arXiv subjects
Publications and source records attributed to Cristian F. Coletti.
In this work we study asymptotic properties of a long range memory random walk known as elephant random walk. First we prove recurrence and positive recurrence for the elephant random walk. Then, we establish the transience regime of the model. Finally, under the Poisson Hypothesis, we study the replica mean field limit for this random walk and we obtain an upper bound for the expected distance of the walker from the origin.
We introduce a system of coalescing random paths with radialbehavior in a subsetof the plane. We call it theDiscrete Radial Poissonian Web. We show that underdiffusive scaling this family converges in distribution toa mapping of a restrictionof the Brownian Web.
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
We study the accessibility percolation model on infinite trees. The model is defined by associating an absolute continuous random variable $X_v$ to each vertex $v$ of the tree. The main question to be considered is the existence or not of an infinite path of nearest neighbors $v_1,v_2,v_3\ldots$ such that $X_{v_1} 0$ is a given constant. We show that there is a percolation threshold at $α_c =1$ such that there is percolation if $α> 1$ and there is absence of percolation if $α\leq 1$. Moreover, we study the event of percolation starting at any vertex, as well as the continuity of the percolation probability function. Finally, we provide a comparison between this model with the well known $F^α$ record model. We also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research.
We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $λ$. There is a threshold for $λ$, which is called $λ_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $λ_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $λ_s$ is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $λ_w$ is the inverse of a certain function of the reproduction rates, which we denote by $K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where $λ_w=1/K_w$ were known; here we provide an example where $λ_w>1/K_w$.
We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory which exhibits a phase transition from diffusive to superdiffusive behaviour. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.
We propose a stochastic model describing a process of awareness, evaluation and decision-making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2}. In this model 0 stands for ignorants, 1 for aware and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate lambda. At rate alpha an agent in aware state becomes an adopter due to the influence of adopters neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate one. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.
In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.
We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.
We consider an interacting particle system representing the spread of a rumor by agents on the $d$-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0,1,2}. Here 0 stands for ignorants, 1 for spreaders and 2 for stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate λ. At rate αa spreader becomes a stifler due to the action of other (nearest neighbor) spreaders. Finally, spreaders and stiflers forget the rumor at rate one. We study sufficient conditions under which the rumor either becomes extinct or survives with positive probability.
We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and sources such that there is a microscopic shock at every time $t$; denote $Z(t)$ its position. We show that the mean and variance of $Z(t)$ are linear functions of $t$ and compute explicitely the respective constants in function of the left and right densities. Furthermore, we describe the dependence of $Z(t)$ on the initial configuration in the scale $\sqrt t$ and, as a corollary, prove a central limit theorem.