Mathematical Sciences: Percolation, Particle Systems, and Other Stochastic Processes

有许多问题需要研究。 三 问题涉及渗流。 团簇大小分布和 有限簇的几何形状的密切相关的方面将 在一个连续模型中， 被占用的键是其长度的函数; 首次通过时间分布 将被研究;和有限集群在Fortuin-Kastelyn 随机聚类模型将被检查，并将结果应用于 假设伊辛模型的磁化强度小于n， 典型的 相互作用粒子系统与细胞研究 自动机将围绕四个模型，有应用在 物理学与生物学：催化表面反应模型 临界林火模型，一个带质量的可消系统 灭绝，以及食物链的奇偶理论模型。 结果要寻求关注的渐近行为，这些 模型和相变。 研究人员将研究小规模的系统， 随机性会产生大规模的现象， 非随机的 这种系统长期以来一直是研究的对象， 数学物理学，以及最近的生物学。 一 一个标准物理例子是磁铁，其中单个铁 原子具有随机排列的磁极， 邻近原子有平行排列的趋势。 取决 在温度上，这种趋势可能会或可能不会反映在 大规模的非随机磁化量的大块 铁.

There are a number of problems to be studied. Three problems concern percolation. The cluster size distribution and closely related aspects of the geometry of finite clusters will be examined for a continuous model in which the probability of a bond being occupied is a function of its length; the spread of th distribution of the passage time in first passage percolation will be studied; and finite clusters in the Fortuin-Kastelyn random cluster model will be examined and the results applied to the Ising model conditioned to be less magnetized thatn is typical. Research in interacting particle systems and cellular automata will center around four model which have applications in physics and biology: a catalytic surface reaction model, a threshold forest fire model, a cancellative system with mass extinction, and a model for the even-odd theory of food chains. Results to be sought concern the asymptotic behavior of these model and phase transitions. The investigator will study systems in which small-scale randomness produces large-scale phenomena which are essentially nonrandom. Such systems have long been an object of study in mathematical physics, and more recently also in biology. A standard physical example is a magnet, in which individual iron atoms have randomly aligned magnetic poles, with a slight tendency for nearby atoms to have parallel alignments. Depending on the temperature, this tendency may or may not be reflected in a large-scale nonrandom amount of magnetization of the chunk of iron.