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arXiv · 2610.38941

Anchored isoperimetry in percolation

Sahar Diskin·Lyuben Lichev·Ritvik Ramanan Radhakrishnan·Benny Sudakov·Laury Verhoeven
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Abstract

We show that, for any infinite transitive graph $G$, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of $G$. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, $\mathbb{Z}^d$, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.

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Sahar Diskin, Lyuben Lichev, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Laury Verhoeven. 2026-10-28. Anchored isoperimetry in percolation. https://arxiv.org/abs/2610.38941

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