Scaling limit of the invasion percolation cluster on a regular tree

TL;DR

This paper establishes the existence of a scaling limit for the invasion percolation cluster on a regular tree, describing it as a random real tree with diffusive properties, and relates it to asymptotic behaviors.

Contribution

It proves the convergence of the invasion percolation cluster to a well-defined random real tree and characterizes its contour and height functions as stochastic processes.

Findings

The invasion percolation cluster converges to a random real tree in the scaling limit.

The contour and height functions are described by diffusive stochastic processes.

The asymptotic behavior of the IPC's level sets is precisely characterized.

Abstract

We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.