Geodesic
Basic properties of the infinite critical-FK random map
Annales de l’Institut Henri Poincaré D, Tome 4 (2017) no. 3, pp. 245-271

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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. 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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/40
Classification : 60-XX, 05-XX
Keywords: Fortuin–Kasteleyn percolation, random planar maps, hamburger–cheeseburer bijection, local limits, recurrent graph, ergodicity of random graphs 
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     author = {Chen, Linxiao},
     title = {Basic properties of the infinite {critical-FK} random map},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {245--271},
     volume = {4},
     number = {3},
     year = {2017},
     doi = {10.4171/aihpd/40},
     mrnumber = {3713017},
     zbl = {1381.60033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpd/40/}
}
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Chen, Linxiao. Basic properties of the infinite critical-FK random map. Annales de l’Institut Henri Poincaré D, Tome 4 (2017) no. 3, pp. 245-271. doi: 10.4171/aihpd/40

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