Particle systems and universality of the cutoff phenomenon

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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.