Scaling of Percolation on Infinite Planar Maps, I

TL;DR

This paper investigates the scaling limits of percolation on random planar maps, deriving explicit formulas for crossing probabilities and revealing symmetries related to Airy-Levy processes.

Contribution

It provides new formulas and insights into the scaling limits of percolation on random maps, connecting crossing probabilities with Airy-Levy processes.

Findings

Explicit formulas for limit crossing probabilities.

Relations between symmetries of maps and Airy-Levy processes.

Connections to stable process results.

Abstract

We consider several aspects of the scaling limit of percolation on random planar triangulations, both finite and infinite. The equivalents for random maps of Cardy's formula for the limit under scaling of various crossing probabilities are given. The limit probabilities are expressed in terms of simple events regarding Airy-Levy processes. Some explicit formulas for limit probabilities follow from this relation by applying known results on stable processes. Conversely, natural symmetries of the random maps imply identities concerning the Airy-Levy processes.