Study on phase transition for interacting particle systems

相互作用粒子系统的相变研究

• 批准号: 12440024
• 负责人: KONNO Norio
• 金额: $ 9.79万
• 依托单位: Yokohama National University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440024/
• 关键词: Interacting particle systems; Contact Process; duality; correlation inequality; quantum random walk; phase transition; パーコレーション; 極限定理; コンタクトプロセス

We study phase transitions of interacting particle systems, in particular, contact process type model, the Domany-Kinzel model by using various correlation inequalities and correlation identities. Recently we obtain the BFKL inequality which is a refinement of a special case of the well-known Harris-FKG inequality. From this inequality, bounds of the survival probability and critical value can be obtained systematically. In general, we can apply the above mentioned method (called correlation inequality method (CIM) to attractive interacting particle systems. However it was not known whether or not the CIM can be applied to even non-attractive models. Some Monte Carlo simulations suggested that some non-attractive models satisfy the Harris-FKG type inequality. In fact, we could prove this fact later. But we do not show that the BFKL inequality can be satisfied for the same model. Furthermore we obtain a necessary and sufficient condition on duality for the Domany-Kinzel model by several methods. One of them is closely related to quantum mechanics. So we move to study quantum walks which have been widely studied in the field of quantum computing. In particular, we study symmetry of distribution, limit theorems, and absorption problems of quantum walks with nearest-neighbor transition in one dimension. Very recently we show a localization of two-dimensional quantum walk including the Grover walk.

我们通过使用各种相关不等式和相关恒等式来研究相互作用粒子系统的相变，特别是接触过程类型模型、Domany-Kinzel 模型。最近我们得到了 BFKL 不等式，它是著名的 Harris-FKG 不等式的特例的改进。从这个不等式中，可以系统地获得生存概率和临界值的界限。一般来说，我们可以将上述方法（称为相关不等式方法（CIM））应用于有吸引力的相互作用粒子系统。然而，尚不清楚 CIM 是否可以应用于甚至非有吸引力的模型。一些蒙特卡洛模拟表明：一些非吸引力模型满足 Harris-FKG 型不等式，但我们并没有证明同一模型可以满足 BFKL 不等式。此外，我们还得到了对偶性的充分必要条件。为Domany-Kinzel 模型有多种方法，其中之一与量子力学密切相关，因此我们开始研究量子计算领域中广泛研究的量子游走，特别是我们研究分布的对称性、极限定理和量子游走。一维中最近邻跃迁的量子行走的吸收问题 最近我们展示了包括 Grover 行走在内的二维量子行走的局域化。

项目成果

N.Konno, R.Schinazi, H.Tanemura: "Coexistence results for a spatial stochastic epidemic model"Markov Processes and Related Fields. (2004)

N.Konno、R.Schinazi、H.Tanemura：“空间随机流行病模型的共存结果”马尔可夫过程和相关领域。

Makoto Katori: "Survival probabilities for discrete-time models in one dimension"Journal of Statistical Physics. 99. 603-612 (2000)

Makoto Katori：“一维离散时间模型的生存概率”统计物理学杂志。

Kazunori Sato: "Parity law for population dynamics of N-species wity cyclic advantage competitions"Applied Mathematics and Computation. Vol.126 Nos.2-3. 255-270 (2002)

Kazunori Sato：“具有循环优势竞争的 N 物种种群动态的奇偶律”应用数学和计算。

N.Konno: "Absorption problems for quantum walks in one dimension"J.Phys.A : Math.Gen. 36. 241-253 (2003)

N.Konno：“一维量子行走的吸收问题”J.Phys.A ：Math.Gen。

N.Konno, T.Namiki, T.Soshi, A.Sudburg: "Absorption problems for quantum random walks in one dimertion"Journal of Physics A : Mathematical and General. 36巻. 241-253 (2003)

N.Konno、T.Namiki、T.Soshi、A.Sudburg：“一维量子随机游走的吸收问题”《物理学杂志 A》：数学与综合。 36. 241-253 (2003)