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Probability and Combinatorics

Tuesday, September 14, 2021 - 3:30pm

Jack Hanson

City College of NY


Temple University

Wachman Hall 617

In their study of percolation, physicists have proposed ``scaling hypotheses'' relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ -- the probability that the open cluster of the origin has Euclidean diameter at least $n$.


As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.