How fast planar maps get swallowed by a peeling process

TL;DR

This paper analyzes the probability decay of a peeling process failing to swallow the origin in a random planar map, providing precise decay rates and implications for random walk behavior.

Contribution

It introduces a new decay rate for the probability that the origin remains unswallowed, derived via a Lamperti representation of a spectrally negative stable Lévy process.

Findings

Probability decays at least as n^{-2c/3} with c ≈ 0.1283

Provides a sharper upper bound for the sub-diffusivity exponent of random walk

Connects peeling process behavior to stable Lévy processes and scaling limits

Abstract

The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [Curien & Le Gall, Scaling limits for the peeling process on random maps, Ann. Inst. Henri Poincar\'e Probab. Stat. 53, 1 (2017), 322-357] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where \[c \approx 0.12831235141783245423674486573872854933142662048339843...\] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable L\'evy process conditioned to remain positive [Chaumont, Kyprianou & Pardo, Some explicit identities associated with positive self-similar Markov processes, Stochastic Process. Appl. 119, 3 (2009), 980-1000] which appears as a scaling limit for the perimeter process. As an application we sharpen the upper bound of the sub-diffusivity exponent for random walk of [Benjamini & Curien, Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points, Geom. Funct. Anal. 23, 2 (2013), 501-531].