On reversing the Simon-Lieb inequality in high-dimensional percolation

Abstract

We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a partial reversal. As a main application, we show that the quantity $φ_{p_c}(S)$, introduced by Duminil-Copin and Tassion (Comm.\ Math.\ Phys., 2016), is uniformly bounded over all $S\subset \mathbb Z^d$. This partial reversal further yields a short and self-contained route to several key results, including near-critical estimates on the two-point function and sharp bounds on the critical one-arm probability.