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Analysis for Probabilistic Models with phase transition

相变概率模型分析

基本信息

批准号：
11640101
负责人：
TANEMURA Hideki
金额：
$ 2.3万
依托单位：
CHIBA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640101/
关键词：
Domany-Kinzel model Skorohod equation vicious walkers particle systems non-attractive systems non-attractive system friendly walkers Dyson model random matrix phase transition Brownian meander Domany-Kinzel model Boolean model 無限

项目摘要

Probabilistic models such as Domany-Kinzel model, Continuum perclation model, systems of interacting Brqwnian particles and vicious walkers, were studied during the term of this research. Result obtained are explained here for each model.1) Domany-Kinzel model is a stochastic model with two parameters and includes oriented site bond percolatin models and rule 90 model as special cases. For this model we obtain the relation between local survival probability and global survival probability. Applying this relation we studied properties of its stationary distributions and show the limit theorem for the model for non-attractive case.2) A Skorohod equation in infinite dimensional was studied. The solution of this equation represents a system of infinte Brownian balls, that is, a system of infinite Brownian motions with hard core interaction. The existence and uniqueness of the solutions was proved for the class of interactions with hard core and short range potential.3) Continuum percolation models which are composed by convex sets were studeid. We show that, the critical covered area fraction of the model does depend on shapes of sets and attains its minimum when the sets are common triangles.4) Vicious walkers model is a system of finite many independent random walks conditioned not to collide with each other. We show functional central limit theorems for the model and obtained a temporally inhomogeneous diffusion process which is associated with "Two matrix model".

在本研究期间，研究了诸如 Domany-Kinzel 模型、连续体渗流模型、相互作用的 Brqwnian 粒子和恶性步行者系统等概率模型。这里对每个模型所获得的结果进行解释。1) Domany-Kinzel 模型是具有两个参数的随机模型，包括定向位点键渗透模型和规则 90 模型作为特例。对于该模型，我们获得局部生存概率和全局生存概率之间的关系。应用这种关系我们研究了其平稳分布的性质，并给出了非吸引情况下模型的极限定理。2)研究了无限维的Skorohod方程。该方程的解表示无限布朗球系统，即具有硬核相互作用的无限布朗运动系统。证明了硬核和短程势相互作用类解的存在性和唯一性。3)研究了由凸集组成的连续渗流模型。我们表明，模型的临界覆盖面积分数确实取决于集合的形状，并且当集合是常见三角形时达到最小值。4）恶性步行者模型是有限多个独立随机游走的系统，条件是不相互碰撞。我们展示了该模型的功能中心极限定理，并获得了与“双矩阵模型”相关的时间非均匀扩散过程。