Related papers: A connection between the Ghirlanda--Guerra identit…

We define a numerical quasi-isometry invariant of a finitely generated group, whose values parametrize the difference between the group being uniformly embeddable in a Hilbert space and the reduced C*-algebra of the group being exact.

The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random…

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the…

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian…

In this paper we show that the ultrametricity property remains valid in random field Ising models with independent disorder whenever the field strength is a small perturbation.

The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, $\beta$-invariant and some…

Let G be a discrete, torsion free group with a finite dimensional classifying space BG. We show that the existence of a gamma-element for such G is a metric, that is, coarse, invariant of G. We also obtain results for groups with torsion.…

We consider the non-Hermitian analogue of the celebrated Wigner-Dyson-Mehta bulk universality phenomenon, i.e. that in the bulk the local eigenvalue statistics of a large random matrix with independent, identically distributed centred…

We study the problem $-\Delta u = \gamma$, where $\gamma$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In…

In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism…

We describe a systematic search for all strange hypergeometric identities up to a certain complexity with sheer brute force that lead us to the discovery of two new infinite families of closed-form evaluations.

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of…

Let G be a finite simple group. We show that the commutator map $a : G \times G \to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we…

We devise a geometric description of bounded systems at criticality in any dimension $d$. This is achieved by altering the flat metric with a space dependent scale factor $\gamma(x)$, $x$ belonging to a general bounded domain $\Omega$.…

We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the…

An unpublished identity of Gosper restates a hypergeometric identity for odd zeta values in terms of an infinite product of matrices. We show that this correspondence runs much deeper, and show that many examples of WZ-accelerated series…

A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal…

We analyze the interplay between labeled trees and the ultrametric spaces they present. We provide characterizations of labeled trees that generate separable ultrametric spaces and those that generate locally finite ultrametric spaces. In…

Let $X$ be a real $(\beta=1)$ or complex $(\beta=2)$ Ginibre ensemble. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re \sigma_i.$ It was proved that $Z_n$ converges weakly to the…