arXiv subjects

Vladas Sidoravicius

Publications and source records attributed to Vladas Sidoravicius.

85 records · Page 5Linked to original sources

arXiv2008

A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains

We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time $t$, we have an "aggregate" consisting of $\Bbb{Z}\cap[0,R(t)]$ [with $R(t)$ a positive integer]. We also have $N(i,t)$ particles at $i$, $i>R(t)$. All these particles perform independent continuous-time symmetric simple random walks until the first time $t'>t$ at which some particle tries to jump from $R(t)+1$ to $R(t)$. The aggregate is then increased to the integers in $[0,R(t')]=[0,R(t)+1]$ [so that $R(t')=R(t)+1$] and all particles which were at $R(t)+1$ at time $t'{-}$ are removed from the system. The problem is to determine how fast $R(t)$ grows as a function of $t$ if we start at time 0 with $R(0)=0$ and the $N(i,0)$ i.i.d. Poisson variables with mean $μ>0$. It is shown that if $μ<1$, then $R(t)$ is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured that $R(t)$ will grow linearly in $t$ if $μ$ is large enough.

math.PR↗

arXiv2008

Percolation for the Vacant Set of Random Interlacements

We investigate random interlacements on Z^d, d bigger or equal to 3. This model recently introduced in arXiv:0704.2560 corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time-shift tending to infinity at positive and negative infinite times. A non-negative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d bigger or equal to 3 the vacant set at level u percolates when u is small. This solves an open problem of arXiv:0704.2560, where this fact has only been established when d is bigger or equal to 7. It also completes the proof of the non-degeneracy in all dimensions d bigger or equal to 3 of the critical parameter introduced in arXiv:0704.2560.

math.PR↗

arXiv2007

A Problem in Last-Passage Percolation

Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $π\subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with the last coordinate increasing along the path, and of length $n$. Define for such paths $W(π) = \text{number of vertices $π_i, 1 \le i \le n$, with}X(π_i) = e^b$. Finally let $N_n(\al) = \text{number of paths $π$ of length $n$ starting at $π_0 = \bold 0$ and with $W(π) \ge \al n$.}$ We establish several properties of $\lim_{n \to \infty} [N_n]^{1/n}$.

math.PR↗

arXiv2006

Polymer pinning in a random medium as influence percolation

In this article we discuss a set of geometric ideas which shed some light on the question of directed polymer pinning in the presence of bulk disorder. Differing from standard methods and techniques, we transform the problem to a particular dependent percolative system and relate the pinning transition to a percolation transition.

math.PR↗

arXiv2006

Pinning of polymers and interfaces by random potentials

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by, for example, having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction of its monomers at 0 with high probability. This critical point may differ for the quenched, annealed and deterministic cases. We show that self-averaging occurs, meaning that the quenched free energy and critical point are nonrandom, off a null set. We evaluate the critical point for a deterministic interaction ($u$ without added randomness) and establish our main result that the critical point in the quenched case is strictly smaller. We show that, for every fixed $u\in\mathbb{R}$, pinning occurs at sufficiently low temperatures. If the excursion length distribution has polynomial tails and the interaction does not have a finite exponential moment, then pinning occurs for all $u\in\mathbb{R}$ at arbitrary temperature. Our results apply to other mathematically similar situations as well, such as a directed polymer that interacts with a random potential located in a one-dimensional defect, or an interface in two dimensions interacting with a random potential along a wall.

math.PR↗

arXiv2006

The spread of a rumor or infection in a moving population

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on $\mathbb{Z}^d$, with jump rate $D_A$. These particles are called $A$-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-),x\in\mathbb{Z}^d$, are i.i.d., mean-$μ_A$ Poisson random variables. In addition, there are $B$-particles which perform continuous-time simple random walks with jump rate $D_B$. We start with a finite number of $B$-particles in the system at time 0. $B$-particles are interpreted as individuals who have heard a certain rumor or who are infected. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if $\widetilde{B}(t):=\{x\in\mathbb{Z}^d:$ a $B$-particle visits $x$ during $[0,t]\}$ and $B(t)=\widetilde{B}(t)+[-1/2,1/2]^d$, then we investigate the asymptotic behavior of $B(t)$. Our principal result states that if $D_A=D_B$ (so that the $A$- and $B$-particles perform the same random walk), then there exist constants $0<C_i<\infty$ such that almost surely $\mathcal{C}(C_2t)\subset B(t)\subset \mathcal{C}(C_1t)$ for all large $t$, where $\mathcal{C}(r)=[-r,r]^d$. In a further paper we shall use the results presented here to prove a full ``shape theorem,'' saying that $t^{-1}B(t)$ converges almost surely to a nonrandom set $B_0$, with the origin as an interior point, so that the true growth rate for $B(t)$ is linear in $t$. If $D_A\ne D_B$, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_1t)$ eventually.

