Related papers: Spectral dimension and random walks on the two dim…

It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution…

We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to…

We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $\Theta(\sqrt{n})$. In particular this result applies to…

We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight…

In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian…

We study the local time of the anisotropic random walk on the two-dimensional lattice Z^2, by establishing the exact asymptotic behavior of the N-step return probability to the origin.

We prove Barlow--Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the…

For a simple (unbiased) random walk on a connected graph with $n$ vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most $O(n^3)$. We consider locally biased random walks, in which the probability…

We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends…

We consider a class of random graphs, called random brushes, which are constructed by adding linear graphs of random lengths to the vertices of Z^d viewed as a graph. We prove that for d=2 all random brushes have spectral dimension d_s=2.…

The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk…

Consider an infinite planar graph with uniform polynomial growth of degree d > 2. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of…

Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two…

We consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the…

Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even…

We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \varpropto…

Let $d$ be a positive integer and $A$ a set in $\mathbb{Z}^d$, which contains finitely many points with integer coordinates. We consider $X$ a standard random walk perturbed on the set $A$, that is, a Markov chain whose transition…

It has been recently suggested that a totally asymmetric exclusion process with two species on an open chain could exhibit spontaneous symmetry breaking in some range of the parameters defining its dynamics. The symmetry breaking is…

We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the…

We consider the branching capacity of the range of a simple random walk on $\mathbb Z^d$, with $d \ge 5$, and show that it falls in the same universality class as the volume and the capacity of the range of simple random walks and branching…