# Markovian explorations of random planar maps are roundish

## TL;DR

This paper demonstrates that Markovian peeling processes on infinite stable Boltzmann maps consistently reveal similar portions of the map, enabling comparisons of distances and control of pioneer points regardless of the exploration algorithm used.

## Contribution

It proves that all Markovian peeling explorations grow roughly metric balls, allowing for robust comparisons of distances and pioneer points on stable Boltzmann maps.

## Key findings

- Peeling processes reveal similar map portions regardless of algorithm
- Enables comparison of distances in the map and its dual
- Provides control over pioneer points of random walks

## Abstract

The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peeling processes. Such a process depends on an algorithm which selects at each step the next edge where the exploration continues. We prove here that, whatever this algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly metric balls. Applied to well-designed algorithms, this easily enables us to compare distances in the map and in its dual, as well as to control the so-called pioneer points of the simple random walk, both on the map and on its dual.