of Theorem 1.2 is very close to the standard proof of large deviations principle for Coulomb gases in R. Many authors proved similar results following the steps of [START_REF] Ben | Large deviations for Wigner's law and Voiculescu's non-commutative entropy[END_REF]. For general b and positive integer θ, theorem 1.2 is a special case of the article [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF]. The proof is organized in several classical steps:

Not much is new in the proof that we present here, hence we will focus on what differs from the usual techniques. The parts of the proof that are omitted can be taken from [START_REF] Anderson | An introduction to random matrices[END_REF] or [START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF]. The classical proof is organized as follows:

Step 1: Study of the rate function;

Step 2: Exponential tightness for the non-normalized measures;

Step 3: Weak upper bound for the non-normalized measures;

Step 4: Weak lower bound for the non-normalized measures;

Step 5: Recover the full large deviations principle for the normalized measures.

We will give the fundamental inequality to prove step 1. Then, the classical proofs of step 2 and 3 apply with no modification. We will give full details about step 4 as it is the difficult part of the proof. Once the large deviations principle is proved for the nonnormalized measures, step 5 just consists in obtaining the asymptotic of the normalizing constants by applying the large deviations inequalities for the whole space of probability measure.

Study of the rate function.

Definition 2.1. We set, for any non-negative x and y:

f (x, y) = - 1 2 log |x -y| - 1 2 log |g(x) -g(y)| + 1 2 [V (x) + V (y)]
Using the inequality:

log |x -y| ≤ log(1 + |x|) + log(1 + |y|)
we obtain:

f (x, y) ≥ - 1 2 log(1 + |x|) - 1 2 log(1 + |g(x)|) + 1 2 V (x) + (5) - 1 2 log(1 + |y|) - 1 2 log(1 + |g(y)|) + 1 2 V (y) . ( 6
)
This inequality shows that the function I is well defined and taxes its values in R∪{∞}. This inequality is the key to prove that I is a good rate function. All the details are given in the reference book [AGZ10, Lemma 2.6.2 p.72].

To prove that the rate function I is strictly convex where it is finite, we observe that the logarithmic energy µ → E(µ) is known to be a strictly convex function where it is finite, see [START_REF] Anderson | An introduction to random matrices[END_REF] or [START_REF] Deift | Orthogonal polynomials and random matrices: a Riemann-Hilbert approach[END_REF]. As the function µ → g * µ is linear, the function µ → E(g * µ) is strictly convex where it is finite. The rate function I is the sum of two strictly convex functions and a linear function, hence it is strictly convex on the set {µ ∈ M 1 (R + ) | I < ∞}. The exponential tightness is also a consequence of the inequality (5), see for instance [AGZ10, p.77].

To prove the upper bound for non-normalized measures, the strategy of the proof is exactly the same as in [START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF]. This proof applies with no modification.

Proof of the lower bound.

The proof of the lower bound from [START_REF] Eichelsbacher | Large deviations for disordered bosons and multiple orthogonal polynomial ensembles[END_REF] does not seem to cover the case where g is not an integer power function. The classical scheme of proof from Ben Arous and Guionnet for the lower bound does not suit well for biorthogonal ensembles. We show that scheme of proof of [START_REF] Hiai | The semicircle law, free random variables and entropy[END_REF] for the lower bound is more robust and allows to deal with more general types of interactions.

We want to prove that we have, for any σ ∈ M 1 (R + ):

lim δ→0 lim n→∞ 1 n 2 log Z n P(µ n ∈ B(σ, δ)) ≥ -I(σ). ( 7
)
The classical technique consists in constructing configurations for which the density of (1) is very close to exp(-I(σ)). It is important to check that the measure of the configurations we created in R n does not decay too fast. Unfortunately, it is not easy to do it for general measure σ. We notice that it suffices to prove the bound for sufficiently regular measures σ.

First

Step: Reduction to "nice" measures.

We will prove that for any sufficiently regular measure σ, we have:

inf G lim n→∞ 1 n 2 log Z n P(µ n ∈ G) ≥ -I(σ) (8)
where the infimum is take over G neighborhood of σ. In order to prove that this bound is sufficient to obtain the lower bound of the large deviations principle, we prove that the function φ : M 1 (R + ) → R given by:

φ(σ) = inf{ lim n→∞ 1 n 2 log Z n P(µ n ∈ G), G neighborhood of σ} is upper semi-continuous. Let σ k → σ in M 1 (R + ).
Let G be a neighborhood of σ, then there exists an integer K such that for all k ≥ K, σ k ∈ G. This implies that for any k ≥ K:

inf G k lim n→∞ 1 n 2 log P(µ n ∈ G k ) ≤ lim n→∞ 1 n 2 log P(µ n ∈ G)
where G k are neighborhoods of σ k . Then if we take the limit superior of this inequality and the infimum over G neighborhood of σ we obtain the upper semi-continuity of φ. If we prove (8) for a dense set of measures, then for any measure σ ∈ M 1 (R + ), there exist measures σ k such that (8) holds and σ k → σ we get:

φ(σ) ≥ lim sup k φ(σ k ) ≥ lim sup k -I(σ k ).
We will consider a specific sequence of measures σ k such that for any k, σ k is absolutely continuous with respect to the Lebesgue measure on R + , with compact support in R + * and density bounded from above and below by positive constants and such that:

-E(σ k ) ≥ -E(σ) lim k→∞ V (x)dσ k (x) = V (x)dσ(x).
Once we obtain this sequence, we will only have to prove the lower bound for the measures satisfying the regularity conditions given above.

Let

σ k = 1 1/k≤x≤k σ([1/k,k]
) σ, then, as f is bounded from below, by the monotone convergence theorem we get:

lim k→∞ f (x, y)dσ k (x)dσ k (y) = f (x, y)dσ(x)dσ(y)
so we can assume that σ has compact support in R + * . Now let φ ε be a C ∞ probability density with support in [0, ε], then we set σ ε = φ ε * σ. The measures σ ε have compact support in R + * with continuous density and converge towards σ as ε goes to zero. Since it is easy to check that V (x)dσ ε (x) ---→ ε→0 V (x)dσ(x), we only have to prove that for any ε -E(φ ε * σ) ≥ -E(σ).

Recall that the function -E is concave, so if we notice that

φ ε * σ = φ ε (y)σ(• -y)dy
then, thanks to the Jensen inequality and the invariance by translation of the logarithmic energy, we obtained the desired inequality. The last thing we want for our "nice" measures is that the density is bounded from above and from below. As the density of the measures σ ε are continuous with compact support, those densities are already bounded from above.

Changing σ ε to δm + (1 -δ)σ ε where m is the uniform measure on the support of σ ε allows us to deal with measures with continuous density bounded from above and from below.

Second step: lower bound for "nice" measures.

From now, σ will be a measure with compact support [a, b] ⊂ R + * , with density h with respect to the Lebesgue measure on R + for which there exist a constant C > 0 such that

∀x ∈ [a, b] , 1 C ≤ h(x) ≤ C.
Let a 0 , . . . , a n be the 1 n -quantiles of σ, with a 0 = a and a n = b. We have that for any k,

1 Cn ≤ a k+1 -a k ≤ C n . ( 9
) Now divide each interval [a k-1 , a k ] in 3 equal parts and let [c k , d k ] be the central interval. If we set ∆ n = n i=1 [c i , d i ],
then for any (z 1 , . . . , z n ) ∈ ∆ n , we have:

d( 1 n n i=1 δ z i , σ) ≤ max k |a k+1 -a k | ≤ C n
where d is the bounded-Lipschitz distance. We are now ready to prove the lower bound. Let ρ 1 be the finite measure on R + x b-1 e -V (x) dx and ρ n = ρ 1 ⊗ • • • ⊗ ρ 1 the finite n-th product measure on (R + ) n .

Z n P(µ n ∈ B(σ, δ)) = 1 µn∈B(σ,δ) exp -n 2 1 2 E = (µ n ) + 1 2 E = (g * µ n ) + n -1 n V (x)dµ n (x) dρ n (x) ≥ 1 ∆n exp -n 2 1 2 E = (µ n ) + 1 2 E = (g * µ n ) + n -1 n V (x)dµ n (x) dρ n (x) ≥ exp -n 2 n -1 n 2 n k=1 max [c i ,d i ] V (x) × exp   -n 2   - 1 n 2 i<j min [c i ,d i ]×[c j ,d j ] log |x -y|     × exp   -n 2   - 1 n 2 i<j min [c i ,d i ]×[c j ,d j ] log |g(x) -g(y)|     1 ∆n dρ n (x).
We notice that:

1 n 2 log 1 ∆n dρ n (x) ---→ n→∞ 0.
Hence, to obtain the lower bound, it is sufficient to prove that we have:

lim n→∞ 1 n n k=1 max [c i ,d i ] V = V (x)dσ(x), (10)
and, using the fact that the functions logarithm and g are increasing:

lim n→∞ 1 n 2 i<j -log(d j -c i ) ≥ 1 2 -log |x -y|dσ(x)dσ(y) = 1 2 E(σ) (11)
and also:

lim n→∞ 1 n 2 i<j log(g(d j ) -g(c i )) ≥ 1 2 log |g(x) -g(y)|dσ(x)dσ(y) = 1 2 E(g * σ). ( 12
)
If we admit temporarily the inequalities (10), ( 11) and ( 12), the proof of the lower bound for regular measures is completed. The last step will consist in proving those three inequalities.

Last step: Proof of the inequalities.

First, (10) is easy to check as we approximate a continuous integrable function on [a, b] by simple functions.

