arXiv subjects
Publications and source records attributed to Leonardo T. Rolla.
We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.
We study an infinite system of moving particles, where each particle is of type A or B. Particles perform independent random walks at rates D_A>0 and D_B>0, and the interaction is given by mutual annihilation A+B->0. The initial condition is i.i.d. with finite first moment. We show that this system is site-recurrent, that is, each site is visited infinitely many times. We also generalize a lower bound on the density decay of Bramson and Lebowitz by considering a construction that handles different jump rates.
We consider a strictly convex billiard table with $C^2$ boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation distribution corresponds to the physical situation where either the scale of the surface irregularities is smaller than but comparable to the diameter of the reflected object, or the billiard ball is not perfectly rigid. We prove that for a large class of such perturbations the resulting Markov chain is uniformly ergodic, although this is not true in general.
Let $S$ be a countable set provided with a partial order and a minimal element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a quasi-stationary distribution. We use Holley inequality to obtain sufficient conditions under which the following hold. The trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on $S$, when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.
In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including $\mathbb{Z}^d$, and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate $λ$ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case $λ< + \infty$ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.
Given a Poisson point process on R, assign either one or two marks to each point of this process, independently of the others. We study the motion of a particle that jumps deterministically from its current location to the nearest point of the Poisson point process which still contains at least one mark, and removes one mark per each visit. A point of the Poisson point process which is left with no marks is removed from the system. We prove that the presence of any positive density of double marks leads to the eventual removal of every Poissonian point.
We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with probability $q$. We study properties of the critical curve which delimits the set of pairs $(p,q)$ for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.
We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs.
In this paper we consider independent site percolation in a triangulation of $\mathbb{R}^2$ given by adding $\sqrt{2}$-long diagonals to the usual graph $\mathbb{Z}^2$. We conjecture that $p_c=\frac{1}{2}$ for any such graph, and prove it for almost every such graph.
We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $λ> 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $ζ$ and the sleeping rate $λ$. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small $λ$) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents ($β= 1$, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.
In this work we introduce the discrete-space broken line process (with discrete and continues parameter values) and derive some of its properties. We explore polygonal Markov fields techniques developed by Arak-Surgailis. The discrete version is presented first and a natural continuum generalization to a continuous object living on the discrete lattice is then proposed and studied. The broken lines also resemble the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and self-dual. For such distributions there is a law of large numbers and the process extends to the infinite lattice. A proof of Burke's theorem emerges from the construction. We present a simple proof of the explicit law of large numbers for last passage percolation as an application. Finally we show that the exponential and geometric distributions are the only non-trivial ones that yield self-duality.
In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed.
In this paper we present an efficient numerical approach based on the Renormalization Group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the Renormalization Group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.
We study the long-time asymptotics of a certain class of nonlinear diffusion equations with time-dependent diffusion coefficients which arise, for instance, in the study of transport by randomly fluctuating velocity fields. Our primary goal is to understand the interplay between anomalous diffusion and nonlinearity in determining the long-time behavior of solutions. The analysis employs the renormalization group method to establish the self-similarity and to uncover universality in the way solutions decay to zero.
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
Lecture Notes. Minicourse given at the workshop "Activated Random Walks, DLA, and related topics" at IMéRA-Marseille, March 2015.
We consider a self-interacting process described in terms of a single-server system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space-time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit. We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.
* ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the lattice, with a Markovian continuous-time evolution. Active particles perform random walks without interaction, and they may as well change their state to passive, then stopping to jump. When particles of both types occupy the same site, they all become active. This model exhibits phase transition in the sense that for low initial densities the system locally fixates and for high densities it keeps active. Though extensively studied in the physics literature, the matter of giving a mathematical proof of such phase transition remained as an open problem for several years. In this work we identify some variables that are sufficient to characterize fixation and at the same time are stochastically monotone in the model's parameters. We employ an explicit graphical representation in order to obtain the monotonicity. With this method we prove that there is a unique phase transition for the one-dimensional finite-range random walk. Joint with V. Sidoravicius. * BROKEN LINE PROCESS * We introduce the broken line process and derive some of its properties. Its discrete version is presented first and a natural generalization to the continuum is then proposed and studied. The broken lines are related to the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and reversible. One application is a simple proof of the explicit law of large numbers for last passage percolation with exponential and geometric distributions. Joint with V. Sidoravicius, D. Surgailis, and M. E. Vares.