Particle systems and universality of the cutoff phenomenon

粒子系统和截止现象的普遍性

• 批准号: RGPIN-2020-04251
• 负责人: Hermon, Jonathan
• 金额: $ 2.11万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716860
• 关键词: Particle; systems; universality; cutoff; phenomenon

The objective of the proposed research program is to develop general techniques for bounding mixing times and establishing universality of the cutoff phenomenon. Our purpose is also to apply these techniques to a wide range of important models of random graphs and particle systems including ones whose analysis is beyond the reach of the techniques currently available in the literature, including: the interchange, exclusion and zero-range processes, random Cayley graphs and a large class of quasi random graphs. For exclusion process, our goal is to establish the cutoff phenomenon when the base graphs are fixed dimensional tori/boxes of diverging side lengths. For the exclusion and the interchange processes we aim to resolve a conjecture of Oliveira that asserts that the mixing time is bounded by that of (the same number of) independent particles. For the interchange process we also aim to prove an operator version of Aldous' spectral-gap conjecture which will have striking consequences not only regarding the mixing time, but also regarding the emergence of macroscopic cycles in the cycle decomposition obtained by running the interchange process (for inverse of the spectral-gap time units). We aim to establish universality of cutoff for the zero range process. Currently even the order of the mixing time is understood in a handful of examples. The proposed method should have numerous applications. We also intend to establish universality of the cutoff phenomenon for a wide range of random graphs including random Cayley graphs. Lastly, we intend to study the important question of robustness of mixing times of (random walks on) vertex-transitive graphs, which is at least conceptually related to the famous question concerning stability of the Liouville property, and may shed some light on it.

拟议的研究计划的目的是开发一般技术的边界混合时间，并建立普遍性的截止现象。我们的目的也是将这些技术应用到广泛的随机图和粒子系统的重要模型，包括其分析是目前文献中可用的技术所无法达到的，包括：交换，排除和零范围过程，随机凯莱图和一大类准随机图。 对于排斥过程，我们的目标是建立截断现象时，基图是固定维环面/盒的发散边长。对于排斥和交换过程，我们的目标是解决奥利维拉的一个猜想，该猜想声称混合时间是由（相同数量的）独立粒子的混合时间所限定的。对于交换过程，我们还旨在证明Aldous的谱隙猜想的一个算子版本，它不仅对混合时间，而且对通过运行交换过程（对于谱隙时间单位的逆）获得的循环分解中宏观循环的出现具有显著的影响。 我们的目标是建立零程过程截止的普适性。目前，甚至在少数示例中理解了混合时间的顺序。所提出的方法应该有许多应用。 我们还打算建立广泛的随机图，包括随机凯莱图的截止现象的普遍性。最后，我们打算研究的重要问题的鲁棒性的混合时间的（随机游动）顶点传递图，这是至少在概念上与著名的问题有关的稳定性的刘维性质，并可能揭示一些光。