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

有许多问题需要研究。与渗流有关的三个问题。在连续模型中，键被占据的概率是其长度的函数；将研究首次通过渗流中通过时间的分布的分布；我们将考察Fortuin-Kastelyn随机星团模型中的有限星团，并将结果应用于Ising模型，条件是磁化程度低于典型的N。相互作用粒子系统和元胞自动机的研究将围绕在物理和生物学中有应用的四个模型进行：催化表面反应模型、阈值森林火灾模型、具有质量灭绝的抵消系统和食物链奇偶理论模型。所寻求的结果涉及到这些模型和相变的渐近行为。研究人员将研究小规模随机性产生大范围现象的系统，这些现象本质上是非随机的。这样的系统长期以来一直是数学物理学的研究对象，最近也是生物学的研究对象。一个标准的物理例子是磁铁，磁铁中的单个铁原子具有随机排列的磁极，附近的原子略有平行排列的趋势。根据温度的不同，这种趋势可能会也可能不会反映在铁块的大规模非随机磁化量上。

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.