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.
Counting coloured planar maps: differential equations
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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.