# Another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$ for random-cluster model

## TL;DR

This paper provides an alternative proof for the critical probability of the random-cluster model on the 2D square lattice for q in [1,4], utilizing parafermionic observables.

## Contribution

It introduces a new proof technique for the critical point formula using parafermionic observables, differing from previous methods.

## Key findings

- Validates the critical probability formula for q in [1,4] on ^2
- Demonstrates the effectiveness of parafermionic observables in percolation theory
- Provides a new perspective on phase transition proofs in statistical mechanics

## Abstract

In this paper we give another proof of $p_c = \frac{\sqrt{q}}{1+\sqrt{q}}$ on $\mathbb{Z}^2$ with $q \in [1,4]$, based on the method of parafermionic observables.