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

TL;DR

This paper studies the scaling limit of a finite system of non-intersecting FK-percolation clusters, showing it converges to a system of non-intersecting Brownian bridges, and provides probability estimates for cluster connections.

Contribution

It establishes the diffusive scaling limit of non-intersecting FK clusters as Brownian watermelons and derives asymptotics for large cluster probabilities.

Findings

Scaling limit is given by non-intersecting Brownian bridges.

Probability estimates for cluster connections at large distances.

Asymptotics for large finite cluster occurrence in supercritical models.

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.