# A short proof of the discontinuity of phase transition in the planar random-cluster model with $q>4$

## TL;DR

This paper presents a concise proof demonstrating the discontinuity of phase transition in the planar random-cluster model for q>4, avoiding complex techniques like the Bethe ansatz and Russo--Seymour--Welsh theory.

## Contribution

It provides a simplified, elementary proof of phase transition discontinuity for q>4 in the random-cluster model, bypassing advanced methods used previously.

## Key findings

- Phase transition is discontinuous for q>4.
- The proof avoids Bethe ansatz and RSW theory.
- Connects to six-vertex model without complex techniques.

## Abstract

The goal of this paper is to provide a short proof of the discontinuity of phase transition for the random-cluster model on the square lattice with parameter $q>4$. This result was recently shown via the so-called Bethe ansatz for the six-vertex model. Our proof also exploits the connection to the six-vertex model, but does not rely on the Bethe ansatz. Our argument is soft and only uses very basic properties of the random-cluster model (for example, we do not need the Russo--Seymour--Welsh theory).