Strict monotonicity, continuity and bounds on the Kert\'{e}sz line for the random-cluster model on $\mathbb{Z}^d$

TL;DR

This paper studies the geometric phase transition in the random-cluster model on ^d, proving strict monotonicity and continuity of the Kerte9sz line, and providing new bounds that complement existing asymptotic results.

Contribution

It establishes strict monotonicity and continuity of the Kerte9sz line and introduces new bounds valid as the external field approaches zero.

Findings

Proves strict monotonicity of the Kerte9sz line.

Shows continuity of the Kerte9sz line.

Provides asymptotically correct bounds for small external field.

Abstract

Ising and Potts models can be studied using the Fortuin-Kasteleyn representation through the Edwards-Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength $h>0$. In this representation, which is also known as the random-cluster model, the Kert\'{e}sz line separates the two regions of parameter space according to the existence of an infinite cluster in $\mathbb{Z}^d$. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kert\'{e}sz line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit $h \to 0$ complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for $h \to \infty$. Finally, using a cluster expansion, we investigate the continuity of the Kert\'{e}sz line phase transition.