The one-arm exponent for mean-field long-range percolation

Abstract

Consider a long-range percolation model on ${\mathbb{Z}}^d$ where the probability that an edge $\{x,y\}\in {\mathbb{Z}}^d\times {\mathbb{Z}}^d$ is open is proportional to $\Vert x-y{\Vert }_2^{-d-\alpha }$ for some $\alpha >0$ and where $d>3\min \{2,\alpha \}$. We prove that in this case the one-arm exponent equals $\min \{4,\alpha \}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $\alpha$ when $\alpha =4$.

28 pages, 1 figure, 1 appendix