We now prove (11) following the proof of [START_REF] Hiai | The semicircle law, free random variables and entropy[END_REF]. We admit temporarily that there exist a constant A > 0 such that for i < j:

A(d j -c i ) ≥ (a j -a i-1 ) (13)
and also that:

lim n→∞ 2 n 2 #{i < j | (a j -a i-1 ) (d j -c i ) ≤ 1 + ε} = 1. (14)
We postpone the proof of the inequalities ( 13) and ( 14) to prove (11). We call:

B n = E(σ) - 2 n 2 i =j min [c i ,d i ]×[c j ,d j ]
log |x -y| and we want to prove that:

lim n→∞ B n ≤ 0 Since log |z -w|dσ(z)dσ(w) ≤ 2 n 2 i<j log |a j -a i-1 | + 1 n 2 n i=1 log |a i -a i-1 |
then for every ε > 0 we have :

B n ≤ 2 n 2 i<j log |a j -a i-1 | - 2 n 2 i<j log |d j -c i | + 1 n 2 n i=1 log |a i -a i-1 | ≤ 2 n 2 #{i < j | (a j -a i-1 ) (d j -c i ) ≤ 1 + ε} log(1 + ε) + 1 n 2 1 - 2 n 2 #{i < j | (a j -a i-1 ) (d j -c i ) ≤ 1 + ε} log A + 1 n 2 n i=1 log |a i -a i-1 |.
Then we take the limit superior in both sides, and the limit when ε → 0

E(σ) -lim n→∞ 2 n 2 i<j log |d j -c i | ≤ 0
which proves (11).

We prove now inequality (13). From inequality (9), we get for any k > 0:

a i+k -a i-1 d i+k -c i ≤ (k + 1)C/n (k + 2/3)/Cn .
We deduce from this inequality that the left part of the inequality is bounded by a constant independent of k and n, which proves (13). In order to prove (14), we start from:

a i+k -a i-1 d i+k -c i = 1 + a i+k -d i+k d i+k -c i + c i -a i-1 d i+k -c i .
If we show that:

a i+k -d i+k d i+k -c i and c i -a i-1 d i+k -c i
can be made as small as desired when k is bigger than a certain constant independent of n, then (14) will be proved. Using (9) we get:

a i+k -d i+k d i+k -c i-1 ≤ C/3n k/Cn and c i -a i-1 d i+k -c i ≤ C/3n k/Cn .
Those two terms can be made as small as desired is k is sufficiently large, independently of n, which proves (14). The proof of the inequality (12) mimics the proof of inequality (11). Like in the previous case, it is sufficient to find a constant A such that for any i < j:

A (g(d j ) -g(c i-1 )) ≥ g(a j ) -g(a i-1 ) (15)
and to prove that:

lim n→∞ 2 n 2 #{i < j | g(a j ) -g(a i-1 ) g(d j ) -g(c i )) ≤ 1 + ε} = 1. (16)
As the support of σ is a compact included in R + * , there exist two constants m and M such that for all

x ∈ [a, b]: m ≤ g (x) ≤ M.
The inequality (15) is a consequence of (13), using the mean value theorem for g and the fact that its derivative is bounded from above and from below. The inequality ( 16) is equivalent to prove that the quantities

g(a i+k ) -g(d i+k ) g(d i+k ) -g(c i ) and g(c i ) -g(a i ) g(d i+k ) -g(c i )
are as small when k is large enough. Using the mean value theorem we get:

g(a i+k ) -g(d i+k ) g(d i+k ) -g(c i ) ≤ M m a i+k -a i-1 d i+k -c i ≤ M m (k + 1)C/n (k + 2/3)/Cn .
The other term is treated in the same way. Now that we have proved (15) and ( 16), the proof of ( 12) is the exactly the same as the proof of (11).

3 Proof of Corollary 1.4 and Corollary 1.5.

As the function I is lower semi-continuous and strictly convex, it has a unique minimizer, called µ b,/theta . Consider the sets:

A ε = M 1 (R + ) \ B(ν, ε).
As I is lower semi-continuous, inf{I ( µ), µ ∈ A ε } > 0, then, thanks to the Borel-Cantelli lemma, we get: d(µ n , ν) ---→ n→∞ 0.

As we already know that when b = 1 and θ = 0 the random sequence (µ n ) n∈N * converges weakly in probability towards the Dykema-Haagerup distribution µ DH . We also know from Corollary 1.4 that (µ n ) n∈N * converges almost surely weakly towards the minimizer of I. Hence we obtain the following characterization of µ DH :

µ DH = inf{ 1 2 E(µ) + 1 2 E(log * µ) + xµ(x), µ ∈ M 1 (R + )}.

Perspectives.

We can extend to any finite number of interactions of type:

i<j |f 1 (x i ) -f 1 (x j )| β 1 i<j |f 2 (x i ) -f 2 (x j )| β 2 • • • i<j |f p (x i ) -f p (x j )| βp
where each of the f k is locally a C 1 diffeomorphism and the β k are positive numbers. Large deviations will be valid if the confining potential V dominates all the functions f k at the same time at infinity. The proof of this result would be similar to the proof of Theorem 1.2. The result of this note can be extended in any dimension if we make additional assumptions on the function g. One could assume that g is continuously differentiable and that on any compact K, there exist a constant m K such that for any x, y ∈ K: g(x) -g(y) ≥ m K x -y . This condition is equivalent to g being locally a C 1 diffeomorphism. The article [START_REF] Bloom | Modified logarithmic potential theory and applications[END_REF] covers the complex case.

The model studied by Götze and Vencker in [START_REF] Götze | Local universality of repulsive particle systems and random matrices[END_REF] is not covered by this note, as they deal with a double interaction term of the type i<j |x i -x j | 2 φ(x i -x j ). This is really the combination of two different interactions whereas our model deals with the usual logarithmic interaction at two different scales. As this model is covered by the study [START_REF] Chafaï | First-order global asymptotics for confined particles with singular pair repulsion[END_REF], one could try to find the optimal conditions of φ so that a large deviations principle is valid.

We would like to thank Dimitris Cheliotis whose work [START_REF] Cheliotis | Triangular random matrices and biorthogonal ensembles[END_REF] is the starting point of this study.

[Mut95] Khandker Abdul Muttalib. Random matrix models with additional interactions. Journal of Physics A: Mathematical and General, 28(5):L159, 1995.

Figure 1 :

 1 Figure 1: Density of the Dykema-Haagerup distribution.


David S Dean

Shamik Gupta
email: shamikg1@gmail.com

Gleb Oshanin

Alberto Rosso

Grégory Schehr

Diffusion in periodic, correlated random forcing landscapes

Keywords: Brownian motion, quenched disorder, anomalous dynamics PACS numbers: 05.40.-a, 02.50.-r, 05.10.Ln

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent ∈ H (0, 1). While the periodicity ensures that the ultimate long time behavior is diffusive, the generalized Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient D L : although one has the typical value

β ∼ - D L exp ( ) L H typ
, we show via an exact analytical approach that the positive moments ( > k 0) scale like ) , and the negative ones as

β 〈 〉 ∼ -′ + D ckL exp [ ( ) ] L k H H 1 (1
β 〈 〉 ∼ ′ - D a kL exp ( ( ) )
Transport in random media is extensively studied due to its practical and fundamental importance [1 3]. In many cases, the dynamics is modelled as a Langevin process, with a drift generated by a quenched disordered potential. In theoretical analysis, the potential landscape is taken to be either infinitely extended or periodic in space. Stochastic dynamics in a periodic potential, both random and deterministic, is commonly encountered in many different contexts, including modulated structures [4], superionic conductors [5], colloids in light fields [6,7], diffusion on regular [8 11] and disordered [12 16] solid surfaces, molecular motors on disordered tracks like DNA/RNA [17 19], and motion in a tilted potential due to a random polymer [20].

Theoretical approaches often assume that the dynamics in a periodic potential reproduces the behavior in an infinitely extended potential. This is implemented by setting the period in the final result, e.g. for the velocity (if any) or the diffusion coefficient, to infinity [21,22]. It is crucial to investigate how far such an assumption holds. Especially in the context of numerical simulations carried out for periodic systems, one may ask how reliably their results may be extrapolated to infinite systems.

In this work, we address these fundamental questions for a Langevin dynamics x(t) in a periodic, quenched random potential V x t ( ( )) (with

+ = V x t L V x t ( ( ) ) ( ( ))): η ξ = - + x t V x x t d d d ( ) d ( ), (1)
with η the friction coefficient, ξ t ( ) a Gaussian white noise with zero mean and correlations

ξ ξ η δ ′ = -′ t t T t t ( ) ( ) 2 (
), the overbar being an average over the noise, and the temperature T is in units of the Boltzmann constant. We consider two cases:

• the ratchet case where V(x) is a fractional Brownian motion (fBm) in time ∈

x L [0, ], with = V (0) 0 and V(L) arbitrary. Thus, V(x) is a Gaussian process with zero mean, 〈 〉 = V x ( ) 0, and variance

- = - ∈ [ ] V x V y V l x y x y L ( ) ( ) ; , [0, ], (2)
H H 2 0 2 2 2
where ∈ H (0, 1) is the Hurst exponent, V 0 and l define respectively the typical amplitude of V(x) and its scale of variation over x. In (2) the angular brackets denote averaging with respect to V(x). Figure 1 (left) shows a realization of V(x), with a jump at x = L.

• the translationally invariant case, where V(x) is a stationary Gaussian process, which at short length scales -≪

x y L | | has the variance (2), and satisfies = V V L (0) ( ), so that all points are statistically equivalent. The particle in this case diffuses on a ring. A realization of such a V(x) is shown in figure 1 (right).

