Related papers: The number of ends of critical branching random wa…

We prove that the trace of a transient branching random walk on a planar hyperbolic Cayley graph has a.s. continuum many ends and no isolated end.

A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We…

Branching processes are widely used to model the viral epidemic evolution. For more adequate investigation of viral epidemic modelling, we suggest to apply branching processes with transport of particles usually called branching random…

We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This…

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still…

We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex

We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the…

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…

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in arXiv:1711.00852.…

We study intersection properties of two or more independent tree-like random graphs. Our setting encompasses critical, possibly long range, Bernoulli percolation clusters, incipient infinite clusters, as well as critical branching random…

We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.

The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of application of theese processes to model certain…

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…

We consider a class of biased random walks on infinite graphs and present several general results on the spectral radius of biased random walk.

Graph sampling is a technique to pick a subset of vertices and/ or edges from original graph. Among various graph sampling approaches, Traversal Based Sampling (TBS) are widely used due to low cost and feasibility for many cases, in which…

In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of…

We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple…