How fast planar maps get swallowed by a peeling process

Abstract

The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [Curien & Le Gall, Scaling limits for the peeling process on random maps, Ann. Inst. Henri Poincaré Probab. Stat. 53, 1 (2017), 322-357] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where \[c \approx 0.12831235141783245423674486573872854933142662048339843...\] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [Chaumont, Kyprianou & Pardo, Some explicit identities associated with positive self-similar Markov processes, Stochastic Process. Appl. 119, 3 (2009), 980-1000] which appears as a scaling limit for the perimeter process. As an application we sharpen the upper bound of the sub-diffusivity exponent for random walk of [Benjamini & Curien, Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points, Geom. Funct. Anal. 23, 2 (2013), 501-531].

arXiv comment11 pages, 3 figures

References in corpus (6)

• Uniform Infinite Planar Triangulations
• Local structure of random quadrangulations
• Recurrence of bipartite planar maps
• Geometry of infinite planar maps with high degrees
• Conformal growth rates and spectral geometry on distributional limits of graphs
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Cited by in corpus (2)

• Infinite stable Boltzmann planar maps are subdiffusive
• Markovian explorations of random planar maps are roundish