Basic properties of the infinite critical-FK random map

Linxiao Chen (1, 2)

1 UP11 - Université Paris-Sud - Paris 11
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We consider 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 probability proportional to the partition function of critical $q$-Potts model on that map. Sheffield introduced the hamburger-cheeseburger bijection which sends 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 using a monotonicity result which compares the model with general value of $q$ to the case $q=0$. When $q=1$, this provides an alternative construction of the UIPQ. In addition, we show that for any $q$, the limit is almost surely one-ended and recurrent for the simple random walk.

Mots clés: hamburger-cheeseburger bijection; random cluster model; random planar maps; local limits

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Linxiao Chen. Basic properties of the infinite critical-FK random map. 2015. ⟨hal-01120810⟩