Related papers: Diffusion at the random matrix hard edge
The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $\beta$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $n\beta \to \infty$, the empirical…
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian…
This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example,…
In this work, we study some statistical properties of the extreme eigenstates of the randomly-weighted adjacency matrices of random graphs. We focus on two random graph models: Erd\H{o}s-R\'{e}nyi (ER) graphs and random geometric graphs…
In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre…
We study probabilities of various rare events for the limiting point process that appears at the random matrix hard edge. We also show a transition from hard edge to bulk behavior. Asymptotic events studied include a central limit theorem…
Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise,…
We consider two non-Gaussian ensembles of large Hermitian random matrices with strong level confinement and show that near the soft edge of the spectrum both scaled density of states and eigenvalue correlations follow so-called Airy laws…
This is an expository account of the edge eigenvalue distributions in random matrix theory and their application in multivariate statistics. The emphasis is on the Painlev\'e representations of these distributions.
We define a class of random matrix ensembles that pertain to random looped polymers. Such random looped polymers are a possible model for bio-polymers such as chromatin in the cell nucleus. It is shown that the distribution of the largest…
For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion…
A distributional route to Gaussianity, associated with the concept of Conservative Mixing Transformations in ensembles of random vector-valued variables, is proposed. This route is completely different from the additive mechanism…
The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of…
In this paper, explicit error bounds are derived in the approximation of rank $k$ projections of certain $n$-dimensional random vectors by standard $k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$, $n$, and a…
Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and…
It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time.…
Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the…
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for…
We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence…
Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed. We trace this consistency to a simple linear effect: the shared Gaussian statistics across…