The only rerooting-invariant Markovian random quadrangulations are Boltzmann maps, and every submap that induces a Markovian decomposition is a 'stopping map,' a class strictly larger than peeling explorations.
Spatial Markov property in Brownian disks
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abstract
We derive a new representation of the Brownian disk in terms of a forest of labeled trees, where labels correspond to distances from a subset of the boundary. We then use this representation to obtain a spatial Markov property showing that the complement of a hull centered at a boundary point of a Brownian disk is again a Brownian disk, with a random perimeter, and is independent of the hull conditionally on its perimeter. Our proofs rely in part on a study of the peeling process for triangulations with a boundary, which is of independent interest. The results of the present work will be applied to a continuous version of the peeling process for the Brownian half-plane in a companion paper.
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Characterisation of Markov property on planar maps
The only rerooting-invariant Markovian random quadrangulations are Boltzmann maps, and every submap that induces a Markovian decomposition is a 'stopping map,' a class strictly larger than peeling explorations.