arXiv subjects
Publications and source records attributed to Luiz Renato Fontes.
We study a random walk in $\mathbb{Z}^d$ in a dynamical random environment constituted of independent null recurrent birth-and-death chains, and establish a scaling limit result for the position of the walker in terms of a sublinear time change of Brownian motion, thus establishing subdiffusivity, in contrast to a diffusivity result established previously in an ergodic version of the model.
We take scaling limits of the Bouchaud and Dean trap model on Parisi's tree in time scales where the dynamics is either ergodic (close to equilibrium) or aging (far from equilibrium). These results follow from a continuity theorem formulated for a certain kind of process on trees, which we call a cascading jump evolution, defined in terms of a collection of jump functions, with a cascading structure given by the tree.
We study a system of random walks, known as the frog model, starting from a profile of independent Poisson($λ$) particles per site, with one additional active particle planted at some vertex $\mathbf{o}$ of a finite connected simple graph $G=(V,E)$. Initially, only the particles occupying $\mathbf{o}$ are active. Active particles perform $t \in \mathbb{N} \cup \{\infty \}$ steps of the walk they picked before vanishing and activate all inactive particles they hit. This system is often taken as a model for the spread of an epidemic over a population. Let $\mathcal{R}_t$ be the set of vertices which are visited by the process, when active particles vanish after $t$ steps. We study the susceptibility of the process on the underlying graph, defined as the random quantity $\mathcal{S}(G):=\inf \{t:\mathcal{R}_t=V \}$ (essentially, the shortest particles' lifetime required for the entire population to get infected). We consider the cases that the underlying graph is either a regular expander or a $d$-dimensional torus of side length $n$ (for all $d \ge 1$) $\mathbb{T}_d(n)$ and determine the asymptotic behavior of $\mathcal{S} $ up to a constant factor. In fact, throughout we allow the particle density $λ$ to depend on $n$ and for $d \ge 2$ we determine the asymptotic behavior of $\mathcal{S}(\mathbb{T}_d(n))$ up to smaller order terms for a wide range of $λ=λ_n$.
We consider the equilibrium surface of the Random Average Process started from an inclined plane, as seen from the height of the origin, obtained in [Ferrari & Fontes, 1998], where its fluctuations were shown to be of order of the square root of the distance to the origin in one dimension, and the square root of the log of that distance in two dimensions (and constant in higher dimensions). Remarkably, even if not pointed out explicitly in [Ferrari & Fontes, 1998], the correlation structure of those fluctuations is given in terms of the Green's function of a certain random walk, and thus corresponds to those of Discrete Gaussian Free Fields. In the present paper we obtain the scaling limit of those fluctuations in one and two dimensions, in terms of Gaussian processes, in the sense of finite dimensional distributions. In one dimension, the limit is given by Brownian Motion; in two dimensions, we get a process with a discontinuous covariance function.
We introduce the following model for the evolution of a population. At every discrete time $j\geq 0$ exactly one individual is introduced in the population and is assigned a death probability $c_j$ sampled from $C$, a fixed probability distribution. We think of $c_j$ as a genetic marker of this individual. At every time $n\geq 1$ every individual in the population dies or not independently of each other with its corresponding death probability $c_j$. We show that the population size goes to infinity if and only if $E(1/C)=\infty$. This is in sharp contrast with the model with constant $c$ and with the model in random environment (same random $c_n$ for all individuals at time $n$). Both of these models are always positive recurrent. Thus, genetic diversity does seem to help survival! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the $c'$s. For certain $C$ distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.
Consider the weakly asymmetric simple exclusion processes on the one-dimensional torus. We study the non-equilibrium fluctuation of a class of additive functionals, and show that its scaling limit is a Gaussian process. The proof is mainly based on the results obtained and techniques developed by Jara and Menezes [Non-equiliburim fluctuations of interacting particle systems, arXiv:1810.09526].
We refine previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any dimension $d \ge 1$ and requires only a moment condition slightly stronger than finite first moment. For heavy-tailed interarrival times, we prove a Complete Convergence Theorem and examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of distribution attracted to a stable law of index 1 for which the critical value vanishes.
