随机场概论

A Survey on Random Fields

二十几年前,诞生了“随机场”这一概率论与统计物理的交叉学科.它一方面大大扩充了概率论的研究领域,另一方面为统计物理提供了一种严格的数学工具.这个学科及共它概率物理分支,代表着当今数学与物理相互渗透的大潮流的一个重要侧面.新近超导研究的突破,为随机场提出了许多迷人的新课题,带来了巨大的推动力.

These notes are presented at a summer school as a preparation of the Nankai's year 1988-1989 on probability and statistics. First some basic facts about weak convergence and probability metrics are reviewed. Then some existence and uniqueness theorems for random fields are introduced. Finally the phase transitions by using Peierls' contour method and reflection positivity are studied. The notes are self-contained in the sense that all details are given.