Abstract
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.
Highlights
In this paper, continuing our earlier work [27], we present a robust method allowing to prove the existence of an exponential rate of convergence of the
The configuration in which all vertices are healthy is absorbing for the dynamics, and in finite graphs it is reached with probability one
The contact process on the supercritical random geometric graph has been previously considered by Ménard and Singh [16], who proved that the critical infection rate is positive, and by Can [5] who obtained sharp bounds on the expected value of the extinction time on Gn when the radius of connectedness goes to infinity
Summary
Bruno Schapira and Daniel Valesin contact process extinction time on various models of random graphs, when the infection parameter is large enough. The contact process on the supercritical random geometric graph has been previously considered by Ménard and Singh [16], who proved that the critical infection rate is positive, and by Can [5] who obtained sharp bounds on the expected value of the extinction time on Gn when the radius of connectedness goes to infinity. Let us mention that for several important random graph models, it would be interesting to obtain results of the form (1.2), but our present techniques are not applicable (at least not directly) These include the configuration model, the Erdős–Renyi random graph, random planar maps, and Delaunay triangulations of the plane (provided that in each case, the parameters defining the graph and the value of λ yield a regime of exponentially large extinction time). The case of Galton–Watson trees is treated separately in the last section, as the proof in this setting presents some substantial differences
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