Analysis on Interacting Particle Systems Based on a New Type of Correlation Inequalities

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We study interacting particle systems (IPSs) based on correlation inequalities, for examples, Harris-FKG inequality, submodularity, and BFKL inequality. In particular, BFKL inequality is a new type of correlation inequality and a refinement of Harris-FKG inequality and submodularity. As a typical example of IPSs, here we focus on one-dimensional contact process. By using above correlation inequalities, we can easily and systematically obtain lower bonds on critical value and upper bounds on survival probability of contact process, compared with other methods, i.e, edge process technique and Harris lemma method. So we call this method correlation inequality method. However there are some limitations on the method in the present stage, The contact process is attractive and two-state process. Therefore we are studying whether or not Harris-FKG inequality and/or BFKL inequality hold even in non-attractive and/or multi-sate IPSs. Concerning non-attractive Domany-Kinzel model, Monte-Carlo simulations suggest that Harris-FKG inequality is correct. For some special cases we could prove it holds very recently. As for multi-state IPSs, for example, 3-state cyclic particle system and successional model in two dimensions, both-Harris-FKG type inequality and BFKL type one would be correct according to results on Monte-Carlo simulations.