Ergodic Theory and Interacting Particle Systems

概率论与分析的各个分支之间的相互作用有着悠久而富有成果的历史。 这一建议的重点是进一步这些连接，无论是applyingrecent结果在概率分析动力系统，并使用遍历理论和其他分支的分析，研究问题的概率。 两个概率模型，其中有很大的贡献，遍历理论的第一次渗透和Richardson的增长模型。本文的工作旨在应用遍历理论中的技巧来研究Richardson增长模型中多重感染的共存问题和首通渗流中单边测地线的存在性问题。概率结果在动力系统研究中的一个应用是使用多米诺骨牌，即有限型二维子移位的可能遍历性质。另一个是利用最近在g测度遍历性的必要条件方面的进展来回答一个长期悬而未决的问题，即环面上Anosov映射的连续模的必要条件，它确保了相应的测度保持系统的遍历性。这些模型的例子包括理解磁化过程的伊辛模型，研究疾病通过性接触传播的随机图，and Richardson理查森's增长model模型to understand理解the spread传播of infection感染.该提案旨在通过使用动力系统数学领域的新技术来分析受物理学和生物学启发的概率模型，为这一领域做出贡献。

AbstractThere is a long and fruitful history of interaction betweenvarious branches of analysis and probability theory. Thisproposal focuses furthering these connections, both by applyingrecent results in probability to analyze dynamical systems, andusing ergodic theory and other branches of analysis, to studyquestions in probability. Two probabilistic models which haveseen large contributions from ergodic theory are first passagepercolation and Richardson's growth model. The work in thisproposal aims to apply techniques from ergodic theory to studyquestions of coexistence of multiple infections in Richardson'sgrowth models and the existence of one sided geodesics in firstpassage percolation. One proposed application of probabilityresults to study dynamical systems is the use of domino tilingsthe possible ergodic properties of two dimensional subshifts offinite type. Another is to use recent advances in necessaryconditions for ergodicity of g measures to answer a longstandingopen question about the necessary conditions modulus of continuityof an Anosov map on the torus which ensures ergodicity of thecorresponding measure preserving system.Mathematicians have long used probabilistic models to gainunderstanding of complex physical, biological and social systems.Examples of these models include the Ising model to understand theprocess of magnetization, random graphs to study the spread ofdisease through sexual contact, and Richardson's growth model tounderstand the spread of infection. This proposal aims tocontribute to this field by using new techniques in themathematical field of dynamical systems to analyze probabilisticmodels inspired by physics and biology.