Abstract
We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$ of Schramm--Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.
Highlights
Introduction and main resultsThe purpose of this work is to study duality properties of Boltzmann planar maps through the Bernoulli bond percolation model
The Boltzmann measures on planar maps are parametrized by a weight sequence q = (q1, q2, . . .) of nonnegative real numbers assigned to the faces of the maps
Building on earlier works of Marckert & Miermont [29] and Le Gall [27], Marzouk proved in [30] that generic critical Boltzmann maps all have the same scaling limit, the Brownian map
Summary
The purpose of this work is to study duality properties of Boltzmann planar maps through the Bernoulli bond percolation model. There is a natural definition of a critical parameter, the percolation threshold pcq ∈ [0, 1], that has been determined in [14, Theorem 11.8] as an explicit function of the weight sequence q This parameter is the almost sure percolation threshold for existence of infinite cluster in a half-plane model of q-Boltzmann maps. This allows to establish Theorem 1.1 in the cases of critical and supercritical percolation. We deal with subcritical percolation, where we prove Theorem 1.1 and Proposition 1.3 by using a peeling process defined on q-Boltzmann maps instead of their half-planar version.
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