Related papers: An elementary solution of Gessel's walks in the qu…

We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational…

Prudent walks are self-avoiding walks on the square lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex is adjacent to the…

We construct a continuous-time non-commutative random walk on $U(\mathfrak{gl}_N)$ with dilation maps $U(\mathfrak{gl}_N)\rightarrow L^2(U(N))^{\otimes\infty}$. This is an analog of a continuous-time non-commutative random walk on the group…

In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a…

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive…

An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the…

We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $R^d$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no…

We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite…

We consider four examples of short step lattice paths confined to the quarter plane. These are the Kreweras, Reverse Kreweras, Gessel, and Mishna-Rechnitzer lattice paths.The Reverse Kreweras are straightforward to solve and thus…

Finding a local minimum or maximum of a function is often achieved through the gradient-descent optimization method. For a function in dimension d, the gradient requires to compute at each step d partial derivatives. This method is for…

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start-…

Many widely used network centralities are based on counting walks that meet specific criteria. This paper introduces a systematic framework for walk enumeration using generating functions. We introduce a first-passage decomposition that…

In this article, we consider several models of random walks in one or several dimensions, additionally allowing, at any unit of time, a reset (or "catastrophe") of the walk with probability $q$. We establish the distribution of the final…

We develop an approach for performing scaling analysis of $N$-step Random Walks (RWs). The mean square end-to-end distance, $\langle\vec{R}_{N}^{2}\rangle$, is written in terms of inner persistence lengths (IPLs), which we define by the…

We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions. The fractional random walk dynamics is governed by a master equation…

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $\lambda_S^{L^2}.$ We…

Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion formula to arbitrary powers of roots of the characteristic equation, that being a…

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…

A n-step Pearson-Gamma random walk in Rd starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained…

A previous paper (hep-lat/9311011) proposed a new kind of random walk on a spherically-symmetric lattice in arbitrary noninteger dimension $D$. Such a lattice avoids the problems associated with a hypercubic lattice in noninteger dimension.…