Universality in Two-Dimensional Enhancement Percolation

TL;DR

This paper establishes that certain monotonic enhancements in two-dimensional percolation models do not alter critical exponents or the scaling limit, demonstrating a form of universality in dependent percolation.

Contribution

It proves that non-essential monotonic enhancements do not change key critical properties or the scaling limit in 2D percolation, extending universality results.

Findings

Critical exponents remain unchanged under non-essential enhancements.

Scaling limit of crossing probabilities is unaffected by certain enhancements.

Full scaling limit for site percolation on the triangular lattice is invariant under non-essential enhancements.

Abstract

We consider a type of dependent percolation introduced by Aizenman and Grimmett, who showed that certain "enhancements" of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this paper we first prove that, for two-dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two-dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit is not affected by any monotonic enhancement that does not shift the critical point.