Scaling limits for the peeling process on random maps

TL;DR

This paper investigates the scaling limits of volume and perimeter in peeling processes on infinite random planar maps, connecting these limits to the hull process of the Brownian plane and exploring various applications.

Contribution

It establishes the scaling limits for the perimeter and volume of discovered regions in peeling processes on infinite random maps, linking them to the Brownian plane's hull process.

Findings

Scaling limits of perimeter and volume are characterized in terms of the Brownian plane.

Results apply to metric exploration and peeling by layers algorithms.

Applications include dual graph exploration and first-passage percolation models.

Abstract

We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or quadrangulation (UIPQ). In particular, our results apply to the metric exploration or peeling by layers algorithm, where the discovered regions are (almost) completed balls, or hulls, centered at the root vertex. The scaling limits of the perimeter and volume of hulls can be expressed in terms of the hull process of the Brownian plane studied in our previous work. Other applications include the metric exploration of the dual graph of our infinite random lattices, and first-passage percolation with exponential edge weights on the dual graph, also known as the Eden model or uniform peeling.