Invariant colorings of random planar maps

TL;DR

This paper develops optimal invariant coloring methods for random planar maps, achieving the best possible decay rates for connection probabilities in percolation models across different dimensions.

Contribution

It introduces a novel equivariant partitioning and matching construction that improves decay bounds for percolation connections in all dimensions.

Findings

Optimal decay rate of cr^{1/2} in 2D percolation

Exponential tail decay in dimensions ≥3

Applicable to Poisson process matching and allocation problems

Abstract

Consider Bernoulli(1/2) percolation on $\mathbb{Z}^d$, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want to find such matching rules that make the probability that the pair of the origin is at distance greater than $r$ decay as fast as possible. For two dimensions, we give a matching of decay $cr^{1/2}$, which is optimal. For dimension at least 3 we give a matching rule that has an exponential tail. This substantially improves previous bounds. The construction has two major parts: first we define a sequence of coarser and coarser partitions of $\mathbb{Z}^d$ in an equivariant way, such that with high probability the cell of a fixed point is like a cube, and the labels in it are i.i.d. Then we define a matching for a fixed finite cell, which stabilizes as we repeatedly apply it for the cells of the consecutive partitions. Our methods also work in the case when one wants to match points of two Poisson processes, and they may be applied to allocation questions.