Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

• 成果类型: Article
• 署名作者: Miller, Jason; Sheffield, Scott; Werner, Wendelin
• 署名单位: University of Cambridge; Massachusetts Institute of Technology (MIT); Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01070-4
• 发表日期: 2021
• 页码: 669-710
• 关键词: erased random-walks; random-cluster; random surfaces; Invariance; geometry

摘要

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE kappa' for kappa' in (4, 8) that is drawn on an independent gamma-LQG surface for gamma(2) = 16/kappa'. The results are similar in flavor to the ones from our companion paper dealing with CLE kappa for kappa 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 CLE kappa' 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: In our paper entitled CLE Percolations described the law of interfaces obtained when coloring the loops of a CLE kappa' independently into two colors with respective probabilities p and 1 - p. This description was complete up to one missing parameter rho. The results of the present paper about CLE on LQG allow us to determine its value in terms of p and (kappa)'. It shows in particular that CLE kappa' and CLE16/kappa' 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 = 4 cos(2)(4 pi/(kappa')). This provides further evidence for the long-standing belief that CLE kappa' and CLE16/kappa' represent the scaling limits of FKq percolation and the q-Potts model when q and kappa' are related in this way. Another consequence of the formula for rho(p, kappa') 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.