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 [15] 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 \[ c \approx 0.1283123514178324542367448657387285493314266204833984375... \] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [12] 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 [4].
Highlights
One of the main tool to study random maps is the so-called peeling procedure which is a step-by-step Markovian exploration of the map
Is defined via an integral equation derived using the Lamperti representation of the spectrally negative 3/2-stable Lévy process conditioned to remain positive [12] 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 [4]
Summary
One of the main tool to study random maps is the so-called peeling procedure which is a step-by-step Markovian exploration of the map. In this work we will study the peeling process on random infinite critical Boltzmann planar maps with bounded face-degrees. Let us note that the proof of Theorem 1.1 shows that for any sub-map e with a unique hole and any edge E ∈ ∂e, if denotes the length of the hole, if we start the peeling process with e0 = e, we have. The results of this paper can be adapted to the case of peeling processes (and random walks on the dual and primal) on Boltzmann maps with large faces [16]
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