Some Problems in Probability Theory: Random Walks and Percolation

9970943将研究各种概率问题。主要的一类是处理弱相互作用随机漫步系统和低渗透系统。在随机漫步系统中，从d维整数晶格的每个位置上的一个粒子开始，让每个粒子在这个晶格上进行连续时间随机漫步。只有当粒子跃迁到一个自身被占据的位置或与被占据的位置相邻时，相互作用才会发生。在这种情况下，刚刚跳跃的粒子可能与其中一个粒子在它跳跃的位置或其附近结合。在另一个版本中，刚刚跳跃的粒子可能会湮灭另一个粒子。我们将研究这类系统的渐近密度。rho渗滤具有与普通现场渗滤相同的设置。格子的位置分别以p或1-p的概率被独立地选择为被占用或空着。一种说法是，如果晶格上存在一条无限的自回避路径，其较低的占位密度至少为，则会发生rho渗透。提议者打算研究已知或推测的普通渗流的各种功率边界的类似物。本研究将研究随机运动的粒子系统的问题，当两个粒子彼此靠近时，系统中存在相互作用，使粒子从系统中消失。研究这类系统的一个动机是，它们可以被视为某些化学反应的粗糙模型。希望这里提出的用于研究的模型的变体比通常考虑的模型更现实一些。提案中另一类不相关的问题考虑的是一个经历相变的系统。几乎所有的材料在一定的温度或磁场下都有相变。尽管在统计物理中有很多关于相变的非严格水平的知识，但在数学水平上对它们的理解仍然很少。希望对这一相对较新的rho-渗流模型进行有意义和有指导意义的计算。

9970943Various probability problems will be studied. The principal ones deal with systems of weakly interacting random walks and with rho-percolation. In the random walk systems one starts with one particle at each site of the integer lattice in d dimensions and lets each particle perform a continuous time random walk on this lattice. Interactions occur only when a particle jumps to a site which is itself occupied or is adjacent to an occupied site. In this case the particle which just jumped may coalesce with one of the particles at or adjacent to the site to which it jumped. In another version the particle which just jumped may annihilate another particle. The asymptotic density of such systems will be investigated. Rho-percolation has the same set-up as ordinary site percolation. The sites of a lattice are independently chosen to be occupied or vacant with probability p or 1-p, respectively. One says that rho-percolation occurs if there exists an infinite self-avoiding path on the lattice whose lower density of occupied sites is at least rho. The proposer intends to investigate analogues of various power bounds which are already known or conjectured for ordinary percolation.This research will study problems about systems of particles which undergo random motion and which have an interaction which removes particles from the system when two particles come close to each other. One motivation for studying such systems is that they can be viewed as crude models for certain chemical reactions. It is hoped that the variants of the model proposed for investigation here are somewhat more realistic than the models usually considered. Another, unrelated, class of problems in the proposal considers a system which undergoes a phase transition. Almost all materials have phase transitions at some temperature or magnetic field. Even though a great deal is known in statistical physics on the non rigorous level about phase transitions, they are still only poorly understood on a mathematical level. It is hoped that interesting and instructive calculations can be carried out for the relatively new model of rho-percolation.