2020/06/25 by Miller, Jason, Sheffield, Scott, Werner, Wendelin · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2006.14605
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLEκ' for κ' in (4,8) that is drawn on an independent γ-LQG surface for γ2=16/κ'. The results are similar in flavor to the ones from our paper dealing with CLEκ for κ in (8/3,4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLEκ' in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLEκ' independently into two colors with respective probabilities p and 1-p. This description was complete up to one missing parameter ρ. The results of the present paper about CLE on LQG allow us to determine its value in terms of p and κ'. It shows in particular that CLEκ' and CLE16/κ' are related via a continuum analog of the Edwards-Sokal coupling between FKq percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if q=4cos2(4π/κ'). This provides further evidence for the long-standing belief that CLEκ' and CLE16/κ' represent the scaling limits of FKq percolation and the q-Potts model when q and κ' are related in this way. Another consequence of the formula for ρ(p,κ') is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.