Basic properties of the infinite critical-FK random map

TL;DR

This paper studies the critical FK random map model, establishing its properties, including local convergence to an infinite map, recurrence, and singularity across different parameters, using a bijection with random words.

Contribution

It provides a detailed proof of local convergence of the critical FK random maps to an infinite map and explores their properties across different parameters.

Findings

Limit is almost surely one-ended.

Limit is recurrent for simple random walk.

Distributions are mutually singular for different q values.

Abstract

We investigate 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 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 $q$.