The dynamics (1) involves a combination of two paradigmatic situations: random motion in a periodic potential and the generalized Sinai dynamics in presence of a force = -F x V x x ( ) d ( ) d that is a time-independent stochastic variable with spatial correlations (except for = H 1 2 when V (x) is the trajectory of a Brownian motion itself so that (1) is the periodic Sinai model [23]). While the latter produces an archetypal subdiffusion with logarithmically-confined trajectories, the periodicity of the random potential enforces a long time diffusive behavior with a diffusion coefficient D L . Here, we show that the trade off between these two competing trends results in a rich statistical behavior of D L . In particular, D L is strongly non-self-averaging, with both negative and positive moments exhibiting an anomalous dependence on the temperature, period L and the order of the moment, and being supported by atypical realizations of V (x). For the ratchet case, we obtain exact analytical results, relying on exact bounds, for both positive and negative moments of D L . We also discuss the full form of the probability distribution of D L , and show that it is characterized by a log normal left tail and a highly singular log stable right tail, reminiscent of a Lifshitz singularity. We finally highlight the issue of sample to sample fluctuations of D L . From standard scaling arguments and physical intuition, one expects that our exact results for the ratchet case also hold for the translationally invariant situation, which is harder to analyze analytically. This is confirmed below by thorough numerical simulations [24].

The dynamics (1) in an infinite system for arbitrary H, where > H 1 2 ( < H 1 2) implies positively (negatively) correlated increments and superdiffusive (subdiffusive) V (x), respectively, was discussed in [25] where it was shown that

〈 〉 ∼ →∞ x t t lim ( ) ln ( ) t H 2 2
(see also [26]). In contrast, in a periodic system, the long time motion is diffusive for any given realization of the potential V (x), so that we have the diffusion coefficient

≡ →∞ D x t t lim ( ) (2 ) L t 2
, with D L given by [ 27 31] (see also [7,8,12,32])

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∫ ∫ = β - - D D x L y L d d e , ( 3 )
L L L V x V y (0) 0 0 [ ( ) ( )] 1
where β is the inverse temperature, and

η = D T (0)
. Clearly, D L is a random variable that fluctuates between realizations of V (x), and has support on

D [0, ] (0)
. The inverse of D L may be regarded as a product of partition functions in potentials V (x) and -V x ( ), respectively. The Brownian version of this quantity finds applications in disordered systems and has been extensively studied, while our results for ≠ H 1 2 apply to more general situations (note that the marginal case H = 0, when V (X) is log-correlated, was studied in [33]). The expression in equation ( 3) also describes the ground state energy in a toy model of localization, and its average value was studied in [34] for = H 1 2. Turning to the discussion of the behavior of D L , we first reduce the number of parameters. In the following, we measure L in units of l [see equation (2)], absorb V 0 into β, and measure D L in units of D (0) , so that D L has support on [0, 1] 5 . Now, the typical behavior of D L is easy to estimate as

τ ∝ D L L typ 2
typ , where the dimensionless L sets the scale of an inhomogeneous region, and τ typ defines the typical (dimensionless) time a particle spends in this region. The random potential V (x) being a fBm, the typical height of the potential barrier over a length L scales like L H , for both the ratchet and the translationally invariant case. Assuming Arrhenius type activation, one expects τ

∼ β e L typ H , which implies β ∼ - ( ) D L L exp . ( 4
) L H typ 2
The average behavior of D L is a much more delicate question because, as can be seen from equation ( 3), computing the statistics of D L is a highly non-trivial task that involves the study of an exponential functional of fBm, for which standard methods like the Feynman Kac formula are of little use for ≠ H 1 2. Let us first summarize our main analytical results obtained for the ratchet case: We find that the average of the logarithm of D L is given, to leading order in L, by 5 l V , 0 and D (0) may be made explicit in the final result by the replacements

→ L L l, β β → V 0 and → D D D (0) . β ∝ - D m L ln 2 , ( 5
) L H with = 〈 〉 ∈ m Vs max ( ) s [0,1]
. The result in equation ( 5) is consistent with the logarithmic growth of the disorder averaged mean square displacement in an infinite system:

〈 〉 ∼ →∞ x t t lim ( ) ln ( ) t H 2 2
, and the argument leading to equation (4). Further on, we obtain sharp bounds for the positive moments > k ( 0) of the random variable D L :

⩽ ⩽ A L D B L ( ) ( ), ( 6
) k L k k
where, in the limit → ∞ L , the bounds satisfy

⎜ ⎟ ⎜ ⎟ ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ ⎥ β β = -+ = -+ + + + + A L H k C H L B L H k c H L ( ) exp (1 ) (2 ) , ( ) exp (1 )
( ) , ( 7
) k H H H k H H H 1 1 1 1 1 1 with constants c and C, < ⩽ < ∞ c C 0
, being independent of L, k and β. Both bounds exhibit the same dependence on β L k , , , from which we infer that the exact asymptotic result has the same functional form. Finally, for the negative moments of D L , we find

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ β ∼ - D a k L exp 4 , ( 8 )
L k H 2 2 2 2
where a is a constant (independent of L k , and β). Our exact results, equations ( 6) and ( 8), thus show that the positive and negative moments are dominated by atypical realizations of V (x), in contrast to 〈 〉 D ln L , see equation (5). We now turn to a derivation of our results. Using equation ( 3), the logarithm of D L can be formally written as

= + + + - D J L J L L ln ln ( ) ln ( ) 2 ln L
, where ± J L ( ) are stationary currents through a finite sample of length L with potentials ±V x ( ) [26]:

∫ β = ± ± - J L x V x ( ) [ d exp [ ( )]] L 0 1
. Statistical properties of these currents for the Sinai problem ( = H 1 2) are known [ 35 40]. Using the results of [41], and noting that with 〈 〉 = V x ( ) 0, + J L ( ) and - J L ( ) have equal moments, we have for arbitrary H and to leading order in L,

β 〈 〉 = 〈 〉 ∝ - + - J L J L m L ln ( ) ln ( )
H , which yields equation ( 5). The proof of the result in equation ( 6) is based on a Theorem due to Monrad and Rootzén [42] on the probability that a fBm V (x), with = V (0) 0, remains within a strip of width ϵ for the time ∈

x L [0, ]. Defining ≡ ⩾ ⩾ M Vx max | ( )| L xL 0
, the Monrad Rootzén theorem, in our notation, states that ϵ ⩽ P M ( )

L satisfies ϵ ϵ ϵ - ⩽ ⩽ ⩽ - - - ( ) ( ) ( ) CL P M cL exp exp , (9)
H L H 1 1 for ϵ < ⩽ L 0
H , c and C being L-independent constants [see equation (6)]. Consider the lower bound in equation ( 6). Suppose we average the positive definite quantity D k L by considering instead of the entire set Ω of all possible paths V (x) only a subset

Ω Ω ′ ⊂ of paths such that ϵ ⩽ M L
. This gives the lower bound

ϵ 〈 〉 ⩾ 〈 〉 ⩽ Ω Ω ′ D D P M ( ) L k L k L
. However, for paths in Ω′, we have

β β ϵ - ⩽ V x V y exp ( [ ( ) ( )]
) exp (2 ), and hence,

∫ ∫ ⩾ β β ϵ - - - ( e ) e L x L L y L V x V y 0 d 0 d [ ( ) ( )] 1 2 . Therefore, we obtain ϵ 〈 〉 ⩾ ⩽ Ω βϵ - D PM e ( ) L k k L 2 .
Making the inequality more stringent by choosing the lower bound in equation ( 9), we get

ϵ ⩾ - Ω βϵ - - ( ) D C L e exp , ( 10
) L k k H 2 1
which holds for any ϵ with ϵ < ⩽ L 0 H . The function on the right-hand side (rhs) of equation ( 10) is a non-monotonic function of ϵ, attaining its maximum at ϵ ϵ ) . Clearly, the best lower bound corresponds to the choice ϵ ϵ = opt , leading to the lower bound in equation (6). Note that to satisfy the conditions of validity of the Monrad Rootzén theorem, we require that ϵ ⩽

β = = + CL k H ( 2 ) H H opt ( 1
L H opt , that is, β ⩾ k L C H 2
, which is easily realized for sufficiently large L. The derivation of the lower bound is an example of the Lifshitz optimal fluctuation method [43], which has been used to bound the survival probability of particles diffusing in the presence of randomly scattered immobile traps (see, e.g., [44]).

We now discuss the derivation of the upper bound. To this end, we discretize x, and write the rhs of equation ( 3) as

∫ ∫ ∑ ∼ β β - = - x L y L d d e e . ( 11
) L L V x V y i j N V j V i 0 0 [ ( ) ( )] , 1 [ ( ) ( )]
Given that the fBm starts at = V (0) 0, at least one term in the double sum on the rhs takes the value βM exp ( ) L , corresponding to the point x = 0 and the point where V x | ( )| attains its maximal value M L . Now, as all the other terms are positive, we have the bound

β ⩾ - D M exp ( ) L L 1
, and, thus,

∫ β ϵ ϵ ⩽ = ⩽ β β ϵ - ∞ - ( ) D k PM e d e . ( 12
) L k k M L k 0 L
The integral in the rhs is dominated, for large L, by the small ϵ region, where we can thus use the upper bound in (9). Performing the remaining integral over ϵ by the saddle-point method and omitting the pre-exponential terms lead to the upper bound in equation ( 6). The result in (6) has several striking features. Namely, the function . This means that the disorder-averaged D L is generically larger than the one expected on the basis of typical realizations of the disorder. In turn, this implies that the behavior of the positive moments of D L is supported by atypical realizations of disorder, reminiscent of the so-called Lifshitz singularities, as discussed above. In conclusion, one cannot infer the dynamical behavior in an infinite system from the positive moments of D L . This is surprising at first glance, as 〈 〉 x t ( ) 2 is linearly proportional to 〈 〉 D L , and shows that the limits → ∞ t and → ∞ L do not commute in this system.

