论文标题
在$ \ mathbb {z}^d $上的随机群集模型的kertész线上的严格单调性，连续性和界限
Strict monotonicity, continuity and bounds on the Kertész line for the random-cluster model on $\mathbb{Z}^d$
论文作者
论文摘要
可以通过Edwards-Kokal耦合使用Fortuin-Kasteleyn代表来研究Ising和Potts模型。这适应了模型暴露于强度$ h> 0 $的外部场的设置。 在此表示形式（也称为随机群集模型）中，Kertész线根据$ \ Mathbb {Z}^d $中的无限群集的存在将参数空间的两个区域分开。这表示即使在没有发生热力学相变的情况下，有序相和无序相之间的几何相变。在本文中，我们证明了Kertész系列的严格单调性和连续性。此外，我们给出了新的严格界限，这些界限在限制$ h \ to 0 $中渐近正确，以补充Ruiz和Wouts的工作中的界限[J. J.数学。物理。 49，053303（2008）]，在$ h \ to \ infty $上是渐近正确的。最后，使用簇扩展，我们研究了Kertész线相变的连续性。
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 Kertész 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 Kertész 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 Kertész line phase transition.
