Nontrivial upper bound for chemical distance on the planar random cluster model

TL;DR

This paper establishes a nontrivial upper bound for the chemical distance in the planar random cluster model with cluster weight between 1 and 4, extending previous results from Bernoulli percolation.

Contribution

It provides the first nontrivial upper bounds for chemical distances in the planar random cluster model for q between 1 and 4, including a complete proof of the arm separation lemma.

Findings

Extended upper bounds for chemical distances in the random cluster model.

Proved the strong arm separation lemma for the model.

Unified approach for percolation and random cluster models.

Abstract

We extend the upper bounds derived for the horizontal and radial chemical distance for 2d Bernoulli percolation in [DHS21, SR20] to the planar random cluster model with cluster weight $1 \le q \le 4$. Along the way, we provide a complete proof of the strong arm separation lemma for the random cluster model.