# Site percolation and isoperimetric inequalities for plane graphs

## TL;DR

This paper establishes new bounds on the site percolation threshold for plane graphs with certain degree conditions, using isoperimetric inequalities, and solves several open problems in hyperbolic and planar graph percolation theory.

## Contribution

It introduces a novel technique combining isoperimetric inequalities to derive bounds on percolation thresholds for plane graphs, including hyperbolic lattices and triangulations.

## Key findings

- Plane graphs with minimum degree ≥7 have percolation thresholds bounded away from 1/2.
- Vertex isoperimetric constants are determined for all triangular and square hyperbolic lattices.
- Progress is made on conjectures regarding critical probabilities for plane triangulations.

## Abstract

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini and Horesh that the critical probability is at most $1/2$ for plane triangulations of minimum degree $6$. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.