Analysis of Critical Phenomena in Random Systems

随机系统中的临界现象分析

• 批准号: 09440079
• 负责人: HIGUCHI Yasunari
• 金额: $ 7.1万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440079/
• 关键词: Glauber dynamics; spectral gap; critical point; Ising model; percolation; shrink of contours; reinforced random walk; コントゥアーの収縮; First passage percolation; fluctuation; コントゥアー; 境界条件; Ising model; Glauber dynamics; spectral gap; reinforced r

We investigated the spectral gap of Glauber dynamics in a finite square in the low temperature case in two dimensions. This is known to have a uniform lower bound independent of the boundary condition when the temperature is higher than the critical point, and goes to zero as the size of the square goes to infinity in the low temperature case.We obtained a sharpe lower bound of the spectral gap for the + boundary condition up to the critical point : gap(L,+)【greater than or equal】exp{-CィイD8LlogLィエD8}, where L is the size of the box. This bound is better than the existing one. It is even better than the estimate which is known only for the low enough temperature.Further, we showed that the uniform spectral gap is equivalent to the strong mixing condition even for unbounded spin systems.

本文研究了二维低温情况下有限正方形中Glauber动力学的谱隙。当温度高于临界点时，它有一个与边界条件无关的统一下界，当温度较低时，当平方的大小趋于无穷大时，它趋于零。我们得到了一个在临界点之前+边界条件下的谱隙的尖锐下界：gap（L，+）[大于或等于]exp{-C D8 LlogL D8}，其中L是盒子的大小。这个界限比现有的界限好。这一结果甚至优于仅在足够低的温度下才知道的估计。进一步，我们证明了即使对于无界自旋系统，均匀谱隙也等价于强混合条件。

项目成果

関口英子: "古典型有界対称領域における Penrose 変換の高次元化と特異ユニタリ表現" リー群と幾何学シンポジウム講演要旨. 1-8 (1998)

Eiko Sekiguchi：“经典有界对称区域中彭罗斯变换的高维化和奇异酉表示”李群和几何研讨会摘要 1-8 (1998)。

Y.Higuchi and N.Yoshida: "Slow relaxation of stochastic Ising models with random and non-random boundary conditions," In : New trends in stochastic analysis, K.D.Elworthy, S.Kusuoka, I.Shigekawa (eds), World Scientific Publishing. 153-167 (1997)

Y.Higuchi 和 N.Yoshida：“具有随机和非随机边界条件的随机 Ising 模型的缓慢松弛”，见：随机分析的新趋势，K.D.Elworthy、S.Kusuoka、I.Shigekawa（编辑），世界科学出版社

M. Katori 他: "Survival probabilities for discrete time models in one dimension"Journal of Statistical Physics. (to appear).

M. Katori 等人：“一维离散时间模型的生存概率”统计物理学杂志（待发表）。

R. Schonmann 他: "Exponential relaxation of Glauber dynamics with some specia boundary conditions"Communications in Mathematical Physics. 189. 299-310 (1997)

R. Schonmann 等人：“具有某些特定边界条件的 Glauber 动力学的指数弛豫”数学物理通讯 189. 299-310 (1997)。

J. Murai: "Percolation on high dimensional Menger sponges"Kobe Journal of Mathematics. 14. 49-61 (1997)

J. Murai：“高维门格尔海绵上的渗透”神户数学杂志。