2025/05/08 by Pablo Araya, Araya, Pablo, Luis Fredes +3
Mathematics · #Characterization (materials science) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Markov property #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Metric (unit) #Planar #Probability (math.PR) #Property (philosophy) #Stochastic processes and statistical mechanics #Variable-order Markov model
paper · pdf · doi:10.48550/arxiv.2505.05447
openalex publication_date 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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,we], 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.