Particle systems and universality of the cutoff phenomenon

拟议的研究计划的目标是开发限制混合时间的通用技术，并建立截止现象的普遍性。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中，包括那些分析超出了目前文献中现有技术的范围的模型，包括：交换、排除和零值过程、随机Cayley图和一大类拟随机图。对于排除过程，我们的目标是建立当基图是固定维环/盒的发散边长时的截断现象。对于排斥和交换过程，我们的目标是解决Oliveira的一个猜想，该猜想断言混合时间是由(相同数量的)独立粒子的混合时间限定的。对于交换过程，我们还旨在证明Aldous的谱隙猜想的算子版本，该猜想不仅在混合时间方面有显著的结果，而且在通过运行交换过程获得的循环分解中出现宏观循环(对于谱隙时间单位的逆)具有显著的结果。我们的目标是建立零程过程截止值的普适性。目前，甚至在少数几个例子中也能理解混合时间的顺序。拟议的方法应该会有很多应用。我们还打算建立包括随机Cayley图在内的广泛随机图的截断现象的普适性。最后，我们打算研究点传递图的(随机游动)混合时的稳健性这一重要问题，它至少在概念上与著名的关于Liouville性质的稳定性的问题有关，并可能对此有所启发。

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.