We present general results for the contact process by a method which applies to all transitive graphs of bounded degree, including graphs of exponential growth. The model's infection rates are varied through a control parameter, for which two natural transition points are defined as: i. $λ_T$, the value up to which the infection dies out exponentially fast if introduced at a single site, and ii. $λ_H$, the threshold for the existence of an invariant measure with a non-vanishing density of infected sites. It is shown here that for all transitive graphs the two thresholds coincide. The method, which proceeds through partial differential inequalities for the infection density, yields also generally valid bounds on two related critical exponents. The main results discussed here were established by Bezuidenhout and Grimmett \cite{BG2} in an extension to the continuous-time process of the discrete-time analysis of Aizenman and Barsky \cite{AB}, and of the partially similar results of Menshikov \cite{M}. The main novelty here is in the direct derivation of the partial differential inequalities by an argument which is formulated for the continuum.
We consider the critical spread-out contact process in $\Zd$ with $d\geq 1$, whose infection range is denoted by $L\geq1$. The two-point function $τ_t(x)$ is the probability that $x\in\Zd$ is infected at time $t$ by the infected individual located at the origin $o\in\Zd$ at time 0. We prove Gaussian behavior for the two-point function with $L\geq L_0$ for some finite $L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $τ_{tT}$ with $t>0$ fixed and $T\to \infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any $b>…
We show that the critical value for the contact process on a vertex-transitive graph G with finitely many edges added and/or removed is the same as the critical value for the contact process on G.
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$…
We consider the critical spread-out contact process in Z^d with d\ge1, whose infection range is denoted by L\ge1. In this paper, we investigate the r-point function τ_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that, for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time t_i by the individual at the origin o\in Z^d at time 0. Together with the results of the 2-point function in [van der Hofstad and Sakai, Electron. J. Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially rely, we prove that the r-point functions converge to the mome…
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular …
We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $λ_c(\mathbb{Z})$, the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally).
In this paper we give an improved upper bound for critical value $λ_c$ of the basic contact process on the lattice $\mathbb{Z}^d$ with $d\geq 3$. As a direct corollary of out result, \[ λ_c\leq 0.384. \] when $d=3$.
We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb{Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $p$ (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric r…
In this paper we are concerned with the two-stage contact process on the lattice $\mathbb{Z}^d$ introduced in \cite{Krone1999}. We gives a limit theorem of the critical infection rate of the process as the dimension $d$ of the lattice grows to infinity. A linear system and a two-stage SIR model are two main tools for the proof of our main result.
Bezuidenhout and Grimmett proved that the critical contact process dies out. Here, we generalize the result to the so called contact process in a random evolving environment (CPREE), introduced by Erik Broman. This process is a generalization of the contact process where the recovery rate can vary between two values. The rate which it chooses is determined by a background process, which evolves independently at different sites. As for the contact process, we can similarly define a critical value in terms of survival for this process. In this paper we prove that this definition is independent o…
In this paper we are concerned with contact processes on open clusters of oriented percolation in $Z^d$, where the disease spreads along the direction of open edges. We show that the two critical infection rates in the quenched and annealed cases are equal with probability one and are asymptotically equal to $(dp)^{-1}$ as the dimension $d$ grows to infinity, where $p$ is the probability of edge `open'.
In the present paper, we study the relation between criticality and information transmission in the one-dimensional contact process with infection parameter $λ.$ To do this we define the {\it sensitivity} of the process to its initial condition. This sensitivity increases for values of $λ< λ_c, $ the value of the critical parameter. The main point of the present paper is that we show that actually it continues increasing even after $ λ_c $ and only starts decreasing for sufficiently large values of $λ.$ This provides a counterexample to the common belief that associates maximal information tra…
We study the behaviour of the contact process with rapid stirring on the lattice $\Z^d$ in dimensions $d\geq 3$. This process was studied earlier by Konno and Katori, who proved results for the speed of convergence of the critical value as the rate of stirring approaches infinity. %Katori has improved the result of Katori in $d\geq 3$ and has produced an interval where the critical value must lie. In this article we improve the results of Konno and Kattori and establish the sharp asymptotics of the critical value in dimensions $d\geq 3$.
