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Register a new userThe Symbiotic Contact Process
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We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $mu le 1$ for each particle at that site. We show the critical birth rate $lambda_c(mu)$ for weak survival is of order $sqrt{mu}$ as $mu to 0$. Mean-field calculations predict that when $mu < 1/2$ there is a discontinuous transition as $lambda$ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to physicists the paper that introduced the model, the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.
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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 $lambda_1$ and $lambda_2$ 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 $max_i a_i le Cn^{1-delta}$ and $log(a_1 cdots a_k)/log n to b$ as $ntoinfty$. We show that the critical value for local survival is asymptotically $sqrt{c (log n)/n}$ where $c=(k-b)/2$. This supports Pemantles claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.
A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that the critical values $lambda_1$ and $lambda_2$ for global and local survival were different. Here, we will consider the case of trees in which the degrees of vertices are periodic. We will compute bounds on $lambda_1$ and $lambda_2$ and for the corresponding critical values $lambda_g$ and $lambda_ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $lambda_2$ of $(1,n)$ trees and generalize that result to the $(a_1,ldots, a_k, n)$ tree when $max_i a_i le n^{1-epsilon}$ and $a_1 cdots a_k = n^b$. We also give results for $lambda_g$ and $lambda_ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.
In this paper we are concerned with the binary contact path process introduced in cite{Gri1983} on the lattice $mathbb{Z}^d$ with $dgeq 3$. Our main result gives a hydrodynamic limit of the process, which is the solution to a heat equation. The proof of our result follows the strategy introduced in cite{kipnis+landim99} to give hydrodynamic limit of the SEP model with some details modified since the states of all vertices are not uniformly bounded for the binary contact path process. In the modifications, the theory of the linear system introduced in cite{Lig1985} is utilized.
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
We show that the quasi-stationary distribution of the subcritical contact process on $mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.
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