The random cluster model on the complete graph via large deviations

TL;DR

This paper analyzes the emergence of the giant component in the random cluster model on complete graphs using large deviations, providing an alternative thermodynamic approach and computing the rate function for large deviations.

Contribution

It introduces a large deviations analysis for the random cluster model on complete graphs, extending thermodynamic methods to this setting.

Findings

Computed the rate function for large deviations of the largest component size.

Provided an alternative analysis to previous studies on the random cluster model.

Enhanced understanding of phase transitions in the model.

Abstract

We study the emergence of the giant component in the random cluster model on the complete graph, which was first studied by Bollob\'as, Grimmett, and Janson. We give an alternative analysis using a thermodynamic/large deviations approach introduced by Biskup, Chayes, and Smith for the case of percolation. In particular, we compute the rate function for large deviations of the size of the largest connected component of the random graph for $q\geq 1$.