Basic properties of the infinite critical-FK random map

Chen, Linxiao

Annales de l’Institut Henri Poincaré D, Tome 4 (2017) no. 3, pp. 245-271.

In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each $q\in [0,\infty ]$ and integer $n\ge 1$, 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. She�eld introduced the hamburger–cheeseburer bijection which maps the cFK random maps to a family of random words, and remarked that one can construct in�finite 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 di�fferent values of $q$.

Classification : 60-XX, 05-XX
Mots-clés : Fortuin–Kasteleyn percolation, random planar maps, hamburger–cheeseburer bijection, local limits, recurrent graph, ergodicity of random graphs