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Motivated by the harmonic analysis of self-affine measures, we introduce a class of representations of the Cuntz algebra associated to random walks on graphs. The representations are constructed using the dilation theory of row…

Consider deterministic random walks F: I x Z -> I x Z, defined by F(x,n)=(f(x), K(x)+n), where f is an expanding Markov map on the interval I and K: I->Z. We study the universality (stability) of ergodic (for instance, recurrence and…

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition…

The study of several naturally arising "nearest neighbours" random walks benefits from the study of the associated orthogonal polynomials and their orthogonality measure. I consider extensions of this approach to a larger class of random…

Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in…

We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This…

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…

Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in…

Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schr\"{o}der, and Motzkin number sequences…

We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative…

Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded…

We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given…

We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq…

Motivated by numerous questions in random geometry, given a smooth manifold $M$, we approach a systematic study of the differential topology of Gaussian random fields (GRF) $X:M\to \mathbb{R}^k$, that we interpret as random variables with…

We provide a direct proof of Cram\'er's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic…

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices,…

We associate two linear categories with two objects to a module over the subalgebra of coinvariants of a Hopf-Galois extension, and prove that they are isomorphic. The structure Theorem for cleft extensions, and the Militaru \cStefan…

Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…

Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…