Interacting Particle Systems

Liggett该项目解决了相互作用粒子系统和概率论相关领域的几个问题。一组问题涉及一维和更高维漂移的排除过程的平稳分布。研究者最近的工作证明了在弱假设下一维平稳阻塞测度的存在性。下一组问题涉及一维的非阻塞平稳轮廓测度和高维的不可逆平稳阻塞测度的存在--在这两种情况下都没有任何已知的例子。其他问题涉及(A)对称排斥过程与负相关性之间的联系，(B)对称排斥过程的已知理论在多大程度上推广到具有一些不对称的过程，(C)由研究者最近关于社会学中一类模型的工作而引起的一些无限系统的分析，以及(D)齐次树上一族可逆增长模型的临界值的计算。自然科学和社会科学中的许多问题都涉及大量个人的行为(人、分子、汽车、病毒等)。以不可预测的方式相互作用。这种缺乏可预测性的现象是由随机性造成的。上个世纪概率论的基本贡献之一是认识到，尽管人口中个体的行为缺乏可预测性，但大量人口显示出相当大的可预测性和秩序。这个项目的目的是为了更好地了解这种一般类型的几种模型。其中包括：(A)排斥过程，它模拟了生物中的粒子运动、交通流量和核糖体的行为；(B)某些生长模型，部分是由肿瘤生长和相互竞争的种群之间的冲突所驱动的；以及(C)社会学模型，其中个人之间的关系通过交换礼物或奖励来改变。在每种情况下，目标都是确定系统的长期行为。

0301795Liggett The project addresses several problems in interacting particle systems and related areas of probability theory. One collection of problems involves the stationary distributions for exclusion processes with drift in both one and higher dimensions. Recent work of the investigator showed the existence of stationary blocking measures in one dimension under weak assumptions. The next set of problems concerns the existence of nonblocking stationary profile measures in one dimension and of nonreversible stationary blocking measures in higher dimensions -- in neither case are there any known examples. Other problems involve (a) connections between the symmetric exclusion process and negative dependence, (b) the extent to which the known theory of the symmetric exclusion process extends to processes with a few asymmetries, (c) the analysis of some infinite systems motivated by the investigator's recent work on a class of models in sociology, and (d) the computation of the critical values for a family of reversible growth models on homogeneous trees. Many problems in the natural and social sciences involve the behavior of large numbers of individuals (people, molecules, cars, viruses, etc.) that interact in unpredictable ways. This lack of predictability is modeled by randomness. One of the fundamental contributions of probability theory in the past century is the realization that large populations display a significant amount of predictability and order in spite of the lack of predictability of the actions of the individuals in the population. This project is aimed at gaining a better understanding of several models of this general type. Among them are: (a) the exclusion process, which models particle motion, traffic flow, and the behavior of ribosomes in biology, (b) certain growth models that are partly motivated by tumor growth and conflicts between competing populations, and (c) models from sociology, in which relationships among individuals are altered by the exchange of gifts or rewards. In each case, the objective is to determine the long time behavior of the system.