arXiv subjects

Tom Claeys

Publications and source records attributed to Tom Claeys.

At least 37 records · Page 2Linked to original sources

arXiv2019

A Riemann-Hilbert approach to the lower tail of the KPZ equation

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. We show that logarithmic derivatives of the Fredholm determinants can be expressed in terms of a 2x2 Riemann-Hilbert problem, and we use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal.

math-ph↗

arXiv2019

How much can the eigenvalues of a random Hermitian matrix fluctuate?

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

math.PR↗

arXiv2019

Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

math-ph↗

arXiv2018

Large gap asymptotics for Airy kernel determinants with discontinuities

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number $m$ of discontinuities. These $m$-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the $m$-point determinants can be expressed asymptotically as the product of $m$ $1$-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.

math-ph↗

arXiv2017

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson's Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.

math.PR↗

arXiv2017

The generating function for the Airy point process and a system of coupled Painlevé II equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k-th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near-extreme eigenvalues, such as the gap between the k-th and the \ell-th largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy-Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.

math-ph↗

arXiv2016

Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer $G$-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a $2\times 2$ Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

math-ph↗

arXiv2016

Propagation of singular behavior for Gaussian perturbations of random matrices

We study the asymptotic behavior of the eigenvalues of Gaussian perturbations of large Hermitian random matrices for which the limiting eigenvalue density vanishes at a singular interior point or vanishes faster than a square root at a singular edge point. First, we show that the singular behavior propagates macroscopically for sufficiently small Gaussian perturbations, and we describe the macroscopic eigenvalue behavior for Gaussian perturbations of critical size. Secondly, for sufficiently small Gaussian perturbations of unitary invariant random matrices, we prove that the microscopic eigenvalue correlations near the singular point are described by the same limiting kernel as in the unperturbed case. We also interpret our results in terms of nonintersecting Brownian paths with random starting positions, and we establish multi-time generalizations of the microscopic results.

math.PR↗

arXiv2016

Thinning and conditioning of the Circular Unitary Ensemble

We apply the operation of random independent thinning on the eigenvalues of $n\times n$ Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as $n\to\infty$.

math-ph↗

arXiv2016

Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlevé II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.

math-ph↗

arXiv2016

Random Matrices with Merging Singularities and the Painlevé V Equation

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^α e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $α>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of $α$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

math-ph↗

arXiv2016

Gaussian perturbations of hard edge random matrix ensembles

We study the eigenvalue correlations of random Hermitian $n\times n$ matrices of the form $S=M+εH$, where $H$ is a GUE matrix, $ε>0$, and $M$ is a positive-definite Hermitian random matrix, independent of $H$, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of $S$ close to $0$ if $ε$ tends to $0$ together with $n\to +\infty$ at a critical speed, depending on the random matrix $M$. In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our general results to the cases where (i) $M$ is a Laguerre/Wishart random matrix, (ii) $M=G^*G$ with $G$ a product of Ginibre matrices, (iii) $M=T^*T$ with $T$ a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of $M$ follow a Muttalib-Borodin biorthogonal ensemble.

math-ph↗

arXiv2015

Correlation kernels for sums and products of random matrices

Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum $H + M$ where $H$ is a GUE matrix and $M$ is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of $H + M$ follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.

math.PR↗

arXiv2015

Asymptotics for the partition function in two-cut random matrix models

We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for $\log Z_N(V)$, up to terms that are small as $N$ goes to infinity. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential $V$. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two-cut potentials $V$. The asymptotic expansion of $\log Z_N(V)$ as $N$ goes to infinity contains terms that depend analytically on the potential $V$ and that have already appeared in the literature. In addition our method allows to compute the $V$-independent terms of the asymptotic expansion of $\log Z_N(V)$ which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann-Hilbert techniques which had so far been successful to compute asymptotics for the partition function only in the one-cut case.

math-ph↗

arXiv2015

Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.

math-ph↗

arXiv2015

Biorthogonal ensembles with two-particle interactions

We investigate determinantal point processes on $[0,+\infty)$ of the form \begin{equation*}\label{probability distribution} \frac{1}{Z_n}\prod_{1\leq i<j\leq n}(λ_j-λ_i)\prod_{1\leq i<j\leq n}(λ_j^θ-λ_i^θ) \prod_{j=1}^n w(λ_j)dλ_j,\qquad θ\geq 1. \end{equation*} We prove that the biorthogonal polynomials associated to such models satisfy a recurrence relation and a Christoffel-Darboux formula if $θ\in\mathbb Q$, and that they can be characterized in terms of $1\times 2$ vector-valued Riemann-Hilbert problems which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated to our model if $w(λ)=e^{-nV(λ)}$ in the one-cut case with and without hard edge.

math-ph↗

arXiv2015

Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogues to the Painlevé III equation, which reduces to the Painlevé III equation itself when the pole is simple.

math-ph↗

arXiv2013

Random matrices with equispaced external source

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 x 1 vector-valued Riemann-Hilbert problems, and to perform an asymptotic analysis of the Riemann-Hilbert problems.

math-ph↗