# Planar percolation and the loop O(n) model

## TL;DR

This paper proves that certain planar graph percolation processes exhibit either no or infinitely many infinite clusters, confirming parts of longstanding conjectures and applying to the loop O(n) model.

## Contribution

It establishes a broad class of percolation processes on planar graphs with zero or infinite clusters, resolving a 1996 conjecture and analyzing the loop O(n) model.

## Key findings

- Percolation processes on planar graphs have either zero or infinitely many infinite components.
- Confirmed the existence of infinitely many loops in the loop O(n) model for specific parameters.
- Proved that the critical probability p_c is at least 1/2 for certain planar graphs.

## Abstract

We show that a large class of site percolation processes on any planar graph contains either zero or infinitely many infinite connected components. The assumptions that we require are: tail triviality, positive association (FKG) and that the set of open vertices is stochastically dominated by the set of closed ones. This covers the case of Bernoulli site percolation at parameter $p\leq 1/2$ and resolves Conjecture 8 from the work of Benjamini and Schramm from 1996. Our result also implies that $p_c\geq 1/2$ for any invariantly amenable unimodular random rooted planar graph. Furthermore, we apply our statement to the loop O(n) model on the hexagonal lattice and confirm a part of the phase diagram conjectured by Nienhuis in 1982: the existence of infinitely many loops around every face whenever $n\in [1,2]$ and $x\in [1/\sqrt{2},1]$. The point $n=2,x=1/\sqrt{2}$ is conjectured to be critical. This is the first instance that this behavior has been proven in such a large region of parameters. In a big portion of this region, the loop O(n) model has no known FKG representation. We apply our percolation result to quenched distributions that can be described as divide and color models.