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Alice Guionnet

Publications and source records attributed to Alice Guionnet.

At least 19 recordsLinked to original sources
arXiv2026

Large deviations of the periodic Toda chain

This work establishes a large deviation principle for the spectral measure of the Lax matrix associated to the periodic Toda chain of $N$ particles, subject to a generalised Gibbs measure. This large deviation principle is governed by a rate function which can be regarded as a generalisation of the free energy of the system. Such a large deviation principle is proven both for the model when the momentum is constrained to be zero and when it is allowed to fluctuate. Moreover, the large deviation principle is proven directly at the level of the representation of the generalised Gibbs partition function given in terms of the variables realising the classical separation of variables, \textit{i.e.} rectifying the equations of motion. As such, this work paves the way towards the computation of the thermodynamic limit of dynamical correlation functions in the Toda chain subject to generalised Gibbs ensemble statistics.

math.PR↗
arXiv2026

One-sided large deviations for the ground-state energy of spin glasses

We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.

math.PR↗
arXiv2026

Macroscopic asymptotics in discrete beta-ensembles and random tilings

We carry out the asymptotic analysis of repulsive ensembles of N particles which are discrete analogues of continuous 1d log-gases or beta-ensembles of random matrix theory. The ensembles that we study have several groups of particles which can have different intensities of repulsion. They appear naturally in models of random domino and lozenge tilings, random partitions, supersymmetric gauge theory, asymptotic representation theory, discrete orthogonal polynomial ensembles, etc. We allow filling fractions to be either fixed, or free, or to vary while respecting affine constraints. We are interested in the macroscopic behavior of the distribution of particles, captured by linear statistics, partition functions, and their finite-size corrections as N is large. We prove the law of large numbers and large deviations for the empirical measure around the equilibrium measure. To reach finite-size correction we assume off-criticality. For fixed filling fractions, we prove an asymptotic expansion for the partition function and for the cumulants of linear statistics, in particular establishing a central limit theorem. For varying filling fractions, we prove that the central limit theorem is perturbed by an additional discrete Gaussian component oscillating with N. We apply our general results to the study of uniformly random lozenge tilings on a large class of domains -- not necessarily planar, simply-connected, nor orientable. When the analogues of filling fractions are fixed and this domain is orientable, we show that the Gaussian fluctuations on the vertical extend to the whole liquid region and are governed there by the Gaussian free field, as predicted by the Kenyon-Okounkov conjecture. We also establish a modification of the Kenyon-Okounkov conjecture in the non-orientable case. Complementarily, we prove discrete Gaussian fluctuations for filling fractions, when they are not fixed.

math.PR↗
arXiv2025

LDP for the largest eigenvalue of Kronecker random matrices

We prove a large deviations principle for the largest eigenvalue of Gaussian Kronecker matrices, namely matrices defined as the sum of tensors of independent Gaussian matrices in the regime where the dimension of the Gaussian matrices goes to infinity.

math.PR↗
arXiv2025

Global law of conjugate kernel random matrices with heavy-tailed weights

We study the asymptotic spectral distribution of the conjugate kernel random matrix $YY^\top$, where $Y= f(WX)$ arises from a two-layer neural network model. We consider the setting where $W$ and $X$ are random rectangular matrices with i.i.d.\ entries, where the entries of $W$ follow a heavy-tailed distribution, while those of $X$ have light tails. Our assumptions on $W$ include a broad class of heavy-tailed distributions, such as symmetric $\alpha$-stable laws with $\alpha \in ]0,2[$ and sparse matrices with $\mathcal{O}(1)$ nonzero entries per row. The activation function $f$, applied entrywise, is bounded, smooth, odd, and nonlinear. We compute the limiting eigenvalue distribution of $YY^\top$ through its moments and show that heavy-tailed weights induce strong correlations between the entries of $Y$, resulting in richer and fundamentally different spectral behavior compared to the light-tailed case.

math.PR↗
arXiv2024

Asymptotic expansion of the partition function for $\beta$-ensembles with complex potentials

In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N \beta V(z_k) } \, \mathrm{d}\mathbf{z}$$ where $V \in \mathbb{C}[X]$, $\beta \in 2 \mathbb{N}^*$ is an even integer and $\Gamma \subset \mathbb{C}$ is an unbounded contour such that the integral converges. For even degree, real valued $V$s and when $\Gamma = \mathbb{R}$, it is well known that the large-$N$ expansion is characterised by an equilibrium measure corresponding to the minimiser of an appropriate energy functional. This method bears a structural resemblance with the Laplace method. By contrast, in the complex valued setting we are considering, the analysis structurally resembles the classical steepest-descent method, and involves finding a critical point \textit{and} a steepest descent curve, the latter being a deformation of the original integration contour. More precisely, one minimises a curve-dependent energy functional with respect to measures on the curve and then maximises the energy over an appropriate space of curves. Our analysis deals with the one-cut regime of the associated equilibrium measure. We establish the existence of an all order asymptotic expansion for $\ln \mathcal{Z}_{N,\Gamma}[V]$ and explicitly identify the first few terms.

math-ph↗
arXiv2024

Large deviations for macroscopic observables of heavy-tailed matrices

We consider a finite collection of independent Hermitian heavy-tailed random matrices of growing dimension. Our model includes the L\'evy matrices proposed by Bouchaud and Cizeau, as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviations principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distributions which is additive for free traffic convolution.

