Basic properties of the infinite critical-FK random map

Abstract

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. Sheffield introduced the hamburger-cheeseburger 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 $q$.

arXiv comment14 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

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