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Probability and Combinatorics

Tuesday, October 23, 2018 - 3:00pm

Julian Gold



University of Pennsylvania


We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove a shape theorem for these random graphs, showing that upon rescaling they tend almost surely to a deterministic shape. This limit shape is itself an isoperimetric set for a norm we construct. In addition, we obtain sharp asymptotics for a modification of the Cheeger constant of $\textbf{C}_\infty \cap [-n,n]^d$, settling a conjecture of Benjamini for this modified Cheeger constant. Analogous results are shown for the giant component in dimension two, where we use the original definition of the Cheeger constant, and a more complicated continuum isoperimetric problem emerges as a result.