Luiz Renato Fontes
Research Subjects
Luiz Renato Fontes has a total of 59 co-authors. Here are the top co-authors by number of shared publications:
- Maria Eulalia Vares
Instituto de Matemática Universidade Federal do Rio de Janeiro RJ Brazil
8 shared publications - Domingos H. U. Marchetti
Universidade de São Paulo
5 shared publications - Pablo Augusto Ferrari
Universidad de Buenos Aires and IMAS-UBA-CONICET 1 , Buenos Aires
5 shared publications - Pierre Mathieu
University of Mons
5 shared publications - Charles M. Newman
New York University
4 shared publications - Errico Presutti
Gran Sasso Science Institute, L’Aquila, Italy
4 shared publications
Latest Publications
- 1 December 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (12)
Scaling limits of the Bouchaud and Dean trap model on Parisi’s tree in ergodic and aging time scales
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.
- 1 December 2025
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 66 (12)
Scaling limits of the Bouchaud and Dean trap model on Parisi’s tree in ergodic and aging time scales
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.
- Open Access
Scaling limit of an equilibrium surface under the Random Average Process
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.
Open AccessScaling limit of an equilibrium surface under the Random Average Process
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.
- Open Access
Does genetic diversity help survival?
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.Open AccessDoes genetic diversity help survival?
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. - Open Access
Additive functionals of exclusion processes from non-equilibrium
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].
Open AccessAdditive functionals of exclusion processes from non-equilibrium
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].
- Open Access
Renewal Contact Processes: phase transition and survival
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.Open AccessRenewal Contact Processes: phase transition and survival
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. - 15 March 2023
- Published by Elsevier in Stochastic Processes and their Applications
- Vol. 161, 102-136
Renewal Contact Processes: Phase transition and survival
- 15 March 2023
- Published by Elsevier in Stochastic Processes and their Applications
- Vol. 161, 102-136
Renewal Contact Processes: Phase transition and survival
- Open Access
Null recurrence and transience for a binomial catastrophe model in random environment
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 $\beta_n$. The sequence $(\beta_n)$ 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 $\beta$. This particular case shows a rather unusual phase transition in $\beta$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.Open AccessNull recurrence and transience for a binomial catastrophe model in random environment
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 $\beta_n$. The sequence $(\beta_n)$ 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 $\beta$. This particular case shows a rather unusual phase transition in $\beta$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence. - 1 January 2023
- Published by Springer Nature in Matemática Contemporânea
- Vol. 58 (7)
Contact Process under renewal cures — An overview of recent results
- 1 January 2023
- Published by Springer Nature in Matemática Contemporânea
- Vol. 58 (7)
Contact Process under renewal cures — An overview of recent results
- Open Access
- 1 January 2023
- Published by Institute of Mathematical Statistics in Electronic Journal of Probability
- Vol. 28 (none) , 1-26
Random walk in a birth-and-death dynamical environment
We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on Zd, and its jump rate at (x,t) is given by a fixed function φ of the state of a birth-and-death (BD) process at x, at 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 perform n consecutive jumps; and we also impose conditions on the initial (product) environmental distribution.
- 1 January 2023
- Published by Institute of Mathematical Statistics in Electronic Journal of Probability
- Vol. 28 (none) , 1-26
Open AccessRandom walk in a birth-and-death dynamical environment
We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on Zd, and its jump rate at (x,t) is given by a fixed function φ of the state of a birth-and-death (BD) process at x, at 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 perform n consecutive jumps; and we also impose conditions on the initial (product) environmental distribution.
- Open Access
Random walk in a birth-and-death dynamical environment
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 $\varphi$ 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 $\varphi$ 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 $\varphi$).Open AccessRandom walk in a birth-and-death dynamical environment
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 $\varphi$ 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 $\varphi$ 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 $\varphi$).