Related papers: Local limits in $p$-adic random matrix theory
We consider random matrices whose shape is the dilation $N\lambda$ of a self-conjugate Young diagram $\lambda$. In the large-$N$ limit, the empirical distribution of the squared singular values converges almost surely to a probability…
This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra $\mathfrak{A}$ and (ii) an auxiliary random sequence $(\pi_n)_{n\ge 1}$ of finite-dimensional representations of…
The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular…
We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random.…
We consider the continuum limit of some products of random matrices in $\text{SL}(d,{\mathbb R})$ that arise as discretisations of incompressible renewing flows -- that is, of flows corresponding to a divergence-free velocity field that…
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in…
We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard's isomorphisms between…
We derive new comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We discuss applications to the concentration of log-concave random vectors and bounds on…
In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first…
Using the matrix product formalism we formulate a natural p-species generalization of the asymmetric simple exclusion process. In this model particles hop with their own specific rate and fast particles can overtake slow ones with a rate…
For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described…
If A_1,...,A_N are real square matrices then the p-radius, generalised Lyapunov exponent or matrix pressure is defined to be the asymptotic exponential growth rate of the sum $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$, where p…
We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of colored labeled noncrossing partitions.…
The Poisson boundary of a finite direct product of affine automorphism groups of homogeneous trees is considered. The Poisson boundary is shown to be a product of ends of trees with a hitting measure for spread-out, aperiodic measures of…
Continuous-time Mallows processes are processes of random permutations of the set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution with parameter $t$. Recently Corsini showed that there exists a unique Markov…
We introduce random matrix ensembles that correspond to the infinite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families…
In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of…
Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\lim_{t \to \infty} t^{3/2}e^{\lambda_0 t} p(t,x,y)=C(x,y),$$ where $\lambda_0$…
The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the…
We study random vectors of the form $(\operatorname {Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(\nu)}$ are…