JournalsaihpdVol. 4, No. 3pp. 245–271

Basic properties of the infinite critical-FK random map

  • Linxiao Chen

    Université Paris-Sud, Orsay, France and CEA Saclay, Gif-sur-Yvette, France
Basic properties of the infinite critical-FK random map cover
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Abstract

In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each q[0,]q \in [0, \infty] and integer n1n \geq 1, this model chooses a planar map of nn edges with a probability proportional to the partition function of critical qq-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=1q = 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 qq, and mutually singular in distribution for different values of qq.

Cite this article

Linxiao Chen, Basic properties of the infinite critical-FK random map. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), no. 3, pp. 245–271

DOI 10.4171/AIHPD/40