We study Bernoulli percolation on 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 \phi_{p_c}(S), introduced by Duminil-Copin and Tassion (Comm. Math. Phys., 2016), is uniformly bounded over all S ⊂ Z^d.This inequality further yields a short and self-contained route to several key results (old and new), including near-critical bounds on the two-point function and the derivation of the one-arm exponent.

Based on a joint work with Bruno Schapira.