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Register a new userThe contact process on random hyperbolic graphs: metastability and critical exponents
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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 $chi in(1,2)$ (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 $chi$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider fini
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We show that the contact process on the rank-one inhomogeneous random graphs and Erdos-R{e}nyi graphs with mean degree large enough survives a time exponential in the size of these graphs for any positive infection rate. In addition, a metastable result for the extinction time is also proved.
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $lambda_2=0$ and (ii) when it is geometric($p$) we have $lambda_2 le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $lambda_c(n)$ for prolonged persistence on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $lambda_c(n) to 0$ we give estimates on the rate of convergence. Physicists tell us that $lambda_c(n) sim 1/Lambda(n)$ where $Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $dge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $beta$ be the inverse temperature, $beta_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $beta > beta_c$ there exists $hat{B}_c(beta)in (0,infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $beta> beta_c$ and $0 leq B < hat{B}_c(beta)$, whereas it exhibits the cut-off phenomenon at $c_star n log n$ with a window of order $n$ when $beta < beta_c$ or $beta > beta_c$ and $B>hat{B}_c(beta)$. Interestingly, $hat{B}_c(beta)$ coincides with the critical external field of the Ising model on the $d$-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists $B_c(beta)$ with $B_c(beta) leq hat{B}_c(beta)$ such that for $beta> beta_c$, the mixing time is at least exponential along some subsequence $(n_k)_{kgeq 1}$ when $0 leq B < B_c(beta)$, whereas it is less than or equal to $Cnlog n$ when $B>hat{B}_c(beta)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.
We consider the discrete-time threshold-$theta ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $theta ge 2$ and $r ge theta+2$, $epsilon_1$ is small and p is at least $p_1(epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2epsilon_1$ up to time $exp(gamma_1(r)n)$ with probability at least $1 - exp(-gamma_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $epsilon_2(p_2)>0$ so that if $p le p_2$ and the number of occupied vertices in the initial configuration is at most $epsilon_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $epsilon_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.
We introduce a method to prove metastability of the contact process on ErdH{o}s-Renyi 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 metastability, and we have explicit exponentially growing lower bounds on the expected extinction time.
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