# Invariant matchings of exponential tail on coin flips in $\Z^d$

## TL;DR

This paper constructs invariant matchings in Bernoulli percolation on ^d that achieve optimal decay in two dimensions and exponential decay in higher dimensions, improving bounds on tail probabilities.

## Contribution

It introduces a new method for invariant matchings with optimal or exponential tail decay, applicable in multiple dimensions and to Poisson process matching.

## Key findings

- Optimal decay of $cr^{1/2}$ in 2D for the matching tail.
- Exponential tail decay in dimensions ^d for $d \u2265 3.
- Method applicable to Poisson process matching and allocation problems.

## Abstract

Consider Bernoulli(1/2) percolation on $\Z^d$, 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 $\Z^d$ 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.