2020/06/25 by Jason Miller, Scott Sheffield⋆, Miller, Jason +4 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Combinatorics #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Geometry #Kappa #Loop quantum gravity #Mathematical Physics (math-ph) #Mathematical physics #Mathematics #Physics #Probability (math.PR) #Pure mathematics #Quantum #Quantum gravity #Quantum mechanics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2006.14605
published in arXiv (Cornell University) (Cornell University) · Dedicated to the memory of Harry Kesten. To appear in Probab. Theory Rel. Fields, 30 pages, 18 figures
openalex publication_date 2020/06/25 · arxiv created 2021/05/28 · arxiv updated 2021/05/31 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the structure of the Liouville quantum gravity (LQG) surfaces that\nare cut out as one explores a conformal loop-ensemble CLE\κ' for\n\κ' in (4,8) that is drawn on an independent \γ-LQG surface for\n\γ2=16/\κ'. The results are similar in flavor to the ones from our\npaper dealing with CLE\κ for \κ in (8/3,4), where the loops of\nthe CLE are disjoint and simple. In particular, we encode the combined\nstructure of the LQG surface and the CLE\κ' in terms of stable\ngrowth-fragmentation trees or their variants, which also appear in the\nasymptotic study of peeling processes on decorated planar maps.\n This has consequences for questions that do a priori not involve LQG\nsurfaces: Our previous paper "CLE percolations" described the law of interfaces\nobtained when coloring the loops of a CLE\κ' independently into two\ncolors with respective probabilities p and 1-p. This description was\ncomplete up to one missing parameter \ρ. The results of the present paper\nabout CLE on LQG allow us to determine its value in terms of p and \κ'.\nIt shows in particular that CLE\κ' and CLE16/\κ' are related\nvia a continuum analog of the Edwards-Sokal coupling between FKq percolation\nand the q-state Potts model (which makes sense even for non-integer q\nbetween 1 and 4) if and only if q=4\cos2(4\π /\κ'). This provides\nfurther evidence for the long-standing belief that CLE\κ' and\nCLE16/\κ' represent the scaling limits of FKq percolation and the\nq-Potts model when q and \κ' are related in this way. Another\nconsequence of the formula for \ρ(p,\κ') is the value of half-plane arm\nexponents for such divide-and-color models (a.k.a. fuzzy Potts models) that\nturn out to take a somewhat different form than the usual critical exponents\nfor two-dimensional models.\n