The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Annales de l’Institut Henri Poincaré D


Full-Text PDF (398 KB) | Metadata | Table of Contents | AIHPD summary
Volume 4, Issue 3, 2017, pp. 245–271
DOI: 10.4171/AIHPD/40

Published online: 2017-09-26

Basic properties of the infinite critical-FK random map

Linxiao Chen[1]

(1) Université Paris-Sud, Orsay, France and CEA Saclay, Gif-sur-Yvette, France

In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each $q \in [0, \infty]$ and integer $n \geq 1$, 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 $q$.

Keywords: Fortuin–Kasteleyn percolation, random planar maps, hamburger–cheeseburer bijection, local limits, recurrent graph, ergodicity of random graphs

Chen Linxiao: Basic properties of the infinite critical-FK random map. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), 245-271. doi: 10.4171/AIHPD/40