Supercritical Contact Process Research Articles

Complete convergence theorem for a two-level contact process

We study a two-level contact process. We think of fleas living on a species of animals. The animals are a supercritical contact process in $\mathbb{Z}^d$. The contact process acts as the random environment for the fleas. The fleas do not affect the animals, give birth at rate $\mu$ when they are living on a host animal, and die at rate $\delta$ when they do not have a host animal. The main result is that if the contact process is supercritical and the fleas survive then the complete convergence theorem holds. This is done using a block construction so as a corollary we conclude that the fleas die out at their critical value.

Exponential growth and continuous phase transitions for the contact process on trees

We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $\lambda_1$ for weak survival, and the survival probability $p(\lambda)$ is continuous with respect to the infection rate $\lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $\lambda_1<\lambda_2$, which confirms a conjecture of Stacey's \cite{Stacey}. We also prove that if the contact process survives strongly at $\lambda$ then it survives strongly at a $\lambda'<\lambda$, which implies that the process does not survive strongly at the critical value $\lambda_2$ for strong survival.

Stochastic domination in space‐time for the contact process

AbstractLiggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for and (for infection rate sufficiently large) d‐ary homogeneous trees Td. In this paper, we prove some space‐time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on Td with . We show that, for large infection rate, there exists a subset Δ of the vertices of Td, containing a “positive fraction” of all the vertices of Td, such that the following holds: The contact process on Td observed on Δ stochastically dominates an independent spin‐flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on observed on certain d‐dimensional space‐time slabs stochastically dominates an i.i.d. Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.

Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process

We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d \ge 2$, in the voter model for the occupation time of the origin in dimensions $d \ge 7$, and for finite time integrals of non-constant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution. The tool developed for these purposes is a version of Stein's method adapted to positively associated random variables. In one dimension, letting $\boldsymbol{\xi}=(\xi_1,\ldots,\xi_m)$ be a positively associated mean zero random vector with components that obey the bound $|\xi_i| \le B, i=1,\ldots,m$, and whose sum $W = \sum_{i=1}^m \xi_i$ has variance 1, it holds that $$ d_1 \left(\mathcal{L}(W),\mathcal{L}(Z) \right) \leq 5B + \sqrt{\frac{8}{\pi}}\sum_{i \neq j} \mathbb{E}[\xi_i \xi_j] $$ where $Z$ has the standard normal distribution and $d_1(\cdot,\cdot)$ is the $L^1$ metric. Our methods apply in the multidimensional case with the $L^1$ metric replaced by a smooth function metric.

Uniformity of hitting times of the contact process

For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times t(x), defined for each site x as the first time at which it becomes infected. First, the family of random variables (t(x) - t(y))/vertical bar x-y vertical bar, indexed by x not equal y in Z(d) , is stochastically tight. Second, for each epsilon > 0 there exists x such that, for infinitely many integers n, t(nx) < t((n + 1)x) with probability larger than 1-epsilon. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchanc (2012).

Continuity of the asymptotic shape of the supercritical contact process

We prove the continuity of the shape governing the asymptotic growth of the supercritical contact process in $\mathbb{Z}^d$, with respect to the infection parameter. The proof is valid in any dimension $d \ge 1$.

Random walks generated by equilibrium contact processes

We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos (arXiv:1209.1511). We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.

Scaling of a random walk on a supercritical contact process

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.

Asymptotic Results for the MPL Estimators of the Contact Process of Order p

Let X be a discrete time contact process (CP) of order p on Z 2 as defined by Durrett and Levin (1994), that is a CP where the function of birth takes p different values. We study the maximum marginal pseudo-likelihood (MPL) estimator of the model based on space-time evolution of X, that is, T + 1 successive observations of X on a finite subset S of sites. When T → ∞, this estimator is consistent and asymptotically normal for a non vanishing supercritical CP. We also propose a statistical test for the parameters of the model and verify the asymptotic results by some simulation studies.

Parameter estimation for the supercritical contact process

Contact processes -- and, more generally, interacting particle processes -- can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. We begin a statistical study of the supercritical contact process that starts with a single infected site at the origin and is conditioned on survival of the infection. We consider the statistical problem of estimating the parameter $\lambda$ of the process on the basis of an observation of the process at a single time $t$. We propose an estimator of $\lambda$ and show that it is consistent and asymptotically normal as $t \rightarrow \infty$.

Decaying correlations for the supercritical contact process conditioned on survival

Contact processes -- and, more generally, interacting particle processes -- can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. In a forthcoming paper we shall begin a statistical study of the supercritical contact process $xi_t ^{\{0\}$ that starts with a single infected site at the origin and is conditioned on survival. There we shall consider the simplest statistical problem imaginable, that is, to find an estimator of the parameter of the process based on observing the set of infected sites at a single time $t$. We shall show that this estimator is consistent as $t \rightarrow \infty$ and establish its limit distribution after proper normalization. First, however, we must push some known properties of the contact process a little further. The present paper is devoted to these matters. In particular, we study the convex hull of the set of infected sites for the conditional $xi_t ^{\{0\}$ process as well as its spatial correlation. We find that under some restrictions this correlation decays faster than any negative power of the distance.

