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Isoperimetric shapes in supercritical bond percolation


This thesis is concerned with scaling limits of sequences of random isoperimetric problems. We first consider progressively larger isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\Z^d$ for $d \geq 3$. We prove a shape theorem for these subgraphs, showing that upon rescaling they tend almost surely to a deterministic shape, which is itself an isoperimetric set for a norm we construct. The norm represents a homogenized surface energy arising from random interfaces between subgraphs of $\textbf{C}_\infty$. 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 the version of the Cheeger constant defined here.

We also study the isoperimetric properties of the giant component in dimension two using the original definition of the Cheeger constant, taking into account the boundary of the large box $[-n,n]^d$. Analogous results are shown here, with the caveat that a more complicated continuum isoperimetric problem emerges due to the presence of the boundary.

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