Study on phase transition for interacting particle systems

我们利用各种关联不等式和关联恒等式研究了相互作用粒子系统的相变，特别是接触过程类型模型和Domany-Kinzel模型。最近，我们得到了BFKL不等式，它是著名的Harris-FKG不等式的一个特例的推广。从这个不等式可以系统地得到生存概率和临界值的界。一般来说，我们可以将上述方法(称为关联不等法(CIM))应用于具有吸引力的相互作用粒子系统。然而，目前还不知道CIM是否可以应用于甚至不具吸引力的模特。一些蒙特卡罗模拟表明，一些不吸引人的模型满足Harris-FKG型不等式。事实上，我们可以稍后证明这一事实。但是，我们并没有证明BFKL不等式在相同的模型下也能满足。此外，我们还用多种方法得到了Domany-Kinzel模型的对偶的一个充要条件。其中之一与量子力学密切相关。因此，我们开始研究在量子计算领域中被广泛研究的量子行走。特别地，我们研究了一维最近邻跃迁量子游动的分布对称性、极限定理和吸收问题。最近，我们展示了包括格罗弗行走在内的二维量子行走的局域化。

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.