Some Problems in Probability Theory: Random Walks and Percolation

9970943将研究各种概率问题。主要研究弱相互作用的随机游动系统和Rho渗流系统。在随机游走系统中，从d维整数格的每个位置上的一个粒子开始，并让每个粒子在该格上执行连续时间的随机行走。只有当粒子跳到本身被占用或与被占用站点相邻的站点时，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到的位置上或附近的一个粒子结合。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渗流具有与普通现场渗流相同的设置。晶格的位置分别以概率p或1-p被独立地选择为占用或空闲。一种说法是，如果晶格上存在一条无限的自回避路径，其占据位置的较低密度至少为Rho，就会发生Rho渗流。这项研究将研究粒子系统的问题，这些粒子系统经历随机运动，并且当两个粒子彼此靠近时，相互作用会使粒子从系统中移出。研究这类体系的一个动机是，它们可以被视为某些化学反应的原始模型。希望这里建议研究的模型的变体比通常考虑的模型更现实一些。提案中另一类无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会发生相变。尽管在统计物理学中关于相变的非严格水平已经知道了很多，但在数学水平上对它们的理解仍然很少。希望能对相对较新的Rho渗流模型进行有趣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.