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

有许多问题需要研究。三个问题与渗透有关。在连续模型中，一个键被占用的概率是其长度的函数，我们将研究簇大小分布和有限簇几何的密切相关方面；研究了第一通道渗流中通过时间分布的扩展；将对Fortuin-Kastelyn随机簇模型中的有限簇进行检验，并将结果应用于Ising模型，使其磁化程度低于典型的磁化程度。相互作用粒子系统和元胞自动机的研究将围绕四个在物理和生物学上有应用的模型：催化表面反应模型、阈值森林火灾模型、大灭绝的取消系统和食物链奇偶理论模型。要寻求的结果涉及这些模型和相变的渐近行为。研究者将研究那些小规模随机产生大规模非随机现象的系统。这样的系统长期以来一直是数学物理研究的对象，最近也是生物学研究的对象。一个标准的物理例子是磁铁，其中单个铁原子具有随机排列的磁极，而附近的原子有轻微的平行排列趋势。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中。

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.