μ β = - k L D ( , , ) ln , (13)
The behavior of the negative moments 〈 〉 -

D L k with = k 1, 2 ,.
.. is determined by essentially the same approach as above. Note that both the lower and the upper bound on - D L k are made tighter for a given realization of V (x) by using

β ∼ - D k S exp ( ) L k
, where S is the span of V (x) (the difference between the maximum and minimum) on the interval L [0, ]. Therefore, in contrast to the positive moments, the negative moments are supported by realizations of V (x) with a large span. Using the result that for large S, = ∼ -P M S S aL ( ) e x p( )

L H 2 2
, with a a constant, integration of equation (12) gives the result announced in equation (8), which displays a super-Arrhenius dependence on the temperature, a superlinear dependence on k, and a strong dependence on L. A similar result was obtained earlier in [45]. We note that, as for the positive moments, one cannot deduce the behavior in an infinite system from that of negative moments of D L in a periodic system, as the latter is supported by atypical realizations of V (x) that have anomalously large span scaling as ∼ S L H 2 , while the typical behavior is ∼ S L H typ . Based on our results for the moments, we now obtain the probability distribution P D ( ) L . As already explained, the behavior of the negative moments is supported by anomalously stretched trajectories of V (x) for which the value of D L is small. One may thus expect in view of the form of the moments in equation ( 8) that for small D L , P D ( )

L is log-normal: ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ β β ∼ - ( ) ( ) P D a L D D a L 1 exp ln . ( 14
) L H L L H 2 2 2 2
To analyze the behavior of P D ( ) L for D L close to 1, we recall the formal definition of the onesided Lévy distribution  ν z ( ), ⩽ < ∞ z 0

, of order ν (see, e.g., [46]):

 ∫ = ν ∞ - -ν z z d e ( ) e . ( 15
)
pz p 0

The asymptotic behavior of  ν z ( ) is well-known [46], and, in particular, one has  ∼ - )), and

ν σ τ - z z b z ( ) exp ( ) for → z 0, where b is a computable constant, σ ν ν = - - (2 ) (2(1
τ ν ν = - (1
). It is important to note that this precise asymptotic form is responsible for the stretched-exponential behavior in equation (15), which is immediately verified by substituting the form in equation (15), and performing the integration by the saddle-point method. Moreover, one realizes by making in equation ( 15) a change of the integration variable β = z D L ln ( 1) 1), and setting β = p k L H that equation (15) becomes identical to the result in equation ( 6), up to numerical factors. It follows that for D L close to 1 (i.e., z close to 0), the distribution function behaves as

L H , choosing ν = +H 1 (
⎡ ⎣ ⎤ ⎦  β β ∼ + - ( ) ( ) P D L D D L 1 ln . ( 16
) L H L H L H 1 1 1
Using the asymptotic  ν given above, we get that P D ( ) L is highly singular near the right edge,

β ∼ - - P D b L D ( ) exp [ (1 )
]

L H L H 1 1
, similar to the Lifshitz singularity. 13) as a function of k for L = 512 and three values of H. In all cases, the symbols denote simulation results for the translationally invariant V (x), while the slopes of the solid lines correspond to the results derived for the ratchet case.

We now consider the translationally invariant case. Here, we expect our above analysis, in particular, result (6) to hold, up to possible numerical factors. To demonstrate this, we now present results of extensive numerical simulations: figure 2

for -〈 〉 D L ln ( ) L 2 , -〈 〉 D L ln ( ) L 2
, and μ β k L ( , , ), and figure 3(a) for P D (ln ) L indeed show a very good agreement that supports our expectations.

To close, we ask: if we have two different realizations of V (x), and correspondingly, two different values, D L and ′ D L , of the diffusion coefficient, how likely are these values equal? We introduce a random variable 

 ≡ ∈ + ′ , [ 0,1] D D D L L L
, and analyze its distribution  P ( ) via numerical simulations. Clearly,  = 1 2 maximizing  P ( ) implies that the two values of D L are most likely very close to one another. Variables such as  play a key role in various scale-independent hypothesis testing procedures, in classical problems in statistics, in signal processing (see, e.g., [47]), and in the analysis of chaotic scattering in few-channel systems [48]. Such variables are used to characterize the effective width of narrow distributions possessing moments of arbitrary order [49,50].

In figure 3(b), we present numerical results for  P ( ) for different values of L and = H 1 3, for the translationally invariant case. We observe an interesting phenomenon of a change in the form of the distribution as L is increased. For relatively small L, the distribution is bell-shaped and centered at  = 1 2. However, on increasing L,  P ( ) broadens, becomes almost flat at a certain critical L (whose value depends on H), and then changes its shape so that  = 1 2 minimizes the distribution. This implies that for sufficiently large L, two values of the diffusion coefficients obtained for two different realizations of V (x) are most likely very different, and the event = ′ D D L L is the least probable. Note that a similar dependence in the distribution of the prefactor in the Sinai law on the strength of disorder was recently observed in [51].

Figure 1 .

 1 Figure 1. Sketch of the potential V (x) for the ratchet (left) and the translationally invariant case (right). Here, = H 1 3.

  grows sub-linearly with k (multifractality), (b) is a non-analytic, sublinear function of β, which implies a rather unusual sub-Arrhenius dependence of the positive moments on the temperature, and (c) exhibits a slower anomalous growth with L as ∼ + L H H 1

Figure 2 .

 2 Figure 2. (a) -〈 〉 D L ln [ ] L 2 versus L for three values of H corresponding to diffusive, subdiffusive, and superdiffusive V (x). The inset shows -〈 〉 D L ln [ ] L 2 as a function of L for three values of H, see equation (5). (b) μ β k L ( , , ) in equation (13) as a function of k for L = 512 and three values of H. In all cases, the symbols denote simulation results for the translationally invariant V (x), while the slopes of the solid lines correspond to the results derived for the ratchet case.

Figure 3 . 3 . 2 ,

 332 Figure 3. (a) Distribution P D (ln ) L for L = 1024 and = H 1 3. The red line denotes numerical results, while analytical predictions for the right and left tails behaving as -a H D exp ( ( ) ln ) H L 1 1 and -b H D exp ( ( ) ln ( )) L 2 , respectively, with a(H) and b(H) being constants, are shown by black lines. (b) Numerical results for  P ( ) for different L and = H 1 3.

7 4604 3 .

 73 SG, AR and GS thank the Galileo Galilei Institute for Theoretical Physics, Florence, Italy for the hospitality and the INFN for partial support during the completion of this work.

Acknowledgments

GS is supported by the ANR grant 2011-BS04 013-01 WALKMAT. This project is partially supported by the Indo-French Centre for the Promotion of Advanced Research under Project


Cédric Lecouvey
email: cedric.lecouvey@lmpt.univ-tours.fr

Kilian Raschel
email: kilian.raschel@lmpt.univ-tours.fr

t-Martin boundary of killed random walks in the quadrant

We compute the t-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete t-harmonic functions. Our approach is uniform in t, and shows that there are three regimes for the Martin boundary.

Introduction and main results

Aim of this paper

This work is concerned with discrete t-harmonic functions associated to Laplacian operators with Dirichlet conditions in the quarter plane. Let {p k, } be non-negative numbers summing to 1. Consider the associated discrete t-Laplacian, acting on functions f defined on the quarter plane N 2 = {0, 1, 2, . . .} 2 by

Our aim is to characterize the functions f = { f (i, j)} i, j≥0 which are (i) t-harmonic in the interior of the quarter plane, i.e., L t ( f )(i, j) = 0 for all i, j ≥ 1;

(ii) positive in the interior of the quarter plane: for all i, j ≥ 1, f (i, j) > 0;

(iii) zero on the boundary and at the exterior of the quarter plane: for all (i, j) ∈ Z 2 such that i ≤ 0 and/or j ≤ 0, f (i, j) = 0.

The probabilistic counterpart of this potential theory viewpoint is the following: tharmonic functions satisfying to (i)-(iii) are t-harmonic for random walks (whose increments have the law {p k, }) killed at the boundary of the quarter plane.

Literature

In general, it is a difficult problem to determine the Martin boundary (essentially, the set of harmonic functions) of a given class of Markov chains, especially for nonhomogeneous processes (in our case, the inhomogeneity comes from the boundary of the quadrant). The t-Martin boundary plays a crucial role to determine the Martin boundary (i.e., the t-Martin boundary with t = 1) of products of transition kernels [START_REF] Picardello | Martin boundaries of Cartesian products of Markov chains[END_REF][START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF]. Moreover, via the procedure of Doob h-transform, discrete harmonic functions have also applications to defining random processes conditioned on staying in given domains of Z d (the latter processes arise a great interest in the mathematical community, as they appear in several distinct domains: quantum random walks [START_REF] Biane | Quantum random walk on the dual of SU(n)[END_REF][START_REF] Biane | Minuscule weights and random walks on lattices[END_REF], random matrices, non-colliding random walks [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF]). Further details and motivations of considering the t-Martin boundary can be found in [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF]Introduction].

For non-zero drift random walks in cones, the Martin boundary has essentially been found for very particular cones, as half spaces N × Z d-1 and orthants N d . In [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF]Corollary 1.1] it has been found in the case of N 2 for random walks with exponential moments, using ratio limit theorems for local processes and large deviation techniques. The Martin boundary was proved to be homeomorphic to [0, π/2]. In [START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF], under the small steps and non-degeneration hypotheses, namely, (a) the p k, are 0 as soon as |k| > 1 and/or | | > 1, (b) in the (clockwise) list p 1,1 , p 1,0 , p 1,-1 , p 0,-1 , p -1,-1 , p -1,0 , p -1,1 , p 0,1 , there are no three consecutive zeros, the exact asymptotics of Green functions was obtained, and a similar result as in [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF] on the Martin boundary was derived. In [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF][START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF], no explicit expressions for the harmonic functions were provided. For random walks with zero drift, the results are rarer, and typically require a strong underlying structure: the random walks are cartesian products in [START_REF] Picardello | Martin boundaries of Cartesian products of Markov chains[END_REF], they are associated with Lie algebras in [START_REF] Biane | Quantum random walk on the dual of SU(n)[END_REF][START_REF] Biane | Minuscule weights and random walks on lattices[END_REF], etc. Last but not least, knowing the harmonic functions for zero drift random walks in N d-1 is necessary for constructing harmonic functions of walks with drift in N d , see [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on Z d[END_REF]. The first systematical result was obtained in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]: under (a)-(c), where (c) is the zero drift hypothesis (c) ∑ k, kp k, = ∑ k, p k, = 0, it was proved that there is a unique discrete harmonic function (up to multiplicative factors). In [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], there is also an explicit expression for the generating function --

0 t 0 1 • • • 0 t 0 = 1 • •
Fig. 1 Location of t 0 in the non-zero drift case (left) and in the zero drift case (right). There are three regimes for the t-Martin boundary: empty (green, left), reduced to one point (red, middle), homeomorphic to a segment (blue, right), see Theorem 1

H(x, y) = ∑ i, j≥1 f (i, j)x i-1 y j-1 (1)
of the values of the harmonic function. Finally, this uniqueness result is extended in [START_REF] Bouaziz | Discrete harmonic functions on an orthant in Z d[END_REF] to a much larger class of transition probabilities and dimension. The are less examples of studies of t-Martin boundary. One of them is [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF], for reflected random walks in half-spaces.

