Ergodic Theory and Interacting Particle Systems

摘要分析和概率论的各个分支之间的相互作用有着悠久而卓有成效的历史。通过应用概率中的最新结果来分析动力系统，以及使用遍历理论和其他分析分支来研究概率中的问题，这一建议集中于进一步加强这些联系。两个对遍历理论有很大贡献的概率模型是首次通过渗流模型和Richardson增长模型。这项工作的目的是应用遍历理论的技巧来研究Richardson增长模型中多重感染共存的问题和首次通过渗流中单边测地线的存在性问题。概率结果在研究动力系统中的一个建议的应用是利用多米诺骨牌来研究二维子移位类型的可能的遍历性质。另一个是利用g测度遍历的必要条件方面的最新进展来回答一个长期悬而未决的问题，即环面上的Anosov映射的连续性的必要条件模确保了相应的测度保持系统的遍历性。数学家们长期以来一直使用概率模型来理解复杂的物理、生物和社会系统。这些模型的例子包括用于理解磁化过程的Ising模型，用于研究性接触传播疾病的随机图，以及用于理解感染传播的Richardson增长模型。这一建议旨在通过使用动力系统数学领域的新技术来分析受物理和生物学启发的概率模型，从而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.