Related papers: Second order term of cover time for planar simple …
The purpose of these short notes is to provide a concise proof of a celebrated theorem by Dembo, Peres, Rosen and Zeitouni, which settles the leading order of cover times in the small-\varepsilon regime.
The deterministic random walk is a deterministic process analogous to a random walk. While there are some results on the cover time of the rotor-router model, which is a deterministic random walk corresponding to a simple random walk,…
Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of…
We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is…
We investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v. For the continuous-time problem (Brownian motion), we provide a simple alternative proof of Newman's recent…
Coalescing-branching random walks, or {\em cobra walks} for short, are a natural variant of random walks on graphs that can model the spread of disease through contacts or the spread of information in networks. In a $k$-cobra walk, at each…
We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a…
A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…
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.…
The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting…
We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by…
We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$…
We present an analytical method for computing the mean cover time of a random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This…
We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is…
The cover time is defined as the time needed for a random walker to visit every site of a confined domain. Here, we focus on persistent random walks, which provide a minimal model of random walks with short range memory. We derive the exact…
The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how fast this "deterministic random walk" covers all…
Sharp estimates for the length of loop erased random walk between two vertices on the [n]^d -torus, d > 4, are established. The mean length is order n^{d/2} . In dimension 4 we have only an upper bound.
Let $(X,d)$ be a geodesic Gromov-hyperbolic space, $o \in X$ a basepoint and $\mu$ a countably supported non-elementary probability measure on $\operatorname{Isom}(X)$. Denote by $z_n$ the random walk on $X$ driven by the probability…
We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the…
A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is…