Entropic repulsion and scaling limit for a finite number of non-intersecting subcritical FK interfaces

Abstract

This article is devoted to the study of a finite system of long clusters of subcritical 2-dimensional FK-percolation with q $\geq$ 1, conditioned on mutual avoidance. We show that the diffusive scaling limit of such a system is given by a system of Brownian bridges conditioned not to intersect: the so-called Brownian watermelon. Moreover, we give an estimate of the probability that two sets of $r$ points at distance $n$ of each other are connected by distinct clusters. As a byproduct, we obtain the asymptotics of the probability of the occurrence of a large finite cluster in a supercritical random-cluster model.