Related papers: The frog model with death revisited
In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature,…
In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs), known as frog model. The dynamics of this process is described as follows: There are some active particles, which perform independent SRWs, and sleeping…
We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph…
We introduce a generalized version of the frog model to describe the invasion of a parasite population in a spatially structured immobile host population with host immunity on the integer line. Parasites move according to simple symmetric…
In this paper, we consider generalized wind-tree models and $\Z^d$-covers over compact translation surfaces. Under suitable hypothesis, we prove recurrence of the linear flow in a generic direction and non-ergodicity of Lebesgue measure.
This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of…
We consider random dynamics on a uniform random recursive tree with $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in…
We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size…
We study the Tangled Nature model of macro evolution and demonstrate that the co-evolutionary dynamics produces an increasingly correlated core of well occupied types. At the same time the entire configuration of types becomes increasing…
In this paper, we show that the first passage time in the frog model on $\Z^d$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not holds at least with the standard diffusive scaling. The proof is…
Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$), $\epsilon…
Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still…
The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump-Mode-Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new…
A symmetric relation in the probabilistic Green's function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green's functions in other contexts. The first uses as primary tool…
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 prove that the edge-reinforced random walk on the ladder ${\mathbb{Z}\times\{1,2\}}$ with initial weights $a>3/4$ is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a…
By considering the B-field dynamical and studying its interaction with Ramond-Ramond (RR) background we observe the breaking of the B-field gauge symmetry in the effective action. This effect takes place due to non-perturbative coupling of…
We study a system of random walks, known as the frog model, starting from a profile of independent Poisson($\lambda$) particles per site, with one additional active particle planted at some vertex $\mathbf{o}$ of a finite connected simple…
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…
We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis…