Mathematical Sciences: Research in Probability With Applications to Statistical Mechanics & Queueing Networks

首席调查员将在大偏差领域进行研究。特别是，他将在三个主题上开展项目：(A)在抽象空间取值的马尔可夫链的大偏差；(B)统计力学中的大偏差、极限定理和相变；(C)具有“不连续统计”的马尔可夫过程的大偏差及其在排队网络中的应用。大偏差理论是概率论的一个分支，涉及随机变量的极限。众所周知，在某些理想化的情况下，随机变量是独立且同分布的，序列的平均值收敛于单个随机变量的平均值(在统计学方面，样本平均值收敛于总体平均值)。大偏差理论表明，随着样本量的增加，样本平均偏离总体平均水平的概率呈指数级减小。主要研究人员正在研究困难得多的相依随机变量的情况，其相依结构遵循马尔可夫过程。这一理论在统计力学和排队网络中有许多应用。

The principal investigator will conduct research in the area of large deviations. In particular, he will conduct projects on three topics: (a) large deviations for Markov chains taking values in an abstract space; (b) large deviations, limit theorems, and phase transitions in statistical mechanics; (c) large deviations for Markov processes with "discontinuous statistics" and applications to queueing networks. Large deviation theory is a branch of probability theory involving limits of random variables. It is well known that in certain idealized cases, where the random variables are independent and identically distributed, that the average of the sequence converges to the average of the individual random variable (in terms of statistics, that the sample average converges to the population average). Large deviation theory shows that the probability that the sample average deviates from the population average by a small amount is exponentially small as the sample size increases. The principal investigator is working in the much more difficult case of dependent random variables, whose dependence structure follows a Markov process. There are many applications of this theory to statistical mechanics and queueing networks.