2015/02/28 by Linxiao Chen
Computer Science · Mathematics · #Mathematical Dynamics and Fractals #Mathematics #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #math.PR #msc:05C81 #msc:60D05 #msc:60F20 #msc:60K35
paper · pdf · doi:10.4171/aihpd/40
published as Ann. Inst. Henri Poincaré D, 4(3):245-271, 2017 · 14 pages, 6 figures. v2: Fixed the proof of main theorem, removed old lemma 5, added results on mutually singular measures and ergodicity. Submitted to Annales de l'Institut Henri Poincaré D
arxiv created 2016/03/02 · openalex publication_date 2017/09/26 · arxiv updated 2021/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each q ∈ [0, ∞] and integer n ≥ 1 , this model chooses a planar map of n edges with a probability proportional to the partition function of critical q -Potts model on that map. Sheeld introduced the hamburger–cheeseburer bijection which maps the cFK random maps to a family of random words, and remarked that one can construct infinite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When q = 1 , this provides an alternative construction of the UIPQ. In addition, we show that the limit is almost surely one-ended and recurrent for the simple random walk for any q , and mutually singular in distribution for different values of q .