Main results

Our main results are on the structure of the t-Martin boundary (Theorem 1) and on the explicit expression of the t-harmonic functions (Theorem 2).

Define t 0 by

t 0 = min a∈R 2 φ (a), (2)
where we have noted

φ (a) = φ (a 1 , a 2 ) = ∑ k, p k, e ka 1 e a 2 . (3)
Notice that t 0 ∈ (0, 1], and t 0 = 1 if and only if (c) holds (Figure 1).

Theorem 1. For any random walk satisfying to (a)-(b), the t-Martin boundary is, (i) for t > t 0 , homeomorphic to a segment S t (with non-empty interior);

(ii) for t = t 0 , reduced to one point;

(iii) for t < t 0 , empty.

For t = 1 and non-zero drift, Theorem 1 (i) is proved in [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF][START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF]; for t = 1 and zero drift, Theorem 1 (ii) is obtained in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Theorem 12]. Theorem 1 (iii) follows from general results on Markov kernels, see, e.g., [START_REF] Pruitt | Eigenvalues of nonnegative matrices[END_REF].

We shall give two proofs of Theorem 1. The first one is based on a functional equation satisfied by the generating function (1) of any t-harmonic function (see [START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process[END_REF]). This method (solving the functional equation via complex analysis) was introduced in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] for the case t = 1 and zero drift. As we shall see, it has the following advantages: it works for any t; the critical value t 0 appears very naturally; finally, it provides an expression for the t-harmonic functions (see our Theorem 2).

The second proof is based on an exponential change of measure, which allows to reduce the general case to the case t = 1. In particular, with this second method, Theorem 1 can be extended to a much larger class than those satisfying to (a)-(b): namely, Theorem 1 (i) to the class of random walks whose increments have exponential moments (thanks to [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF]Corollary 1.1]), and Theorem 1 (ii) to random walks with bounded symmetric jumps (thanks to [START_REF] Bouaziz | Discrete harmonic functions on an orthant in Z d[END_REF]Theorem 1]).

Our second theorem provides an explicit expression for the generating function [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF]. We recall that a t-harmonic function f > 0 is said to be minimal if for any t-harmonic function g > 0, the inequality g ≤ f implies the equality g = c • f , for some c > 0. Notice that

H(x, 0) = ∑ i≥1 f (i, 1)x i-1 , H(0, y) = ∑ j≥1 f (1, j)y j-1 (4)
are the generating functions of the values of the harmonic function above/on the right of the coordinate axes. Introduce the second order (in x and y) polynomial, called the kernel,

L(x, y) = xy ∑ -1≤k, ≤1 p k, x -k y --t . ( 5
)
The kernel is fully characterized by the jumps {p k, }.

Theorem 2. Let {p k, } be any jumps satisfying to (a)-(b), and let S t be the segment in Theorem 1.

In case (i) (t ∈ (t 0 , ∞)), there exists a universal function w (see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] and (20)), i.e., a function depending only on the kernel L (and therefore also on t), such that for any minimal t-harmonic function { f (i, j)}, there exist p ∈ S t and two constants α, β (see [START_REF] Kurkova | Martin boundary and elliptic curves. Markov Process[END_REF] and ( 14)), with

H(x, 0) = 1 L(x, 0) α w(x) -w(p) + β .
In case (ii) (t = t 0 ), the t-harmonic function is unique, up to multiplicative factors. Its expression can be obtained either as the limit of the above expression when t → t 0 , or directly with Equations (19) and (23).

A similar expression holds for H(0, y), and finally the functional equation [START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process[END_REF] gives the announced expression for H(x, y). Theorem 2 will be stated in full details in Section 3.

Organization of the paper

In Section 2 we state the functional equation ( 6), we introduce some notation, we compute the growth of t-harmonic functions (Lemmas 2 and 3), and we finally show that the generating function (4) satisfies a simple boundary value problem (Lemma 4). In Section 3 we solve this boundary value problem, by introducing the notion of conformal gluing functions (Definition 1). In Section 4 we extend our Theorem 1 to a larger class of jumps {p k, }, by making an exponential change of jumps (Corollary 1). In Section 5 we propose some remarks and a conjecture around our results. In Appendix 6 we give an explicit expression for the conformal gluing function w of Theorem 2.

Our paper is self-contained. However, for some technical aspects of our work, specially those concerning random walks in the quarter plane, we decided to state the results without proof, referring the readers to the large existing literature (see, e.g., [START_REF] Fayolle | Random walks in the quarter plane[END_REF][START_REF] Fayolle | Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process[END_REF][START_REF] Kurkova | Martin boundary and elliptic curves. Markov Process[END_REF][START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF][START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]).

Boundary value problem for the generating functions of harmonic functions

Our approach extends the one in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], and consists in using the generating function H(x, y) (see ( 1)) of the harmonic function. The key point is that this function H(x, y) satisfies the functional equation

L(x, y)H(x, y) = L(x, 0)H(x, 0) + L(0, y)H(0, y) -L(0, 0)H(0, 0), ( 6
)
where L is defined in [START_REF] Fayolle | Random walks in the quarter plane[END_REF].

The proof of ( 6) simply comes from multiplying the relation L t ( f )(i, j) = 0 by x i y j and then from summing w.r.t. i, j ≥ 1. In ( 6), the variables x and y can be seen as formal variables, but they will mostly be used as complex variables.

This section is organized as follows: we first study important properties of the kernel [START_REF] Fayolle | Random walks in the quarter plane[END_REF]. Then we are interested in the regularity (as complex functions) of H(x, 0) and H(0, y), which is related to the exponential growth of harmonic functions. Then we state a boundary value problem satisfied by these generating functions.

Notations

The kernel L(x, y) in (5) can also be written

L(x, y) = α(x)y 2 + β (x)y + γ(x) = α(y)x 2 + β (y)x + γ(y), (7)
where (without loss of generality, we assume that p 0,0 = 0)

               α(x) = p -1,-1 x 2 + p 0,-1 x + p 1,-1 , β (x) = p -1,0 x 2 -tx + p 1,0 , γ(x) = p -1,1 x 2 + p 0,1 x + p 1,1 , α(y) = p -1,-1 y 2 + p -1,0 y + p -1,1 , β (y) = p 0,-1 y 2 -ty + p 0,1 , γ(y) = p 1,-1 y 2 + p 1,0 y + p 1,1 .
We also define

δ (x) = β (x) 2 -4α(x)γ(x), δ (y) = β (y) 2 -4 α(y) γ(y), (8)
which are the discriminants of the polynomial L(x, y) as a function of y and x, respectively. The following facts regarding the polynomial δ are proved in [5, Chapter 2] for t = 1, their proof for general values of t ≥ t 0 (t 0 being defined in (2)) would be similar: under (a)-(b), δ has degree (in x) three or four. We denote its roots by {x } 1≤ ≤4 , with

|x 1 | ≤ |x 2 | ≤ |x 3 | ≤ |x 4 |, (9)
and x 4 = ∞ if δ has degree three. We have

x 1 ∈ [-1, 1), x 4 ∈ (1, ∞)∪{∞}∪(-∞, -1],
and

x 2 , x 3 > 0. Further δ (x) is negative on R if and only if x ∈ (x 1 , x 2 ) ∪ (x 3 , x 4
). The polynomial δ in ( 8) and its roots {y } 1≤ ≤4 satisfy similar properties.

In what follows, we define the algebraic functions X(y) and Y (x) by L(X(y), y) = 0 and L(x,Y (x)) = 0. With ( 7) and ( 8) we have the obvious expressions

X(y) = -β (y) ± δ (y) 2 α(y) , Y (x) = -β (x) ± δ (x) 2α(x) . (10)
The functions X(y) and Y (x) both have two branches, called X 0 , X 1 and Y 0 ,Y 1 , which are meromorphic on the cut planes

C \ ([y 1 , y 2 ] ∪ [y 3 , y 4 ]) and C \ ([x 1 , x 2 ] ∪ [x 3 , x 4 ]),
respectively. The numbering of the branches can be chosen so as to satisfy |X 0 (y

)| ≤ |X 1 (y)| (resp. |Y 0 (x)| ≤ |Y 1 (x)|
) on the whole of the cut planes, see [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Theorem 5.3.3]. Note that except α, γ, α, γ, all quantities defined above depend on t.

