Related papers: Multi-excited random walks on integers

Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…

We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16].…

The random walk process underlies the description of a large number of real world phenomena. Here we provide the study of random walk processes in time varying networks in the regime of time-scale mixing; i.e. when the network connectivity…

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…

We consider random walks on dynamical networks where edges appear and disappear during finite time intervals. The process is grounded on three independent stochastic processes determining the walker's waiting-time, the up-time and down-time…

We consider a state-dependent, time-dependent, discrete random walks $X_t^{\{a_n\}}$ defined on natural numbers $\mathbb{N}$ (bent to a "stair" in $\mathbb{N}^2$) where the random walk depends on input of a positive deterministic sequence…

We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov…

We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right.…

We base ourselves on the construction of the two-dimensional random interlacements [12] to define the one-dimensional version of the process. For this constructions we consider simple random walks conditioned on never hitting the origin,…

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and…

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk…

In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a…

We consider a random walk in $\mathbb Z^d$ which jumps from a site $x$ to a nearest neighboring site $x+e$ (where $e\in V:=\{x\in\mathbb Z^d: |x|_1=1\}$) with probability $p_0(e)+\epsilon\xi(x,e)$. Here $\sum_e p_0(e)=1$, $p_0(e)> 0$,…

We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This…

We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating…

We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $\omega_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection…

Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous…

In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but…

We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n.

We consider random walks on edge coloured random graphs, where the colour of an edge reflects the cost of using it. In the simplest instance, the edges are coloured red or blue. Blue edges are free to use, whereas red edges incur a unit…