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

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.

27 pages, 4 figures, comments welcome