Related papers: Localization of reinforced random walks

A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study…

In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra A to the study of a polynomial of degree one, evaluated on the enlarged algebra A x M r…

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting…

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this…

Locally Markov walks are natural generalizations of classical Markov chains, where instead of a particle moving independently of the past, it decides where to move next depending on the last action performed at the current location. We…

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem…

We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and…

We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schr\"odinger operator, this exponential…

Random walks with memory typically involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the statistics of first-passage and…

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In…

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we…

The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph, and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked…

Continuous time random walks have been developed as a straightforward generalisation of classical random walk processes. Some 10 years ago, Fogedby introduced a continuous representation of these processes by means of a set of Langevin…

We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then…

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a…

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic…

A number of papers have examined various aspects of "random random" walks on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this…

A distinguishing property of communities in networks is that cycles are more prevalent within communities than across communities. Thus, the detection of these communities may be aided through the incorporation of measures of the local…

A distinguishing property of communities in networks is that cycles are more prevalent within communities than across communities. Thus, the detection of these communities may be aided through the incorporation of measures of the local…

We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where…