论文标题
在$ \ mathbb {z}^d $上的Blume-Capel模型的三级临界点的存在
Existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$
论文作者
论文摘要
我们证明了Blume-Capel模型在$ \ mathbb {z}^d $上的每一个$ d \ geq 2 $上的三级点。 $ d \ geq 3 $中的证明依赖于新颖的组合映射到较大图，艾森曼（Aizenman），杜米尼尔·波辛（Duminil-Copin）和sidoravious的技术（Comm。Math。Phys，2015年）以及著名的红外线绑定。在$ d = 2 $中，证明依赖于对蓝贝贝尔稀释的随机簇表示的交叉概率的定量分析。特别是，我们开发了二次切开术，其精神是Duminil-Copin and Tassion（MoscowMath。J.，2020），这使我们能够在$ d = 2 $中获得相位图的精细图片，包括所有地区的相关性渐近行为。最后，我们表明，用于建立稀释的随机簇模型的亚临界锐度的技术扩展到任何$ d \ geq 2 $。
We prove the existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$ for every $d\geq 2$. The proof in $d\geq 3$ relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In $d=2$, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in $d=2$, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any $d\geq 2$.
