Related papers: Triple collisions on a comb graph

In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical…

We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from Z^2 by removing all horizontal edges off the X-axis,…

In this paper, we give some sufficient conditions for the infinite collisions of independent simple random walks on a wedge comb with profile $\{f(n), n\in \ZZ\}$. One interesting result is that if $f(n)$ has a growth order as $n\log n$,…

The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of…

We study the path behavior of the symmetric walk on some special comb-type subsets of ${\mathbb Z}^2$ which are obtained from ${\mathbb Z}^2$ by generalizing the comb having finitely many horizontal lines instead of one.

We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by…

This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here,…

A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex.…

For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider…

We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers.

In this paper, we study (1,2) and (2,1) random walks in varying environments on the lattice of positive half line. We assume that the transition probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$ For (1,2)…

A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in…

We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation…

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…

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call…

We study the path behavior of the simple symmetric walk on some comb-type subsets of Z^2 which are obtained from Z^2 by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation…

Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved…

Let $X_1$, $X_2$, $...$ be a sequence of independently and identically distributed random variables with $\mathsf{E}X_1=0$, and let $S_0=0$ and $S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote $\tau={cases}\inf\{t>1: S_t\leq0\},…

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $\alpha \in (0,1)$, the union of any $n$-vertex graph with minimum degree $\alpha n$ and the binomial random…

We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random…