Statistics for the contact process

A d‐dimensional contact process is a simplified model for the spread of an infection on the lattice Zd. At any given time t≥0, certain sites x∈Zd are infected while the remaining once are healthy. Infected sites recover at constant rate 1, while healthy sites are infected at a rate proportional to the number of infected neighboring sites. The model is parametrized by the proportionality constant λ. If λ is sufficiently small, infection dies out (subcritical process), whereas if λ is sufficiently large infection tends to be permanent (supercritical process).In this paper we study the estimation problem for the parameter λ of the supercritical contact process starting with a single infected site at the origin. Based on an observation of this process at a single time t, we obtain an estimator for the parameter λ which is consistent and asymptotically normal as t→∞

An Extension of Kuczek's Argument to Nonnearest Neighbor Contact Processes

The right edge of a nearest neighbor supercritical contact process satisfies a central limit theorem.(8, 9) In this paper, a block construction is created to extend the argument by Kuczek(9) to the nonnearest neighbor case. The proof of the following fact in the nonnearest neighbor case is the key to the extension: There is a positive chance that the rightmost particle at time 0 infects the rightmost particle at every time.

Growth Profile and Invariant Measures for the Weakly Supercritical Contact Process on a Homogeneous Tree

It is known that the contact process on a homogeneous tree of degree $d+1\geq3$ has a weak survival phase, in which the infection survives with positive probability but nevertheless eventually vacates every finite subset of the tree. It is shown in this paper that in the weak survival phase there exists a spherically symmetric invariant measure whose density decays exponentially at infinity, thus confirming a conjecture of Liggett. The proof is based on a study of the relationships between various thermodynamic parameters and functions associated with the contact process initiated by a single infected site. These include (1) the growth profile, which determines the exponential rate of growth in space-time on the event of survival, (2) the exponential rate $\beta$ of decay of the hitting probability function at infinity (also studied by the author) and (3) the exponential rate $\beta$ of decay in time $t$ of the probability that the initial infected site is infected at time $t$. It is shown that $\beta$ is a strictly increasing function of the infection rate $\lambda$ in the weak survival phase, and that $\beta = 1/\sqrt{d}$ at the upper critical point $\lambda_2$ demarcating the boundary between the weak 2 and strong survival phases. It is also shown that $\eta < 1$ except at $\lambda_2$, where $\eta =1$.

EXISTENCE OF A CONSTANT FOR FINITE SYSTEM EXTINCTION

We show the existence of a constant γ∈(0, ∞) such that if τn is the extinction time for a supercritical contact process on [1, n]d starting from full occupancy, then {log(E[τn])}/nd tend to γ as n tends to infinity.

Limit set of a weakly supercritical contact process on a homogeneous tree

A conjecture of Liggett (9) concerning the regime of weak survival for the contact process on a homogeneous tree is proved. The conjecture is shown to imply that the Hausdor dimension of the limit set of such a contact process is no larger than half the Hausdor dimension of the space of ends of the tree. The conjecture is also shown to imply that at the boundary between weak survival and strong survival, the contact process survives only weakly, a theorem previously proved by Zhang (14). Finally, a stronger form of a theorem of Hawkes and Lyons concerning the Hausdor dimension of a Galton-Watson tree is proved.

Metastability of thed-dimensional contact process

We prove that thed-dimensional supercritical contact process exhibits metastable behavior, in the pathwise sense. This is done by proving the property of thermalization and using Mountford's theorem. We also extend some previous results on the loss of memory of the process.

SMALL DENSITY FLUCTUTAION FOR ONE DIMENSIONAL CONTACT PROCESSES UNDER NONEQUILIBRIUM

This paper shows that for a supercritical contact process in one dimension, if the initial distribution satisfies reasonable hypothdsis, then the first hitting time of certain set with anomalous small density is asymptotically exponentially distributed. We also show the similar result for the contact process on the finite set [0, N].

The critical contact process seen from the right edge

Durrett (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to it held starting in any semi-infinite initial state. We prove the same for the critical contact process. We also prove that the process starting with one particle, conditioned to survive until timet, converges to the unique invariant measure ast→∞.

Large-density fluctuations for the one-dimensional supercritical contact process

We consider the one-dimensional supercritical contact process. LetTv be the first time the process reaches a densityq larger than the equilibrium oneρ in the region [1⋯N]. We prove that, starting from equilibrium,TN/E(TN) converges to an exponential random time of mean one. In this way we extend previous results of Lebowitz and Schonmann.