Growth of t-harmonic functions

By definition, the exponential growth of a sequence {u i } of positive real numbers is lim sup i→∞ u 1/i i . We first identify (Lemma 1) the exponential growth of { f (i, 1)} and { f (1, j)} for t = 1, and then (Lemma 2) we treat the general case in t. |∇φ (a)| ∈ S 1 (the unit circle). If the drift is non-zero, the function q is a homeomorphism between ∂ D 1 and S 1 , see [START_REF] Hennequin | Processus de Markoff en cascade[END_REF]Proposition 4.4] or [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF]Introduction]. Define finally

S 1 + = S 1 ∩ R 2 + as well as Γ + 1 = {a ∈ ∂ D 1 : q(a) ∈ S 1 + }.
The following result is proved in [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF]. Lemma 1 ( [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF]). For any non-zero minimal 1-harmonic function f , there exists a

= (a 1 , a 2 ) ∈ Γ + 1 such that the exponential growth of { f (i, 1)} (resp. { f (1, j)}) is a 1 (resp. a 2 ).
(And reciprocally, any a ∈ Γ + 1 is the growth of a minimal 1-harmonic function.) Lemma 1 follows from Equation (1.3) in [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF], which gives the structure of any minimal harmonic function. It holds a priori only in the case of a non-zero drift, but it turns out to be also true in the zero drift case, as there is then no exponential growth, i.e., a 1 = a 2 = 0, which is guaranteed by [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lemma 2].

General case in t

We introduce D t = {a ∈ R 2 : φ (a) ≤ t} as well as (with obvious notation) ∂ D t and

Γ + t .
The function φ is strictly convex on R 2 , and due to the hypothesis (b) it admits a global minimum on R 2 . Let t 0 as in [START_REF] Biane | Quantum random walk on the dual of SU(n)[END_REF]. Note that t 0 ≤ 1 (evaluate φ at 0) and that t 0 = 1 if and only if the drift is zero (see [START_REF] Hennequin | Processus de Markoff en cascade[END_REF]Proposition 4.3]). The following result extends Lemma 1 to t-harmonic functions.

Lemma 2. Let t ≥ t 0 . For any non-zero minimal t-harmonic function f , there exists a ∈ Γ + t such that the exponential growth of { f (i, 1)} (resp. { f (1, j)}) is a 1 (resp. a 2 ).

(And reciprocally, any a ∈ Γ + t is the growth of a minimal t-harmonic function.) Lemma 2 could be proved along the same lines as Lemma 1, but it can also be obtained thanks to the exponential change of the parameters {p i, j } presented in Section 4. As Lemma 1, Lemma 2 holds for any value of the drift.

Reformulation in terms of the kernel

Lemma 2 can be reformulated as follows, in terms of quantities related to the kernel ( 5). This will be more convenient for our analysis. Lemma 3. Let t ≥ t 0 . For any non-zero minimal t-harmonic function f , there exists

p ∈ [x 2 , X(y 2 )] (resp. p ∈ [y 2 ,Y (x 2 )]) with p = Y 0 (p) (or p = X 0 (p )), such that the exponential growth of { f (i, 1)} (resp. { f (1, j)}) is 1/p (resp. 1/p ).
Proof. We first notice that φ (a) = t if and only if L(1/e a 1 , 1/e a 2 ) = 0, see [START_REF] Biane | Minuscule weights and random walks on lattices[END_REF] and [START_REF] Fayolle | Random walks in the quarter plane[END_REF]. Moreover, the real and positive points of {(x, y) ∈ C 2 : L(x, y) = 0} are (see [START_REF] Kurkova | Martin boundary and elliptic curves. Markov Process[END_REF][START_REF] Fayolle | Random walks in the quarter plane[END_REF])

P = {(x,Y 0 (x)) : x ∈ [x 2 , x 3 ]} ∪ {(x,Y 1 (x)) : x ∈ [x 2 , x 3 ]}.
The fact that a ∈ Γ + t implies that x ∈ [x 2 , X(y 2 )], since the normal to the curve P at x 2 (resp. X(y 2 )) is (-1, 0) (resp. (0, -1)).

As a consequence of Lemmas 2 and 3, we obtain a proof of Theorem 1 (i). We shall give more details on the proof of Theorem 1 in Section 3.

A boundary value problem

In this section we prove that the function L(x, 0)H(x, 0) satisfies a simple boundary value problem. A boundary value problem is composed of a boundary condition (Lemma 4) and a regularity condition (Lemma 5).

With the previous notation we introduce

M = X([y 1 , y 2 ]) = X 0 ([y 1 , y 2 ]) ∪ X 1 ([y 1 , y 2 ]).
This curve is symmetrical w.r.t. the real axis, since δ is non-positive on [y 1 , y 2 ], and hence the two branches X 0 and X 1 are complex conjugate on that interval. See Figure 2 for an example of curve M . Denote by x the complex conjugate of x ∈ C.

Lemma 4. We have the boundary condition: for all x in M , L(x, 0)H(x, 0) -L(x, 0)H(x, 0) = 0.

We have a similar equation for L(0, y)H(0, y) on the curve

L = Y ([x 1 , x 2 ]).
Proof. Lemma 4 is classical; see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Section 2.6] for the original proof in the zero drift case. The main idea is to evaluate (6) at (X 0 (y), y), and then to make the difference of the two equations obtained by letting y go to [y 1 , y 2 ] from above and below in C (i.e., with y having a positive and then a negative imaginary part).

Lemma 3 implies that: to any minimal t-harmonic function { f (i, j)} we can associate a number p ∈ [x 2 , X(y 2 )] such that 1/p (resp. 1/p , with p = Y 0 (p)) is the exponential growth of { f (i, 1)} (resp. { f (1, j)}). We write { f p (i, j)} and H p (x, y) to emphasize this exponential growth.

Let D M be the interior domain delimited by the curve M (containing x 2 on Figure 2) and D M = D M ∪ M be its closure. The lemma hereafter gives the regularity of the complex function H p (x, 0) (a similar result holds for H p (0, y)).

Lemma 5. For any t ∈ [t 0 , ∞), the generating function H p (x, 0) is meromorphic in D M , and has in D M a unique singularity, at p. The singularity is on the boundary

- 6 X(y 2 ) X(y 1 ) x 2 S t M • • • Fig. 2 The curve M = X([y 1 , y 2 ]
) is symmetrical w.r.t. the real axis. It is smooth everywhere except at X(y 2 ), where it may have a corner point (if and only if t = t 0 ). Any 1/p, with p point of the red segment S t , is the exponential growth of a harmonic function M if and only if t = t 0 and p = X(y 2 ) or if t = t 0 . In case (i) (t = t 0 ), the singularity is polar. In case (ii) (t = t 0 ), it can be polar or not polar.

Proof. The case t = t 0 is rather special. In the case of a zero drift (t 0 = 1), it has been proved in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lemma 3]. For other values of t = t 0 , the proof would be completely similar.

We therefore assume that t = t 0 . It follows from Lemma 3 that the function H p (x, 0) is analytic in the open disc D(0, p) centered at 0 and of radius p. The same holds for H p (0, y) in D(0, p ). Consider the identity L(x, 0)H p (x, 0) + L(0,Y 0 (x))H p (0,Y 0 (x)) -L(0, 0)H p (0, 0) = 0, [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF] which is the functional equation (6) evaluated at (x,Y 0 (x)). The fact that (11) holds on a non-empty set is not clear a priori, and follows from Lemma 6, with x on the circle of radius p. A consequence of ( 11) is that H p (x, 0) can be continued on the whole of D M . Indeed, writing

D M = D(0, p) ∪ (D M \ D(0, p)),
the generating function is defined through its power series in the first domain, and thanks to H p (0,Y 0 (x)) in the complementary domain. Further, we have that lim x→p |H p (x, 0)| = ∞ (independently of the way that x → p), so that p is indeed a pole, and not an essential singularity.

The following result has been used in the proof of Lemma 5. For the proof we refer to [14, Lemma 28], which is a very close statement. Lemma 6. Let x ∈ [x 2 , X(y 2 )]. Then for all |u| = x, we have |Y 0 (u)| ≤ Y 0 (|u|). Furthermore, the inequality is strict, except for u = x.

3 Resolution of the boundary value problem: proof of Theorems 1 and 2

Conformal gluing functions

Lemmas 4 and 5 imply that the function L(x, 0)H(x, 0) belongs to the set of functions f which are meromorphic in D M and satisfy on M the equality f (x) = f (x). This set of functions is too large to work on: for instance, P • f still belongs to this set for any polynomial P. The good idea is to impose a minimality condition on f , and to introduce the notion of conformal gluing functions (our general reference for this is the book of Litvinchuk [START_REF] Litvinchuk | Solvability theory of boundary value problems and singular integral equations with shift[END_REF], and more specifically its second chapter).

Definition 1. A conformal gluing function w for D M is a function meromorphic and injective on D M , continuous on D M except at a finite number of points, and such that w(x) = w(x) for x ∈ M .

As stated in the lemma below, conformal gluing functions exist. They must have a unique singularity on D M (of order 1 if the singularity is in the interior D M ), and are essentially characterized by the location of this singularity.

Lemma 7 ([16]

). Let p ∈ D M . Up to additive and multiplicative constants, there exists a unique conformal gluing function w for D M with a pole at p. Further, for any two conformal gluing functions w 1 and w 2 , there exist a, b, c, d ∈ C with adbc = 0 such that w 2 = aw 1 +b cw 1 +d .

Complete statement of Theorem 2

Let w be a conformal mapping as in Lemma 7 with a pole at x 0 ∈ (X(y 1 ), x 2 ) \ {0} (this reference point x 0 is arbitrary), see Figure 2 for its location. Subtracting by w(0), we may assume that w(0) = 0. The singularity of L(x, 0)H p (x, 0) is not located anywhere in D M , but on the segment [x 2 , X(y 2 )], see Lemma 5. Let us call α w-w(p) + β the class of conformal gluing functions with a pole at p ∈ [x 2 , X(y 2 )], see Lemma 7. Our Theorem 2 will be restated as:

L(x, 0)H p (x, 0) = α w(x) -w(p) + β . ( 12
)
In other words, the conformal gluing functions parametrized by p ∈ [x 2 , X(y 2 )] offer a complete solution to our problem. Notice that expressions for the constants α and β will follow from a one or two term(s) expansion of the equality [START_REF] Ignatiouk-Robert | Martin boundary of a killed random walk on a quadrant[END_REF].

