Continuity of the critical value and a shape theorem for long-range percolation

Abstract

Consider supercritical long-range percolation on ${\mathbb{Z}}^d$ where two vertices $x,y\in {\mathbb{Z}}^d$ are connected with probability asymptotic to $\Vert x-y{\Vert }^{-s}$ for some $s>2d$. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within $n$ steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\ge 2$, the critical value depends continuously on the precise specifications of the model.

45 pages, 4 figures. Accepted in Stochastic Processes and their Applications

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