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Article Dans Une Revue Journal of Mathematical Physics Année : 2008

On the Kertész line: Some rigorous bounds

Résumé

We study the Kertész 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 Kertész 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 Kertész 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.
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Dates et versions

hal-00254740 , version 1 (13-02-2008)

Identifiants

Citer

Jean Ruiz, Marc Wouts. On the Kertész line: Some rigorous bounds. Journal of Mathematical Physics, 2008, 49, pp.053303. ⟨10.1063/1.2924322⟩. ⟨hal-00254740⟩
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