We now state the theorem in full details. Define

α = -f (1, 1) ×                p 0,1 w(p) 2 w (0) if p 1,1 = 0 and p 0,1 = 0, 2p -1,1 w(p) 2 w (0) if p 1,1 = 0 and p 0,1 = 0, (w(X 0 (0)) -w(p))w(p) w(X 0 (0)) if p 1,1 = 0, (13)
and

β = p 1,1 f (1, 1) + α w(p) . ( 14
)
Theorem 2 (complete version). Let α and β be defined in ( 13) and ( 14). We have

H p (x, 0) = 1 L(x, 0) α w(x) -w(p) + β , H p (0, y) = p 1,1 f (1, 1) -L(X 0 (y), 0)H p (X 0 (y), 0) L(0, y) .

Proof of Theorem 2

Proof. The proof of Theorem 2 is based on the following remark (see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lemma 4], taken from [START_REF] Litvinchuk | Solvability theory of boundary value problems and singular integral equations with shift[END_REF]): if a function f is analytic on D M , continuous on D M and satisfies f (x) = f (x) for x ∈ M , then it must be a constant function.

Let us begin with the case t > t 0 . Lemma 5 implies that L(x, 0)H p (x, 0) has a unique pole at p. Let us assume for a while that this pole is of order 1 (which will always be the case, except if p = X(y 2 ), in which case the pole has order 2). Then by choosing suitably the value of α, the function

L(x, 0)H p (x, 0) - α w(x) -w(p) -β (15)
has no pole in D M (since w is injective in D M , see Definition 1, the function 1 w(x)-w(p) has a pole of order 1 at p, as soon as p ∈ D M ) and is continuous on D M . The function (15) also satisfies the condition f (x) = f (x) on the boundary. Hence we can use the above remark to conclude that ( 15) is a constant function. The value of β can be adapted so as to have that ( 15) is 0. To compute the exact values of the constants α and β , a series expansion of (12) around 0 is enough.

In the case t > t 0 but p = X(y 2 ), the pole of 1 w(x)-w(p) at p is of order 2 (indeed, around p ∈ {X(y 1 ), X(y 2 )}, the equality w(x) = w(x) yields w (p) = 0), and the same expression as ( 15) can be obtained.

The fact that L(x, 0)H p (x, 0) has a pole of order 1 or 2 at p follows essentially from that we are looking for positive harmonic functions { f (i, j)}. If the pole were of higher order, then the solution would have negative coefficients in its expansion near 0, see [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Lemma 11], which is impossible.

See [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] for the proof of Theorem 2 in the case t = t 0 = 1, which can be immediately adapted to the case t = t 0 = 1.

Proof of Theorem 1

Proof. In the small steps case (assumptions (a)-(b)), (i) and (ii) of Theorem 1 follow independently from Theorem 2 or from Lemma 2 (or its reformulation Lemma 3). Theorem 1 (iii) is a consequence of classical results, see [START_REF] Pruitt | Eigenvalues of nonnegative matrices[END_REF].

Extension and second proof of Theorem 1

In this section we consider weights {p k, } having exponential moments. This assumption implies that the function φ introduced in (3) is well defined on R 2 .

We note { f [p k, ]} the set of 1-harmonic functions, once the jumps {p k, } have been fixed. For any a such that φ (a) = t, we define new weights as follows (with •, • denoting the standard scalar product in R 2 )

p a k, = p k, e a,(k, ) t -1 . (16)
The identity φ (a) = t implies that ∑ k, p a k, = 1, and thus the {p a k, } can be interpreted as transition probabilities. Our main result in Section 4 is the following: Proposition 1. Assume that the {p k, } have all exponential moments. Then the set of t-harmonic functions is equal to

{(i, j) → e a t ,(i, j) f [p a t k, ](i, j)},
for any a t such that φ (a t ) = t.

Proposition 1 is a direct consequence of the following simple correspondence between t-harmonic and 1-harmonic functions. Hereafter, we shall denote by f a (a ∈ R 2 ) the function

f a (i, j) = f (i, j)e -a,(i, j) . (17)
Lemma 8. For any a t such that φ (a t ) = t and any t-harmonic function f , f a t is 1-harmonic w.r.t. the weights {p a t k, }. As a consequence of Proposition 1 and Lemma 8, we can reprove and extend Theorem 1.

Corollary 1. Theorem 1, initially proved for small steps random walks, can be generalized as follows:

• Theorem 1 (i) to random walks whose increments have exponential moments, • Theorem 1 (ii) to random walks with bounded symmetric jumps.

Proof. We first assume that the equation φ (a t ) = t has a unique solution. In this case a t is the global minimizer of φ on R 2 and the new weights {p a t k, } have zero drift (this corresponds to φ (a t ) = 0). For random walks with bounded symmetric jumps, there exists a unique f [p a t k, ] which is 1-harmonic (up to multiplicative factors), see [START_REF] Bouaziz | Discrete harmonic functions on an orthant in Z d[END_REF]Theorem 1]. Corollary 1 follows in this case.

We now suppose that the equation φ (a t ) = t has more than one solution (and then in fact, infinitely many). In this case the {p a t k, } have non-zero drift (independently of a t ). For any choice of a t , we can use the result [12, Corollary 1.1] for t = 1 (valid for random walks whose increments have exponential moments), and then with [START_REF] Picardello | Martin boundaries of Cartesian products of Markov chains[END_REF] and Lemma 8 we transfer it to other values of t > t 0 .

We now present some remarks and consequences of Proposition 1:

• Proposition 1 is independent of the choice of a t .

• Proposition 1 is not only a result on the structure of the Martin boundary, it also provides an expression of the t-harmonic functions in terms of the 1-harmonic functions.

• The exponential factor in [START_REF] Picardello | Martin boundaries of Cartesian products of Markov chains[END_REF] does not affect the fact that on the boundary of the quadrant, the functions f and f a are 0. Incidentally, this explains that the simple exponential change ( 17) cannot be used in other situations than killed random walks, like reflected random walks on a quadrant (see [START_REF] Ignatiouk-Robert | t-Martin boundary of reflected random walks on a half-space[END_REF] for the study of the t-Martin boundary of reflected random walks on a half-space).

Miscellaneous

Stable Martin boundaries

According to [START_REF] Picardello | Martin boundaries of Cartesian products of Markov chains[END_REF]Definition 2.4], the Martin boundary is stable if the Martin compactification does not depend on the eigenvalue t (with a possible exception at the critical value) and if the Martin kernels are jointly continuous w.r.t. space variable and eigenvalue. The first item is clearly satisfied in our context (see our Theorem 1). As for the second one, it does not formally come from our results. However, it is most probably true (in this direction, see Section 6.3, where we show that the harmonic functions are continuous w.r.t. the eigenvalue t). For small steps random walks and t = 1, it is proved in [START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF]Remark 29] that the Martin kernel is continuous w.r.t. the space variable.

Transformations of the step set and consequences on harmonic functions

It is natural to make some transformations of the step set, as {p k, } → {p ±k,± }, and to see the effect on the harmonic functions { f (i, j)}. In fact, the consequence will be simpler to read on the generating functions H(x, 0) and H(0, y), without obvious implications on the coefficients { f (i, j)}.

The starting point of all our approach is the functional equation ( 6), and the difference between two functional equations associated with different jumps is all contained in the kernel [START_REF] Fayolle | Random walks in the quarter plane[END_REF].

Consider first the transformation {p k, } → {p -k, }. The new kernel is x 2 L(1/x, y), with new branch points in x equal to the 1/x , while the y remain the same. The new roots of the kernel are 1/X(y) and Y (1/x). The curve L is the same, and the new M is obtained by an inversion.

The new conformal mapping is an algebraic function of w. To find it we can proceed as in the proof of Theorem 2, by compensating the poles of w in the new curve M (see [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF]). Changing accordingly the values of the constants α and β and replacing L(x, 0) by x 2 L(1/x, 0) yields the correct statement of Theorem 2 for the step set {p -k, }.

Regarding the transformation {p k, } → {p k,-}, w takes the same value but L(x, 0), which is equal to γ(x) in the case {p k, }, should be α(x).

Similar facts can be obtained for other transformations, or for the symmetry {p k, } → {p ,k }.

Simple random walks

If p 0,1 + p 1,0 + p 0,-1 + p -1,0 = 1, the minimal t-harmonic functions take the form (with p ∈ [x 2 , X(y 2 )] and p = Y 0 (p))

f p (i, j) =        1 p i - p -1,0 p 1,0 p i j 1 p j if p = x 2 , 1 p i - p -1,0 p 1,0 p i 1 p j - p 0,-1 p 0,1 p j if p ∈ (x 2 , X(y 2 )), i 1 p i 1 p j - p 0,-1 p 0,1 p j if p = X(y 2 ). (18)
In the particular case t = 1, Equation ( 18) is obtained in [14, Section 5.1]. By using techniques coming from representation theory, the authors of [START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF] have obtained an explicit expression for one 1-harmonic function, the one equal to the probability of never hitting the cone. The explicit expression (18) could also be obtained from Section 6 (via the computation of the generating functions H p (x, y)), where we derive an expression for the function w.

Generating functions of discrete harmonic functions as Tutte's invariants

The equality L(x, 0)H(x, 0) = L(x, 0)H(x, 0) for x ∈ M (Lemma 4) implies that, in the terminology of [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF], L(x, 0)H(x, 0) is a Tutte's invariant (these invariants were introduced in the 1970s, when studying properly q-colored triangulations, see [START_REF] Tutte | Chromatic sums for rooted planar triangulations[END_REF]).

In [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF] the authors identity some models of quadrant walks such that their generating function can be expressed in terms of such Tutte's invariants. This illustrates that our results do not only concern Martin boundary theory, but also combinatorial problems as the enumeration of walks in the quarter plane.