math.PR↗

arXiv2005

Percolation theory

This is a survey article to be part of the Encyclopedia of Mathematical Physics, to be published by Elsevier in the beginning of 2006.

math.PR↗

arXiv2005

Nonuniqueness for specifications in $\ell^{2+ε}$

For every $p>2$, we construct a regular and continuous specification ($g$-function), which has a variation sequence that is in $l^p$ and which admits multiple Gibbs measures. Combined with a recent result of Johansson and Oberg, this determines the optimal modulus of continuity for a specification which admits multiple Gibbs measures.

math.PR↗

arXiv2004

A phase transition in a model for the spread of an infection

We show that a certain model for the spread of an infection has a phase transition in the recuperation rate. The model is as follows: There are particles or individuals of type A and type B, interpreted as healthy and infected, respectively. All particles perform independent, continuous time, simple random walks on Z^d with the same jump rate D. The only interaction between the particles is that at the moment when a B-particle jumps to a site which contains an A-particle, or vice versa, the A-particle turns into a B-particle. All B-particles recuperate (that is, turn back into A-particles) independently of each other at a rate lamda. We assume that we start the system with N_A(x,0-) A-particles at x, and that the N_A(x,0-), x in Z^d, are i.i.d., mean mu_A Poisson random variables. In addition we start with one additional B-particle at the origin. We show that there is a critical recuperation rate lambda_c > 0 such that the B-particles survive (globally) with positive probability if lambda < lamda_c and die out with probability 1 if lambda > \lamda_c.

math.PR↗

arXiv2003

A shape theorem for the spread of an infection

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous time simple random walk on the d-dimensional lattice. These particles are called A-particles and move independently of each other. We assume that we start the system with a Poisson number of particles at each lattice site x, with the number of particles at different x's i.i.d. In addition, there are a finite number of B-particles which perform the same continuous time simple random walks as the A-particles. A- and B-particles are interpreted as individuals who are healthy or infected, respectively. The B-particles move independently of each other. The only interaction is that when a B-particle and an A-particle coincide, the latter instantaneously turns into a B-particle. Let B(t) be the set of sites visited by a B-particle during [0,t]. We show that B(t) grows linearly in time and has an asymptotic shape; more precisely, there exists a non-random convex, compact set B_0 such that almost surely, for all 0 < a <1, (1-a)tB_0 is contained in B(t) and B(t) is contained in (1+a)tB_0 eventually.

math.PR↗

arXiv2003

A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models

We study families of dependent site percolation models on the triangular lattice ${\mathbb T}$ and hexagonal lattice ${\mathbb H}$ that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of Aizenman-Burchard [3]) as independent site percolation on ${\mathbb T}$.

math.PR↗

arXiv2001

Cardy's Formula for some Dependent Percolation Models

We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices); it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.

cond-mat.stat-mech↗

arXiv2001

Approach to Fixation for Zero-Temperature Stochastic Ising Models on the Hexagonal Lattice

We investigate zero-temperature dynamics on the hexagonal lattice H for the homogeneous ferromagnetic Ising model with zero external magnetic field and a disordered ferromagnetic Ising model with a positive external magnetic field h. We consider both continuous time (asynchronous) processes and, in the homogeneous case, also discrete time synchronous dynamics (i.e., a deterministic cellular automaton), alternating between two sublattices of H. The state space consists of assignments of -1 or +1 to each site of H, and the processes are zero-temperature limits of stochastic Ising ferromagnets with Glauber dynamics and a random (i.i.d. Bernoulli) spin configuration at time 0. We study the speed of convergence of the configuration $σ^t$ at time t to its limit $σ^{\infty}$ and related issues.

math.PR↗