A phase transition in block-weighted random maps

2023/02/03 by William Fleurat, Fleurat, William, Zéphyr Salvy +1
Mathematics · #05C12 #05C80 #60D05 #60F17 #68R05 #82B41 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2302.01723

openalex publication_date 2023/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the model of random planar maps of size n biased by a weight u>0 per 2-connected block, and the closely related model of random planar quadrangulations of size n biased by a weight u>0 per simple component. We exhibit a phase transition at the critical value u_C=9/5. If u u_C, the largest block is of size Θ(log(n)), the scaling order for distances is n^1/2, and the scaling limit is the Brownian tree. Finally, for u=u_C, the largest block is of size Θ(n^2/3), the scaling order for distances is n^1/3, and the scaling limit is the stable tree of parameter 3/2.

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