Related papers: Random braids and random walks on finite groups
Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erd\"os-R\'enyi model). In this paper, we extend this model to study random general Coxeter groups…
Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random…
In this paper, we study random walks on groups that contain superlinear divergent geodesics, in the line of thoughts of Goldsborough-Sisto. The existence of a superlinear divergent geodesic is a quasi-isometry invariant which allows us to…
The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence…
We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random…
We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method…
Consider a collaborative dynamic of $k$ independent random walks on a finite connected graph $G$. We are interested in the size of the set of vertices visited by at least one walker and study how the number of walkers relates to the…
The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and…
We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\Z^d, d\geq5$. Analogous to the regular discrete capacity, branching capacity is closely related to the…
We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the…
We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a…
We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em…
In this work, we are interested in the set of visited vertices of a tree $\mathbb{T}$ by a randomly biased random walk $\mathbb{X}:=(X_n,n \in \mathbb{N})$. The aim is to study a generalized range, that is to say the volume of the trace of…
To analyze the security of code-based cryptosystems, the smoothing parameter, which is closely related to the total variation distance of codes, has been investigated. While previous studies have bounded this distance using the Fourier…
We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and…
In thermally fluctuating long linear polymeric chain in solution, the ends come from time to time into a direct contact or a close vicinity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a…
On a finite graph, there is a natural family of Boltzmann probability measures on cycle-rooted spanning forests, parametrized by weights on cycles. For a certain subclass of those weights, we construct Gibbs measures in infinite volume, as…
Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the…
Consider a random walk on a tree $G=(V,E)$. For $v,w \in V$, let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at $v$ to reach $w$, and let $\pi_v = \mathrm{deg}(v)/2|E|$ denote the…
This note contains old instead of new results about random walks on groups, which may serve as a small supplement to the author's monograph ``Random Walks on Infinite Graphs and Groups'' (Cambridge Univ. Press 2000/2009). First, we exhibit…