Aizenman らの論文, Random Currents and Continuity of Ising Model's Spontaneous Magnetization¶

本文¶

田崎さんから 11/8 に Aizenman が Ising の磁化の連続性の証明を出したことを教えてもらった.

アブストラクト¶

Random Currents and Continuity of Ising Model's Spontaneous Magnetization

Michael Aizenman, Hugo Duminil-Copin, Vladas Sidoravicius

The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on $Z^d$ in $d=3$ dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. It allows to conclude similar continuity results for one dimensional systems provided the decay is slower than $1/r^2$ (at which the transition is known to be discontinuous). The proof employs in an essential way an extension of Ising model's random current representation to the model's infinite volume limit. This extension enables one to reduce the continuity statement to a simple criterion on the decay of correlation in the Gibbs state with free boundary conditions. For reflection positive models, this criterion may be established through the related infrared bound.

今すぐには読めないが, 忘れないようメモしておこう.

ラベル¶

数学, 物理, 数理物理, Ising, 統計力学, 相転移