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Alain Rouault

Publications and source records attributed to Alain Rouault.

At least 19 recordsLinked to original sources
arXiv2025

Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle

Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been established between the large theory of spectral measures built on random matrices and sum rules. In this work, we extend this approach by studying sum rules within random matrix models with polynomial potentials on the unit circle, with a particular focus on cases where the equilibrium measure lacks full support.

math.PR↗
arXiv2025

Extension of a theorem of Wschebor to free and matrix Brownian motions

In 1992, M. Wschebor proved a theorem on the convergence of small increments of the Brownian motion. Since then, it has been extended to various processes. We prove a version of this theorem for the Hermitian Brownian motion and the free Brownian motion. Since these theorems deal with a convergence to a deterministic limit, we prove also the convergence in distribution of the corresponding fluctuations.

math.PR↗
arXiv2023

On some gateways between sum rules

We present correspondences induced by some classical mappings between measures on an interval and measures on the unit circle. More precisely, we link their sequences of orthogonal polynomial and their recursion coefficients. We also deduce some correspondences between particular equilibrium measures of random matrix ensembles. Additionally, we show that these mappings open up gateways between the sum rules associated with some classical models, leading to new formulations of several sum rules.

math.PR↗
arXiv2021

Large deviations for spectral measures of some spiked matrices

We prove large deviations principles for spectral measures of perturbed (or spiked) matrix models in the direction of an eigenvector of the perturbation. In each model under study, we provide two approaches, one of which relying on large deviations principle of unperturbed models derived in the previous work "Sum rules via large deviations" (Gamboa-Nagel-Rouault, JFA, 2016).

math.PR↗
arXiv2020

Invariance principles for clocks

We show an invariance principle for rescaled clocks of positive semi-stable Markov processes, proving a conjecture presented in Remark 4 in Demni, Rouault, Zani [11], 2015.

math.PR↗
arXiv2020

Sum rules via large deviations: extension to polynomial potentials and the multi-cut regime

A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have showed sum rules by using only probabilistic tools (namely the large deviations theory). Here, we prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: firstly, when the equilibrium measure is not necessarily supported by a single interval and secondly, when the potential is a nonnegative polynomial. The rate functions can be expressed as functions of the Jacobi coefficients. These new large deviation results lead to original sum rules both for the one and the multi-cut regime and also answer a conjecture stated in Gamboa, Nagel and Rouault (2016) concerning general sum rules.

math.PR↗
arXiv2018

Sum rules and large deviations for spectral matrix measures in the Jacobi ensemble

We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [20]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten-McKay law. As an important intermediate step, we derive the distribution of canonical moments of the matrix Jacobi ensemble.

math.PR↗
arXiv2017

Asymptotic results in solvable two-charge models

Rider, Sinclair and Xu (2013) introduced a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied by Shum and Sinclair (2014) and Forrester (2010). In this paper we explore the sharp asymptotic behavior of the number of unit charge particles on the line and on the circle, as the total charge goes to infinity. We establish extended central limit theorems, Berry-Esseen estimates and precise moderate deviations using the machinery of the mod-Gaussian convergence. Also a large deviation principle is derived using the Gärtner-Ellis theorem.

math.PR↗
arXiv2017

Mod-Gaussian convergence for random determinants and random characteristic polynomials

The aim of this paper is to give a precise asymptotic description of some eigenvalue statistics stemming from random matrix theory. More precisely, we consider random determinants of the GUE, Laguerre, Uniform Gram and Jacobi beta ensembles and random characteristic polynomials evaluated at 1 for matrices in the Circular and Circular Jacobi beta ensembles. We use the framework of mod-Gaussian convergence to provide quantitative estimates of their logarithmic behavior, as the size of the ensemble grows to infinity. We establish central limit theorems, Berry-Esseen bounds, moderate deviations and local limit theorems. Furthermore, we identify the scale at which the validity of the Gaussian approximation for the tails breaks. With the exception of the Gaussian ensemble, all the results are obtained for a continuous choice of the Dyson parameter, that is for general $β>0$. The proofs rely on explicit computations which are possible thanks to closed product formulas of the Laplace transforms.

math.PR↗
arXiv2017

Asymptotic properties of the density of particles in $β$-ensembles

We extend recent results on the Asymptotic Equipartition Property for the density of $n$ particles in $β$-ensembles, as $n$ tends to infinity. We prove the Large Deviation Principle of the log-density for a general potential and the mod-gaussian convergence in the classical examples.

math.PR↗
arXiv2017

Addendum to " Sum rules via large deviations "

In these notes we fill a gap in a proof in Section 4 of Gamboa, Nagel, Rouault [Sum rules via large deviations, J. Funct. Anal. 270 (2016), 509-559]. We prove a general theorem which combines a LDP with a convex rate function and a LDP with a non-convex one. This result will be used to prove LDPs for spectral matrix measures and for spectral measures on the unit circle.

math.PR↗
arXiv2017

Sum rules and large deviations for spectral measures on the unit circle

This work is a companion paper of Gamboa, Nagel, Rouault (J. Funct. Anal. 2016). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two matrix models. The first one is the Gross-Witten ensemble. In the gapped regime we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua-Pickrell ensemble. Unlike the Gross-Witten ensemble the potential is here infinite at one point. Surprisingly, but as in the above mentioned paper, we obtain a completely new sum rule for the deviation to the equilibrium measure of the Hua-Pickrell ensemble. The extension to matrix measures is also studied.

math.PR↗
arXiv2016

Sum rules and large deviations for spectral matrix measures

A sum rule relative to a reference measure on R is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example Killip and Simon 2003). In this paper, using only probabilistic tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover the earlier result of Damanik, Killip and Simon (2010) when the reference measure is the (matrix-valued) semicircle law and obtain a new sum rule when the reference measure is the (matrix-valued) Marchenko-Pastur law.

math.PR↗
arXiv2016

Large deviations for near-extreme eigenvalues in the beta-ensembles

For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.

math.PR↗
arXiv2015

Sum rules via large deviations

In this paper, a sum rule means a relationship between a functional defined on a subset of all probability measures on $\mathbb{R}$ involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. The first sum rule is the Szegő-Verblunsky theorem (see Simon (2011) Theorem 1.8.6 p. 29) and concerns the case of trigonometrical polynomials on the torus. More recently, Killip and Simon (2003) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution. Indeed, this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law. In this paper, we recover this sum rules by using only large deviations tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Pastur-Marchenko or Kesten-McKay distributions. As in the semicircular case, these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure. From the point of view of large deviations, we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to Gamboa Rouault (2011) for the recursion coefficients.

math.PR↗
arXiv2014

Large deviations for clocks of self-similar processes

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a Lévy process drifting to $\infty$ and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani.

math.PR↗