Some Problems in Probability Theory: Random Walks and Percolation

9970943将研究各种概率问题。 主要的处理弱相互作用的随机游动和ρ-渗流系统。 在随机游走系统中，我们从d维整数格点的每个点上的一个粒子开始，让每个粒子在这个格点上进行连续时间的随机游走。 只有当一个粒子跳到一个被占据的位置或与被占据的位置相邻的位置时，才会发生相互作用。 在这种情况下，刚刚跳跃的粒子可能会与它跳跃到的位置处或附近的粒子之一合并。 在另一个版本中，刚刚跳跃的粒子可能会湮灭另一个粒子。 将研究这类系统的渐近密度。 Rho渗流与普通的站点渗流具有相同的设置。 一个格子的位置被独立地选择为被占据或空置，概率分别为p或1-p。 有人说，如果在晶格上存在一个无限的自回避路径，其较低的占据位置密度至少为ρ，则发生ρ-渗流。 本研究的目的是研究各种已知的或为普通渗流所确定的幂边界的类似物。本研究将研究进行随机运动的粒子系统的问题，当两个粒子相互靠近时，系统中的粒子会相互作用。 研究这些系统的一个动机是，它们可以被视为某些化学反应的粗糙模型。 我们希望，这里提出的研究模型的变体比通常考虑的模型更现实一些。 另一个不相关的问题，在建议中考虑一个系统，经历了相变。 几乎所有的材料在一定的温度或磁场下都会发生相变。 尽管统计物理学在非严格的层面上对相变有很多了解，但在数学层面上对它们的理解仍然很少。 人们希望，有趣的和有指导意义的计算可以进行相对较新的模型的ρ-渗流。

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.