We consider a discrete time population model for which each individual alive at time $n$ survives independently of everybody else at time $n+1$ with probability $β_n$. The sequence $(β_n)$ is i.i.d. and constitutes our random environment. Moreover, at every time $n$ we add $Z_n$ individuals to the population. The sequence $(Z_n)$ is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts). We apply our results to a particular $(Z_n)$ distribution and deterministic $β$. This particular case shows a rather unusual phase transition in $β$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.
We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb Z}^d$ , and its jump rate at $({\mathbf x},t)$ is given by a fixed function $φ$ of the state of a birth-and-death (BD) process at $\\mathbf x$ on time $t$; BD processes at different sites are independent and identically distributed, and $φ$ is assumed non increasing and vanishing at infinity. We derive a LLN and a CLT for the particle position when the environment is 'strongly ergodic'. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give $n$ jumps; and we also impose conditions on the initial (product) environmental initial distribution. We also present results on the asymptotics of the environment seen by the particle (under different conditions on $φ$).
We derive the scaling limit for the Hierarchical Random Hopping dynamics for the non cascading 2-GREM at low temperatures and time scales where the dynamics is close to equilibrium. The {\em fine tuning} phenomenon plays a role (under certain choices of parameters of the model), yielding three dynamical regimes. In contrast to the {\em cascading} case, the pairs of first and second level energies have fluctuations which scale with the volume, and this leads to a family of time scales where we see the dynamics moving through only a part of the full low temperature energy landscape.
We establish central limit theorems for the position and velocity of the charged particle in the mechanical particle model introduced in the paper "Limit velocity for a driven particle in a random medium with mass aggregation" (https://doi.org/10.1016/S0246-0203(00)01059-1).
We consider a random walk with catastrophes which was introduced to model population biology. It is known that this Markov chain gets eventually absorbed at $0$ for all parameter values. Recently, it has been shown that this chain exhibits a metastable behavior in the sense that it can persist for a very long time before getting absorbed. In this paper we study this metastable phase by making the parameters converge to extreme values. We obtain four different limits that we believe shed light on the metastable phase.
We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph G=(V,E): an individual is attached to each site x in V, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate lambda>0; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution mu such that mu(t,infty) = t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if |V| < 2 + (2 alpha -1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda; and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive probability to survive
We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution $μ$ is heavier than $t^{-α}$ for some $α<1$ (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if $μ$ has decreasing hazard rate and tail bounded by $t^{-α}$ with $α>1$, then the critical value is positive in the one-dimensional case. A more robust and much simpler argument shows that the critical value is positive in any dimension whenever the interarrival distribution has a finite second moment.
We propose a variation of the GMS model of evolution of species. In this version, as in the GMS model, at each birth, the new species in the system is labeled with a random fitness mark, but in our variation, to each extinction event is associated a random threshold mark and all species with fitness lower than the threshold are removed from the system. We present necessary and suficient criteria for the recurrence and transience of the empty configuration of species; we show the existence of a long time limit distribution of species in the system, and present necessary and suficient criteria for the finiteness of the number of species in that distribution. There is a remarkable symmetry between both sets of criteria.
We propose a metric space of coalescing pairs of paths on which we are able to prove (more or less) directly convergence of objects such as the persistence probability in the (one dimensional, nearest neighbor, symmetric) voter model or the diffusively rescaled weight distribution in a silo model (as well as the equivalent output distribution in a river basin model), interpreted in terms of (dual) diffusively rescaled coalescing random walks, to corresponding objects defined in terms of the Brownian web.
We study a pure death process. At each discrete time every individual dies or not independently of each other with a constant probability. We give examples showing that in a certain limit extinction happens along a path where one and only one individual is lost at a time. We also exhibit an example for which such a path goes from infinity to 0 in a finite time. This is what we call a process implosion.
We derive scaling limit results for the Random Hopping Dynamics for the cascading two-level GREM at low temperature at extreme time scales. It is known that in the cascading regime there are two static critical temperatures. We show that there exists a (narrow) set of fine tuning temperatures; when they lie below the static lowest critical temperature, three distinct dynamical phases emerge below the lowest critical temperature, with three different types of limiting dynamics depending on whether the temperature is (well) above or below, or at a fine tuning temperature, all of which are given in terms of K processes. We also derive scaling limit results for temperatures between the lowest and he highest critical ones, as well as aging results for all the limiting processes mentioned above, by taking a second small time limit.