Analyticity results in Bernoulli Percolation

TL;DR

This paper proves that the percolation density in Bernoulli percolation on integer lattices is an analytic function in the supercritical phase, and introduces techniques with broader implications for percolation theory.

Contribution

It establishes analyticity of percolation density and susceptibility across different regimes and models, and addresses conjectures related to critical probabilities in specific triangulations.

Findings

Percolation density is analytic in the supercritical interval.

Susceptibility is analytic in the subcritical interval for various models.

Proves $p_c^{bond} < 1/2$ for certain triangulations.

Abstract

We prove that for Bernoulli percolation on $\mathbb{Z}^d$, $d\geq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini \& Schramm conjectured that $p_c^{site} \leq 1/2$.