Related papers: Local limits in $p$-adic random matrix theory

We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the…

We consider a random process with discrete time formed by singular values of products of truncations of Haar distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives…

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on…

We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1) ... X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n \rightarrow \infty$ and the limit is Dirichlet…

We investigate singular value statistics for products of independent rectangular complex Ginibre matrices. When the rectangularity parameters of the matrices converge to a common limit in the asymptotic regime, the limiting spectral density…

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a…

We study p-adic counterparts of stable distributions, that is limit distributions for sequences of normalized sums of independent identically distributed p-adic-valued random variables. In contrast to the classical case, non-degenerate…

We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise…

We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and…

In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one dimensional lattice Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of…

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…

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline…

We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number $n$ of particles tends to infinity we obtain the limiting local correlation kernel…

We introduce a random matrix model where the entries are dependent across both rows and columns. More precisely, we investigate matrices of the form $\X=(X_{(i-1)n+t})_{it}\in\R^{p\times n}$ derived from a linear process $X_t=\sum_j c_j…

It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence…

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class…

Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical…

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in \mathbb{Z}_p[t]$ be monic polynomials whose reductions modulo $p$ in $\mathbb{F}_p[t]$ are…

The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$…

It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an…