Particle systems and universality of the cutoff phenomenon

提出的研究计划的目的是发展混合时间边界的一般技术和建立截断现象的普遍性。我们的目的也是将这些技术应用于广泛的随机图和粒子系统的重要模型，包括那些目前在文献中可用的技术无法分析的模型，包括：交换，排斥和零范围过程，随机Cayley图和一大类拟随机图。在排除过程中，我们的目标是建立当基图是边长发散的固定维环面/盒时的切断现象。对于不相容和交换过程，我们的目标是解决Oliveira的一个猜想，该猜想断言混合时间由（相同数量的）独立粒子的时间限制。对于交换过程，我们还旨在证明Aldous谱隙猜想的算子版本，这不仅会对混合时间产生显著影响，而且还会对通过运行交换过程获得的循环分解中宏观循环的出现产生显著影响（对于谱隙时间单位的逆）。我们的目的是建立零距离过程的截止的通用性。目前，即使是混合时间的顺序，在少数例子中也是可以理解的。所提出的方法应该有许多应用。我们还打算建立包括随机Cayley图在内的各种随机图的截止现象的普遍性。最后，我们打算研究点传递图的混合时间的鲁棒性问题，这个问题至少在概念上与著名的Liouville性质的稳定性问题有关，并可能对它有所启发。

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.