On the Kert\'esz line: Some rigorous bounds

TL;DR

This paper investigates the properties of the Kertész line in the q-state Potts model, establishing bounds and relationships with phase transitions and percolation thresholds using rigorous mathematical techniques.

Contribution

It provides rigorous bounds for the Kertész line and shows it remains below the first order phase transition line, with explicit asymptotic behavior at large fields.

Findings

Kertész line remains below the first order phase transition line.

Explicit asymptotic form of the Kertész line at large fields.

Proved that a jump in the infinite cluster density occurs at the first order transition.

Abstract

We study the Kert\'esz line of the $q$--state Potts model at (inverse) temperature $\beta$, in presence of an external magnetic field $h$. This line separates two regions of the phase diagram according to the existence or not of an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is known that the Kert\'esz line $h_K (\beta)$ coincides with the line of first order phase transition for small fields when $q$ is large enough. Here we prove that the first order phase transition implies a jump in the density of the infinite cluster, hence the Kert\'esz line remains below the line of first order phase transition. We also analyze the region of large fields and prove, using techniques of stochastic comparisons, that $h_K (\beta)$ equals $\log (q - 1) - \log (\beta - \beta_p)$ to the leading order, as $\beta$ goes to $\beta_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond percolation.