Related papers: Ladder Chains: A Variation of Random Walks

We study the winding behavior of random walks on two oriented square lattices. One common feature of these walks is that they are bound to revolve clockwise. We also obtain quantitative results of transience/recurrence for each walk.

First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of…

The recurrence properties of random walks can be characterized by P\'{o}lya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random…

We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required…

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{{\rm d}}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient…

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…

Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal…

In this paper, we make a few random explorations that relate directly to the items mentioned in the title. We define transient chains and recurrent chains with "killing", the Green's function, the Laplacian operator, and harmonic functions.…

Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = \Theta \left(\|x\|^{-s}\right)$ for some $s>d$. We show that the random walk…

We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random…

The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented…

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not…

A random walk with counterbalanced steps is a process of partial sums $\check S(n)=\check X_1+ \cdots + \check X_n$ whose steps $\check X_n$ are given recursively as follows. For each $n\geq 2$, with a fixed probability $p$, $\check X_n$ is…

The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of…

We consider a branching random walk in a random space-time environment of disasters where each particle is killed when meeting a disaster. This extends the model of the "random walk in a disastrous random environment" introduced by [15]. We…

We investigate excited random walks on $\Z^d, d\ge 1,$ and on planar strips $\Z\times\{0,1,...,L-1\}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of the…

In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $\alpha\in (1, 2)$ and obtain the asymptotic form of the…

We consider multidimensional random walks in pyramids, which by definition are cones formed by finite intersections of half-spaces. The main object of interest is the survival probability $\mathbb{P}(\tau>n)$, $\tau$ denoting the first exit…

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…

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…