Study on phase transition for interacting particle systems

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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.