Tree embeddings and nonuniqueness in site percolation

Abstract

We prove a nonuniqueness theorem for Bernoulli site percolation on properly embedded planar graphs, and we obtain a general connectivity principle beyond planarity. Let $G$ be an infinite connected graph properly embedded in $\RR^2$ with minimum degree at least $7$. Then \[ p_c^{\mathrm{site}}(G)<\tfrac12, \] and for every \[ p\in \bigl(p_c^{\mathrm{site}}(G),\,1-p_c^{\mathrm{site}}(G)\bigr), \] Bernoulli$(p)$ site percolation on $G$ has almost surely infinitely many infinite open clusters. In particular, this verifies a conjecture of Benjamini and Schramm for properly embedded planar graphs. The core new ingredient is an explicit embedded-tree separation mechanism for planar nonuniqueness. We construct embedded trees and an embedded forest whose separation properties yield exponential decay of two-point connection probabilities in the matching graph. To treat the high-density regime, we introduce a binary-tree version of uniform percolation and prove stability of infinite clusters under edge additions, without any bounded-degree assumption. Beyond the planar theorem, we prove a general lower bound on two-point connectivity under uniqueness for arbitrary infinite locally finite graphs. As a consequence, if \[ p_c^{\mathrm{site}}(G)<p<p_{\mathrm{conn}}(G), \] then Bernoulli site percolation on $G$ has almost surely infinitely many infinite open clusters.