Related papers: Branching capacity of a random walk range

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 consider a continuous-time symmetric branching random walk on the $d$-dimensional lattice, $d\ge 1$, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk…

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…

We revisit the problem of deriving the mean-field values of avalanche critical exponents in systems with absorbing states. These are well-known to coincide with those of an un-biased branching process. Here, we show that for at least 4…

Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on…

We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{\"i} [KMSS94]. This settles, in…

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the…

A random walk starts from the origin of a d-dimensional lattice. The occupation number n(x,t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sums M(t) of observables…

Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching random walk started from the origin. Write $Z_n(S)$ for the number of particles located in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by…

We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, the endpoint…

In this paper, we consider random walk in random environment on $\mathbb{Z}^{d}\,(d\geq1)$ and prove the Strassen's strong invariance principle for this model, via martingale argument and the theory of fractional coboundaries of Derriennic…

In this work, we study the large deviation properties of random walk in a random environment on $\mathbb{Z}^d$ with $d\geq1$. We start with the quenched case, take the point of view of the particle, and prove the large deviation principle…

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a "mating-of-trees" type bijection. This class includes the…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and…

Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices…

We present some exact results on the behavior of Branching and Annihilating Random Walks, both in the Directed Percolation and Parity Conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the…

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized…

In this paper, we study the dynamics of a random walker diffusing on a disordered one-dimensional lattice with random trappings. The distribution of escape probabilities is computed exactly for any strength of the disorder. These…

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Z^d that is an extension of a result of Bolthausen, Sznitman and Zeitouni (2003). We use this result, along with the lace expansion for…