Percolation on uniform infinite planar maps

TL;DR

This paper constructs the uniform infinite planar map (UIPM), introduces a peeling sampling process, and determines precise percolation thresholds for various types of percolation on UIPM and related maps.

Contribution

It provides a new construction and sampling method for UIPM and establishes exact percolation thresholds for bond and site percolation on these maps.

Findings

Percolation threshold for bond percolation on UIPM is 1/2.

Percolation threshold for site percolation on UIPM is 2/3.

Percolation threshold for bond percolation on uniform infinite quadrangulation is 1/3.

Abstract

We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is p_c^bond=1/3.