Related papers: Universality classes for general random matrix flo…
We show that eigenvalue correlations in unitary-invariant ensembles of large random matrices adhere to novel universal laws that only depend on a multicriticality of the bulk density of states near the soft edge of the spectrum. Our…
We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It…
We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study…
Eigenvalue correlations of random matrix ensembles as a function of an external perturbation are investigated vis the Dyson Brownian Motion Model in the situation where the level density has a hard edge singularity. By solving a linearized…
Let $\mathcal{M}_n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset\mathbb{R}$ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in…
The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct…
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve…
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed…
We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the…
In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a…
The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows which are families of probability distributions on the space of solutions to the associated ODEs, which no longer satisfy…
We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models,…
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from…
The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the…
Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $\sigma$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper…
In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank…
We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the…
Interacting random matrix systems are fundamental to modern theoretical physics and data science, yet a unified framework for their analysis has been lacking. This work introduces such a universal framework, built upon two novel concepts:…
This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary…
In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let $V_n=(v_{ij})_{i \leq n,\, j\leq m}$ be a $n\times m$ random matrix, where $(n/m)\to…