Approximations of standard equivalence relations and Bernoulli percolation at p\_u

TL;DR

This paper explores how standard equivalence relations can be approximated by increasing sequences of sub-relations and applies these results to understand Bernoulli percolation behavior at the critical threshold p_u.

Contribution

It introduces new methods for approximating p.m.p. standard equivalence relations and applies these to analyze Bernoulli percolation at the percolation threshold p_u.

Findings

Approximation of equivalence relations by increasing sub-relations

Application to Bernoulli percolation at p_u

Insights into orbit equivalence theory

Abstract

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\'esum\'e en Fran\c{c}ais. Approximations de relations d'\'equivalence standards et percolation de Bernoulli \`a p\_u.