论文标题
$φ^4 $的随机纠结电流：翻译不变的吉布斯测量和相变的连续性
Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition
论文作者
论文摘要
我们证明，多项式增长图上$φ^4 $模型的一组自动形态不变的Gibbs度量最多具有$β$的所有值。我们还提供足够的条件，以确保所有Gibbs措施的集合都是单身人士。作为一个应用程序，我们表明，$ \ Mathbb {z}^d $在$ d \ geq 3 $的关键时消失的最近的neighbour $φ^4 $模型的自发磁化。在Aizenman，Duminil-Copin和Sidoravicius（Comm。Math。Phys。，2015年）和Raoufi（Ann。prob。，2020）的开创性作品中，为ISING模型建立了类似的结果。本文的主要贡献之一是开发$φ^4 $模型的相应几何表示，称为随机纠结电流表示。
We prove that the set of automorphism invariant Gibbs measures for the $φ^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $β$. We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour $φ^4$ model on $\mathbb{Z}^d$ vanishes at criticality for $d\geq 3$. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the $φ^4$ model called the random tangled current representation.
