Related papers: Random walks and orthogonal polynomials: some chal…
The foundational work of Karlin and McGregor established a powerful connection between random walks with tridiagonal transition matrices and the theory of orthogonal polynomials. We consider a particular extension of this framework, where…
In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study…
A number of papers have examined various aspects of "random random" walks on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this…
A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in…
Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the…
Random walks as well as diffusions in random media are considered. Methods are developed that allow one to establish large deviation results for both the `quenched' and the `averaged' case.
The random walk with choice is a well known variation to the random walk that first selects a subset of $d$ neighbours nodes and then decides to move to the node which maximizes the value of a certain metric; this metric captures the number…
A random walk is known as a random process which describes a path including a succession of random steps in the mathematical space. It has increasingly been popular in various disciplines such as mathematics and computer science.…
Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and…
In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.
We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…
This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give…
We show connection between Dyck paths with peaks of bounded height and random walks. The correspondence between a certain class of random walks and such Dyck paths allows us to develop a probabilistic perspective on Chebyshev polynomials.
Modeling of polymer chains has received a lot of attention in mathematics. In fact, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems…
We consider an urn model leading to a random walk that can be solved explicitly in terms of the well known Jacobi polynomials.
A notion of random walks for circle packings is introduced. The geometry behind this notion is discussed, together with some applications. In particular, we obtain a short proof of a result regarding the type problem for circle packings,…
There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk conditioned on its survival up to time $t$. In…
Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence…
This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichm\"uller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the…
We define a new stochastic process on general simplicial complexes which allows to study their spectral and homological properties. Some results for random walks on graphs are shown to hold in this general setting. As an application, the…