Infinite stable Boltzmann planar maps are subdiffusive

TL;DR

This paper proves that simple random walks on infinite stable Boltzmann planar maps exhibit subdiffusive behavior with an exponent less than 1/3, using stationarity and peeling process techniques.

Contribution

It introduces new geometric estimates and stationarity methods to analyze random walks on these generalized random planar maps, extending understanding beyond classical models.

Findings

Random walk on these maps is subdiffusive

Subdiffusive exponent is less than 1/3

Peeling process provides key geometric estimates

Abstract

The infinite discrete stable Boltzmann maps are generalisations of the well-known Uniform Infinite Planar Quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than 1/3. Our method is based on stationarity and geometric estimates obtained via the peeling process which are of own interest.