A conjecture

Our conjecture is that Theorem 1, a priori valid only for a subclass of jumps {p k, } with exponential moments, can be extended as follows:

Conjecture 1. Theorem 1 is valid for any {p k, } such that ∑ k, (k 2 + 2 )p k, < ∞ (i.e., with moments of order 2).

In the particular case of zero drift jumps {p k, } (i.e., t = t 0 = 1), this conjecture is stated in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF]Conjecture 1].

In this section we simply state the expression of w, and we refer to [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Chapter 5] or to [START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF]Section 3] for the details. The expression is w(x) = u(x 0 ) u(x)u(x 0 ) -u(x 0 ) u(0)u(x 0 )

(the second term u(x 0 ) u(0)-u(x 0 ) in ( 19) is to ensure that w(0) = 0), where the function u is different in the two cases t ∈ (t 0 , ∞) and t = t 0 .

6.1 Case t ∈ (t 0 , ∞)

In that case, u can be expressed in terms of Weiertrass elliptic functions, with the formula u(x) = ℘ 1,3 (s -1 (x)ω 2 /2),

where

• ℘ 1,3 is the Weierstrass elliptic function associated with the periods ω 1 and ω 3 defined in (21), i.e., ℘ 1,3 (ω) = 1

ω 2 + ∑ n 1 ,n 3 ∈Z 1 (ω -n 1 ω 1 -n 3 ω 3 ) 2 - 1 (n 1 ω 1 + n 3 ω 3 ) 2 ,
• ω 1 and ω 2 are defined as below (with δ as in ( 8)):

ω 1 = i x 2 x 1 dx -δ (x)
, ω 2 =

• s(ω) = g -1 (℘ 1,2 (ω)), where ℘ 1,2 is the Weierstrass elliptic function associated with the periods ω 1 and ω 2 , and g -1 is the reciprocal function of

g(x) =        δ (x 4 ) 6 + δ (x 4 ) x -x 4 if x 4 = ∞, δ (0) 6 + δ (0)x 6 if x 4 = ∞, (22)
• x 0 ∈ (X(y 1 ), x 2 ) \ {0} is arbitrary.

with g as in ( 22) and θ = arccos -∑ -1≤i, j≤1 i j p i, j x i 2 y j 2 2 α(x 2 ) α(y 2 ) .

Remarks

It can be shown that:

• The expressions given in ( 20) and ( 23) are a priori complicated, but it may happen that for some {p k, }, they become much simpler. If p 0,1 + p 1,0 + p 0,-1 + p -1,0 = 1 for instance, the function u is rational. More generally, if ω 2 /ω 3 ∈ Q, then u is an algebraic function. See [START_REF] Kurkova | Random walks in Z 2 + with non-zero drift absorbed at the axes[END_REF]Proposition 15 and Remark 16] for further remarks on u. • The function u is is continuous w.r.t. the eigenvalue t ∈ [t 0 , ∞), see [6, Section 2.2]. • At t = t 0 we have x 2 = x 3 and y 2 = y 3 (in fact t 0 = inf{t > 0 : x 2 = x 3 } = inf{t > 0 : y 2 = y 3 }).

First case: t = 1

 1 Consider on R 2 the function φ defined by (3), and define the set D 1 = {a ∈ R 2 : φ (a) ≤ 1} and its boundary ∂ D 1 = {a ∈ R 2 : φ (a) = 1}. If the drift is zero (hypothesis (c)), the set D 1 is reduced to {0}, see[START_REF] Hennequin | Processus de Markoff en cascade[END_REF] Proposition 4.3]. If not, it is homeomorphic to the unit disc. More precisely, for a ∈ ∂ D 1 , let q(a) = ∇φ (a)

6. 2

 2 Case t = t

Acknowledgments

We would like to thank Irina Ignatiouk-Robert for very interesting discussions, in particular for suggesting Proposition 1 and Lemma 8. We acknowledge support from the project MADACA of the Région Centre. The second author would like to thank Gerold Alsmeyer and the Institut für Mathematische Statistik (Universität Münster, Germany), where the project has started. We finally thank a referee for useful comments.

6 Appendix: Explicit expressions for w It turns out that a suitable expression for w has been found in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Section 5.5.2]. Let us briefly explain why such a function appears in [START_REF] Fayolle | Random walks in the quarter plane[END_REF]. The main goal of [START_REF] Fayolle | Random walks in the quarter plane[END_REF] is to develop a theory for solving a functional equation satisfied by the stationary probabilities generating function of reflected random walks in the quarter plane. The functional equation in [START_REF] Fayolle | Random walks in the quarter plane[END_REF] is closed to ours (compare [START_REF] Fayolle | Random walks in the quarter plane[END_REF]Equation (1.3.6)] with ( 6)). Roughly speaking, the general solution of [START_REF] Fayolle | Random walks in the quarter plane[END_REF] can be expressed as

w (y) w(x)w(y) dy for some function f , with the same function w as ours. Our situation is therefore simpler, since we can express the solutions directly in terms of w, without any integral.


Jérémy Frey
email: jeremy.frey@inria.fr

Comparison of a consumer grade EEG amplifier with medical grade equipment in BCI applications

Introduction:

The price of the hardware prevents the dissemination of non invasive BCI. Recently, more affordable EEG amplifiers appeared on the market. Among them, the OpenBCI board (http://www.openbci.com/) claims to bring BCI to the many. Enthusiasts and laboratories have started to use this board, but the quality of the recordings and the reliability of the resulting systems have yet to be assessed. In this study, we compare side by side the OpenBCI board with the g.tec g.USBamp amplifier (http://www.gtec.at/), a device commonly used in BCI research. Both OpenBCI and g.USBamp amplifiers can record up to 16 electrodes. This number of channels is sufficient to setup various BCI. We compared OpenBCI with g.USBamp for, on the one hand, P300 speller and, on the other hand, EEG-based workload monitoring using the n-back task [START_REF] Mühl | EEG-based workload estimation across affective contexts[END_REF]. Doing so, we could study respectively temporal and spectral features.

Material and Methods: As opposed to most of the literature, that compares electrodes (e.g. wet vs dry), in the present work we study amplifiers. Therefore, instead of complex montages [START_REF] Tautan | Framework for evaluating EEG signal quality of dry electrode recordings[END_REF] or between-subjects experimental designs [START_REF] Nijboer | Usability of Three Electroencephalogram Headsets for Brain-Computer Interfaces: A Within Subject Comparison[END_REF], we used the same electrodes during simultaneous recordings to compare both. This setup ensures that the signal coming to each amplifier is exactly the same, avoiding any offset or bias regarding the source of the measures. For that purpose we crafted two adapters, with and without a circuit made of ideal diodes. We called those connections "direct" and "isolated" respectively. The ideal diodes montage, placed before amplifiers' inputs, prevented any current to flow in reverse direction from either amplifier to the adapter; it ensured that one set of recordings would not bias the other. One recording session occurred for each application and each condition. We acquired 16 EEG channels all over the scalp using the active g.Ladybird electrodes from g.tec -ground set to "bias" pin on the OpenBCI. Two kinds of analyses were performed. One compared how the amplifiers behaved in practice, when used for classification. The second then looked at the Pearson correlation between the acquired signals, on par with the literature -e.g. [START_REF] Zander | A Dry EEG-System for Scientific Research and Brain-Computer Interfaces[END_REF]. OpenViBE 1.0 was used to acquire signals for both amplifiers -512Hz sampling rate for the g.USBamp, 125Hz for the OpenBCI. The visual P300 speller came from OpenViBE, with default settings. The protocol inducing workload was implemented following [START_REF] Mühl | EEG-based workload estimation across affective contexts[END_REF]. The signal processing pipeline for temporal and spectral features were analogous to [START_REF] Frey | Framework for Electroencephalography-based Evaluation of User Experience[END_REF].

Results: Direct connection: we tested classification accuracy for significance using Wilcoxon signed-rank tests -we repeated the classification ten times using randomized 4-fold cross-validation. There was a significant difference between amplifiers for the P300 tasks (p < 0.01, 48 target trials, 240 distractors). The AUROCC mean score for the g.USBamp was 0.961 vs 0.918 for the OpenBCI. There were no significance (p = 0.079) for the workload monitoring application, AUROCC scores were 0.89 vs 0.90 (180 trials in each 0-back and 2-back task). The Pearson correlation between temporal features of the P300 speller was statistically significant (p < 0.001), with a mean R score of 0.9965 over the 16 channels. There was also a significant correlation (p < 0.001) for the spectral features from the workload application, with a mean R score of 0.9983 for the 0-back task and 0.9979 for the 2-back task. Isolated connection: there was no significant differences between amplifiers' AUROCC scores, neither with the P300 speller nor for the workload application. As for the signals, there was significant correlations for temporal and spectral features (p < 0.001); R score of 0.8847 for temporal features, 0.9976 for spectral features during 0-back task and 0.9987 during the 2-back task. Note that for all applications and conditions AUROCC scores were far beyond chance level (which is 0.5).

Discussion: The correlation between temporal and spectral features tends so show that the signals acquired from the g.USBamp and the OpenBCI are, if not identical, very closely related. While there were no significant differences in classification of spectral features, the g.USBamp performed slightly better than the OpenBCI during the P300 speller task. The results with the "isolated" connections seems to support the similarities between both amplifiers. The weaker correlation compared to "direct" connections may be due to noise added by the ideal diodes circuit, as it did not translate into different classification accuracy -working BCI applications were produced still.

Significance: Overall, the results suggest that the OpenBCI board -or a similar solution also based on the Texas Instrument ADS1299 chip -could indeed be an effective alternative to traditional EEG amplifiers. Even though medical grade equipment possesses certification and still outperforms the OpenBCI board in terms of classification, the latter gives very close EEG readings. In practice, the obtained classification accuracy may be suitable for reliable BCI, widening the realm of applications and increasing the number of potential users.


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