arXiv subjects

Maria Eulalia Vares

Publications and source records attributed to Maria Eulalia Vares.

18 recordsLinked to original sources

arXiv2024

Survival of one dimensional renewal contact process

The renewal contact process, introduced in $2019$ by Fontes, Marchetti, Mountford, and Vares, extends the Harris contact process in $\mathbb{Z}^d$ by allowing the possible cure times to be determined according to independent renewal processes (with some interarrival distribution $μ$) and keeping the transmission times determined according to independent exponential times with a fixed rate $λ$. We investigate sufficient conditions on $μ$ to have a process with a finite critical value $λ_c$ for any spatial dimension $d \geq 1$. In particular, we show that $λ_c$ is finite when $μ$ is continuous with bounded support or when $μ$ is absolutely continuous and has a decreasing hazard rate.

math.PR↗

arXiv2021

Polynomial ballisticity conditions in mixing environments

We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger, Drewitz, and Ram\'ırez (\emph{Comm. Pure Appl. Math,} {\bf 67}, (2014) 1947--1973). The novel idea therein was the construction of several simultaneous renormalization steps, providing more flexibility for seed estimates. For our proof, we indeed follow a similar path, and introduce a new mixing effective criterion which will be implied by the polynomial condition. This allows us to prove, in a mixing framework, the RWRE conjecture concerning the equivalence between each condition $(T^γ)|\ell$, for $γ\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work of Guerra (\emph{Ann. Probab.} {\bf 47} (2019) 3003--3054) and completes the answer about the meaning of condition $(T')|\ell$ in a mixing setting, an open question posed by Comets and Zeitouni (\emph{Ann. Probab.} {\bf 32} (2004) 880--914).

math.PR↗

arXiv2019

Contact process under renewals II

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.

math.PR↗

arXiv2019

Fast-reaction limit for Glauber-Kawasaki dynamics with two components

We consider the Kawasaki dynamics of two types of particles under a killing effect on a $d$-dimensional square lattice. Particles move with possibly different jump rates depending on their types. The killing effect acts when particles of different types meet at the same site. We show the existence of a limit under the diffusive space-time scaling and suitably growing killing rate: segregation of distinct types of particles does occur, and the evolution of the interface between the two distinct species is governed by the two-phase Stefan problem. We apply the relative entropy method and combine it with some PDE techniques.

math.PR↗

arXiv2018

Contact process under renewals I

Motivated by questions regarding long range percolation, we investigate a non-Markovian analogue of the Harris contact process in $\mathbb{Z}^d$: an individual is attached to each site $x \in \mathbb{Z}^d$, 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 $λ$; nevertheless, the possible recovery times for an individual are given by the points of a renewal process with heavy tail; the renewal processes are assumed to be independent for different sites. We show that the resulting processes have a critical value equal to zero.

math.PR↗

arXiv2015

Extinction time for a random walk in a random environment

We consider a random walk with death in $[-N,N]$ moving in a time dependent environment. The environment is a system of particles which describes a current flux from $N$ to $-N$. Its evolution is influenced by the presence of the random walk and in turn it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in $N$) for the survival probability up to time $t$ which goes as $c\exp\{-bN^{-2}t\}$, with $c$ and $b$ positive constants.

math.PR↗

arXiv2015

Exponential rate of convergence in current reservoirs

In this paper, we consider a family of interacting particle systems on $[-N,N]$ that arises as a natural model for current reservoirs and Fick's law. We study the exponential rate of convergence to the stationary measure, which we prove to be of the order $N^{-2}$.

math.PR↗

arXiv2015

Layered systems at the mean field critical temperature

We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)\sim γJ(γ(x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $ε$ and prove that for every $ε>0$ the systems exhibits phase transition provided $γ>0$ is small enough.

math.PR↗

arXiv2014

Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)=γJ(γ(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $γ^{A}$ (where $A\ge 2$ is fixed) and prove that for any $β$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $γ$ small enough.

math.PR↗

arXiv2013

First passage percolation and escape strategies

Consider first passage percolation on $\mathbb{Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_π(u,v)$ be the time from $u$ to $v$ for a path $π$ and $t(u,v)$ the minimal time among all paths from $u$ to $v$. We ask whether or not there exist points $x,y \in \mathbb{Z}^d$ and a semi-infinite path $π=(y_0=y,y_1,\dots)$ such that $t_π(y, y_{n+1})<t(x,y_n)$ for all $n$. Necessary and sufficient conditions on $F$ are given for this to occur. When the support of $F$ is unbounded, we also obtain results on the number of edges with large passage time used by geodesics.

math.PR↗

arXiv2012

Oriented percolation in a random environment

On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+ \colon y=i}$ are first declared to be bad or good with probabilities $\de$ and $1-\de$ respectively, independently of each other. Given the configuration of lines, sites on good lines are open with probability $p_{_G}>p_c$, the critical probability for the standard oriented site percolation on $\mathbb Z_+ \times \mathbb Z_+$, and sites on bad lines are open with probability $p_{_B}$, some small positive number, independently of each other. We show that given any pair $p_{_G}>p_c$ and $p_{_B}>0$, there exists a $δ(p_{_G}, p_{_B})>0$ small enough, so that for $δ\le δ(p_G,p_B)$ there is a strictly positive probability of oriented percolation to infinity from the origin.

math.PR↗

arXiv2012

Non equilibrium stationary state for the SEP with births and deaths

We consider the symmetric simple exclusion process in the interval $\La_N:=[-N,N]\cap\mathbb Z$ with births and deaths taking place respectively on suitable boundary intervals $I_+$ and $I_-$, as introduced in De Masi et al. (J. Stat. Phys. 2011). We study the stationary measure density profile in the limit $N\to\infty$

math-ph↗

arXiv2011

Current reservoirs in the simple exclusion process

We consider the symmetric simple exclusion process in the interval $[-N,N]$ with additional birth and death processes respectively on $(N-K,N]$, $K>0$, and $[-N,-N+K)$. The exclusion is speeded up by a factor $N^2$, births and deaths by a factor $N$. Assuming propagation of chaos (a property proved in a companion paper "Truncated correlations in the stirring process with births and deaths") we prove convergence in the limit $N\to \infty$ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.

math-ph↗

arXiv2011

Truncated correlations in the stirring process with births and deaths

We consider the stirring process in the interval $\La_N:=[-N,N]$ of $\mathbb Z$ with births and deaths taking place in the intervals $I_+:=(N-K,N]$, $K>0$, and respectively $I_-:=[-N,-N+K)$. We prove bounds on the truncated moments uniform in $N$ which yield strong factorization properties.

math.PR↗

arXiv2011

Study of a long range perturbation of a one-dimensional Kac model

We consider a one dimensional ferromagnetic Ising spin system with interactions that correspond to a $1/r^2$ long range perturbation of the usual Kac model. We apply a coarse graining procedure, widely used for higher-dimensional finite range Kac potentials, to describe the basic properties of the system and the relation with the mean field theory.

math-ph↗

arXiv2009

A system of grabbing particles related to Galton-Watson trees

We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree.

math.PR↗

arXiv2009

On a randomized PNG model with a columnar defect

We study a variant of poly-nuclear growth where the level boundaries perform continuous-time, discrete-space random walks, and study how its asymptotic behavior is affected by the presence of a columnar defect on the line. We prove that there is a non-trivial phase transition in the strength of the perturbation, above which the law of large numbers for the height function is modified.

math.PR↗

arXiv2004

One-dimensional random field Kac's model: localization of the phases

We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of the two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other.

math.PR↗