math.PR↗
arXiv2024

Asymptotics of Symmetric Polynomials: A Dynamical Point of view

In this paper we study the asymptotic behavior of the (skew) Macdonald and Jack symmetric polynomials as the number of variables grows to infinity. We characterize their limits in terms of certain variational problems. As an intermediate step, we establish a large deviation principle for the $\theta$ analogue of non-intersecting Bernoulli random walks. When $\theta=1$, these walks are equivalent to random Lozenges tilings of strip domains, where the variational principle (with general domains and boundary conditions) has been proven in the seminal work by Cohn, Kenyon, and Propp. Our result gives a new argument of this variational principle, and also extends it to non-intersecting $\theta$-Bernoulli random walks for any $\theta \in (0,\infty)$. Remarkably, the rate functions remain identical, differing only by a factor of $1/\theta$.

math.PR↗
arXiv2024

Large deviation principle for the largest eigenvalue of random matrices with a variance profile

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N \in \mathbb N$, we consider random $N \times N$ symmetric matrices $H^N$ which are such that $H_{ij}^{N}=\frac{1}{\sqrt{N}}X_{i,j}^{N}$ for $1 \leq i,j \leq N$, where the $X_{i,j}^{N}$ for $1 \leq i \leq j \leq N$ are independent and centered. We then denote $\Sigma_{i,j} ^N = \text{Var} (X_{i,j}^{N}) ( 1 + \textbf{1}_{ i =j})^{-1}$ the variance profile of $H^N$. Our large deviation principle is then stated under the assumption that the $\Sigma^N$ converge in a certain sense toward a real continuous function $\sigma$ of $[0,1]^2$ and that the entries of $H^N$ are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving $\sigma$. This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries.

math.PR↗
arXiv2023

About universality of large deviation principles for conjugacy invariant permutations

We prove the universality of the large deviations for conjugacy invariant permutations with few cycles. As an application, we establish the universality of large deviation at speeds $n$ and $\sqrt{n}$ for the length of monotone subsequences in conjugacy invariant permutations, with a sharp control over the total number of cycles. This universality class includes the well-known Ewens measures.

math.CO↗
arXiv2023

Spectral Phase Transitions in Non-Linear Wigner Spiked Models

We study the asymptotic behavior of the spectrum of a random matrix where a non-linearity is applied entry-wise to a Wigner matrix perturbed by a rank-one spike with independent and identically distributed entries. In this setting, we show that when the signal-to-noise ratio scale as $N^{\frac{1}{2} (1-1/k_\star)}$, where $k_\star$ is the first non-zero generalized information coefficient of the function, the non-linear spike model effectively behaves as an equivalent spiked Wigner matrix, where the former spike before the non-linearity is now raised to a power $k_\star$. This allows us to study the phase transition of the leading eigenvalues, generalizing part of the work of Baik, Ben Arous and Pech\'e to these non-linear models.

math.PR↗
arXiv2023

Estimating rank-one matrices with mismatched prior and noise: universality and large deviations

We prove a universality result that reduces the free energy of rank-one matrix estimation problems in the setting of mismatched prior and noise to the computation of the free energy for a modified Sherrington-Kirkpatrick spin glass. Our main result is an almost sure large deviation principle for the overlaps between the truth signal and the estimator for both the Bayes-optimal and mismatched settings. Through the large deviations principle, we recover the limit of the free energy in mismatched inference problems and the universality of the overlaps.

math.PR↗
arXiv2023

Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices

We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $\lambda_1$ of Wigner matrices with sub-Gaussian entries. In contrast to the case of heavier tails, where deviations of $\lambda_1$ are due to the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that the general sub-Gaussian case is determined by a mixture of localized and delocalized effects. Our key result is a finite-$N$ approximation for the upper tail of $\lambda_1$ by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This new type of argument allows us to derive full large deviation principles when the log-Laplace transform of the entries' distribution $\mu$ has bounded second derivative, whereas previous results required much more restrictive assumptions, namely sharp sub-Gaussianity and symmetry, or only covered certain ranges of deviations. We show that the sharp sub-Gaussian condition characterizes measures $\mu$ for which the rate function coincides with that of the Gaussian Orthogonal Ensemble (GOE). When $\mu$ is not sharp sub-Gaussian, at a certain distance from the bulk of the spectrum there is a transition from the GOE rate function to a non-universal rate function depending on $\mu$, and this transition coincides with the onset of a localization phenomenon for the associated eigenvector.

math.PR↗
arXiv2022

Matrix models at low temperature

In this article we investigate the behavior of multi-matrix unitary invariant models under a potential $V_\beta=\beta U+W$ when the inverse temperature $\beta$ becomes very large. We first prove, under mild hypothesis on the functionals $U,W$ that as soon at these potentials are "confining" at infinity, the sequence of spectral distribution of the matrices are tight when the dimension goes to infinity. Their limit points are solutions of Dyson-Schwinger's equations. Next we investigate a few specific models, most importantly the "strong single variable model" where $U$ is a sum of potentials in a single matrix and the "strong commutator model" where $U = -[X,Y]^2$.

math.PR↗
arXiv2022

Low-rank Matrix Estimation with Inhomogeneous Noise

We study low-rank matrix estimation for a generic inhomogeneous output channel through which the matrix is observed. This generalizes the commonly considered spiked matrix model with homogeneous noise to include for instance the dense degree-corrected stochastic block model. We adapt techniques used to study multispecies spin glasses to derive and rigorously prove an expression for the free energy of the problem in the large size limit, providing a framework to study the signal detection thresholds. We discuss an application of this framework to the degree corrected stochastic block models.

math.PR↗
arXiv2021

Bernoulli Random Matrices

Random Matrix theory has become a field on its own with a breadth of new results, techniques, and ideas in the last thirty years. In these proceedings of the 8ECM 2021, I illustrate some of these advances by describing what is known about the spectrum and the eigenvectors of Bernoulli matrices.

math.PR↗
arXiv2021

Large Deviations Asymptotics of Rectangular Spherical Integral

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.

math.PR↗