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This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function…

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great…

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum…

The paper analyzes a specific class of random walks on quotients of $X:=\text{SL}(k,{\Bbb R})/ \Gamma$ for a lattice $\Gamma$. Consider a one parameter diagonal subgroup, $\{g_t\}$, with an associated abelian expanding horosphere, $U\cong…

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent…

Let X be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the "forward" neighbours of the actual state, i.e., it does not go back along the…

We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite…

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's…

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\gamma-\epsilon$, where $\gamma$…

Why do we move forward when we walk? Our legs undergo periodic motion and thus possess no net change in position; however, our bodies do possess a net change in position and we are propelled forward. From a geometric perspective, this…

Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random…

As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of…

We show under weak hypotheses that the pushforward $\{Z_no\}$ of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either…

Let $\Gamma$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $\mu$ a probability measure on $\Gamma$ whose support generates a non-elementary subsemigroup. Under the assumption that $\mu$ has a finite…

L\'{e}vy walks are a particular type of continuous-time random walks which results in a super-diffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a…

The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary…

Diffusion is a central phenomenon in almost all fields of natural science revealing microscopic processes from the observation of macroscopic dynamics. Here, we consider the paradigmatic system of a single atom diffusing in a periodic…

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study…

Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to…

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products $ A \wr B = \bigoplus_B A \rtimes B $, where $A$ is any countable group and…