关于Kertész线：一些严格的界限
On the Kertész line: Some rigorous bounds
摘要：
研究了$q$ -态Potts模型在（逆）温度$β$，外加磁场$h$下的Kertész线.这条线根据模型的Fortuin-Kasteleyn表示中是否存在无限团簇来分隔相图的两个区域。当$q$足够大时，Kertész线$h_K (β)$与小场的一阶相变线重合.在这里，我们证明了一级相变意味着一个跳跃的无限集团的密度，因此Kertész线仍然低于一级相变线。我们还分析了该地区的大领域，并证明，使用随机比较技术，$h_K (β)$等于$\log (q - 1) - \log (β- β_p)$的领先顺序，$β$去$β_p = - \log (1 - p_c)$，其中$p_c$是键渗流的阈值。
英文摘要：
We study the Kertész line of the $q$--state Potts model at (inverse) temperature $β$, 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 $h_K (β)$ 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 $h_K (β)$ equals $\log (q - 1) - \log (β- β_p)$ to the leading order, as $β$ goes to $β_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond percolation.
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