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Sharpness of the Percolation Phase Transition for the Contact Process on $\mathbb{Z}^d$. (arXiv:1807.05591v1 [math.PR])
来源于:arXiv
We study percolation properties of the upper invariant measure of the contact
process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase
transition with exponentially small clusters throughout the subcritical regime
and a mean-field lower bound for the infinite cluster density in the
supercritical regime. This generalizes and simplifies an earlier result of Van
den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase
transition on $\mathbb{Z}^d$. Our proof relies on the OSSS inequality for
Boolean functions and is inspired by a series of papers by Duminil-Copin,
Raoufi and Tassion in which they prove similar sharpness results for a variety
of models. 查看全文>>