Related papers: On the range of a two-dimensional conditioned simp…

We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^\alpha$, with $\alpha\in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that…

We prove sharp asymptotic estimates for the rate of escape of the two-dimensional simple random walk conditioned to avoid a fixed finite set. We derive it from asymptotics available for the continuous analogue of this process (cf…

We study a $d$-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function $h$ for the killed process and then…

This thesis examines linearly edge-reinforced random walks on infinite trees. In particular, recurrence and transience of such random walks on general (fixed) trees as well as on Galton-Watson trees (i.e. random trees) is characterized, and…

We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the…

We prove a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $\gamma$. First, we establish that if…

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,…

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes…

The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx…

We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience,…

We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h which is roughly the size of each step of the walk. We prove uniform bounds with…

Deterministic walks over a random set of points in one and two dimensions (d=1,2) are considered. Points (``cities'') are randomly scattered in R^d following a uniform distribution. A walker (a ``tourist''), at each time step, goes to the…

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that…

We study the once-reinforced random walk on $\mathbb Z^d$, which is a self-interacting walk that has a higher probability to cross edges that were already visited. We prove that the walk is transient when $d\ge 6$ and when the reinforcement…

Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence,…

It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on…

Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $\alpha \not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we…

We develop a general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via…