Critical probabilities for site and bond percolation models

Abstract

.Any infinite graph $G = (V, E)$ has a site percolation critical probability $p_c^{\rm site}$ and a bond percolation critical probability $p_c^{\rm bond}$. The well-known weak inequality $p_c^{\rm site} \geq p_c^{\rm bond}$ is strengthened to strict inequality for a c c broad category of graphs $G$, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality $p_c^{\rm site} \leq 1 - (1 - p_c^{\rm bond})^{\Delta - 1}$ is proved also, where $\Delta$ denotes the supremum of the vertex degrees of $G$.