Characterisation of Markov property on planar maps

TL;DR

This paper explores the Markov property in planar maps, characterizing laws satisfying it, introducing stopping maps, and defining decorated metric maps with Markovian properties, expanding understanding of probabilistic structures in planar geometry.

Contribution

It provides a comprehensive characterization of Markovian laws on planar maps, introduces new classes of stopping maps, and extends the Markov property to decorated metric maps.

Findings

Boltzmann-type laws characterized by Markov and rerooting invariance

Stopping maps are more general than peeling-based maps

Decorated metric maps satisfy Markov property even with mid-edge exploration

Abstract

We revisit, in a self contained way, the Markov property on planar maps and decorated planar maps from three perspectives. First, we characterize the laws on these planar maps that satisfy both the Markov property and rerooting invariance, showing that they are Boltzmann-type maps. Second, we provide a comprehensive characterization of random submaps, that we call stopping maps, satisfying the Markov property, demonstrating that they are not restricted to those obtained through a peeling procedure. Third, we introduce decorated metric planar maps in which edges are replaced by copies of random length intervals $[0,w_e]$, and the decorations are given by continuous functions on the edges. We define a probability measure on them that is the analogue of the Boltzmann map and show that it satisfies the Markov property even for sets that halt exploration mid-edge.