In this paper we are concerned with contact processes with random edge weights on rooted regular trees. We assign i.i.d weights on each edge on the tree and assume that an infected vertex infects its healthy neighbor at rate proportional to the weight on the edge connecting them. Under the annealed measure, we define the critical value λ_c as the maximum of the infection rate with which the process will die out and define λ_e as the maximum of the infection rate with which the process dies out at exponential rate. We show that these two critical values satisfy an identical limit theorem and gi…
We consider the super-critical contact process on $\mathbb{Z}^d$. It is known that measures which dominate the upper invariant measure $μ$ converge exponentially fast to $μ$. However, the same is not true for measures which are below $μ$, as the time to infect a large empty region is related to its diameter. The result of this paper is the existence of a spectral gap in $L^2(μ)$, that is, the spectrum of the generator is empty inside an open strip $\{z\in\mathbb{C}: -λ<\Im(z)<0\}$ of the complex plane. This is equivalent to the fact that the variance of the semi-group of the contact process de…
We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $χ\in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $χ$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of…
Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $λ_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $ξ$ has an exponential tail. In this paper, we derive the first-order asymptotics of $λ_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $ξ$ is appropriately concentrated around its mean, we demonstrate that $λ_1(ξ) \sim 1/\mathbb{E} ξ$ as $\mathbb{E}ξ\rightarrow \infty$, which matches with the known asymptotics on the $d$-regular…
We introduce a method to prove metastability of the contact process on Erdős-Rényi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metast…
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.
In this paper we study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ_c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ_c^{T^n}\rightarrow1 and 2dλ_c^{Z^d}\rightarrow1 as n,d\rightarrow+\infty. Our result is a development of the conclusion that λ_c^{Z^d}\leq\frac{2.18}{d} shown in \cite{Dur1991}. To prove our main result, a crucial lemma about the probability of a simple random walk on a latt…
If we consider the contact process with infection rate $λ$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $λ_c$ of the infection rate is positive if the power $α>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $λ_c$ is zero for any value of $α>3$, and the contact process starting from all vertices infected, with a probability…
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 $λ_1$ for weak survival, and the survival probability $p(λ)$ is continuous with respect to the infection rate $λ$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $λ…
In this paper, we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connecting radius tending to infinity. We obtain that for any infection rate $λ>0$, the contact process on these graphs survives a time super-exponential in the number of vertices.
This paper studies contact processes on general countable groups. It is shown that any such contact process has a well-defined exponential growth rate, and this quantity is used to study the process. In particular, it is proved that on any nonamenable group, the critical contact process dies out.
A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $λ_1$ and $λ_2$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generation is $(n,a_1,\ldots, a_k)$ with $\max_i a_i \le Cn^{1-δ}$ and $\log(a_1 \cdots a_k)/\log n \to b$ as $n\to\infty$. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n}$ where $c=(k…
We consider the contact process with infection rate $λ$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $τ_{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $λ$ is smaller or larger than $λ_1(\mathbb{T}^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $λ< λ_1(\mathbb{T}^d)$, $τ_{G_n}$ grows logarithmically with $n$, while if $λ> λ_1(\mathbb{T}^d)$, it grows exponentially with $n$. This result differs f…
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
This paper is concerned with contact process with random vertex weights on regular trees, and study the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices $x$ and $y$ at rate $λρ(x)ρ(y)$ for some $λ>0$, where $\{ρ(x),{x\in T^d}\}$ are $i.i.d.$ vertex weights. We show that when $d$ is large enough there is a phase transition at $λ_c(d)\in(0,\infty)$ such that for $λ<λ_c(d)$ the contact process dies out, and for $λ>λ_c(d)$ the contact process survives with a positive pr…
Recently, by introducing the notion of cumulatively merged partition, Ménard and Singh provide a sufficient condition on graphs ensuring that the critical value of the contact process is positive. In this note, we show that the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.
We propose the following model for speciation and extinction. Birth and deaths occur according to spatially inhomogeneous contact rates. We assume that the ratio of the birth rate over the death rate at a site converges to some limit as the site goes to infinity. We show that the existence of a phase transition depends critically on the value of this limit.
