Recent Developments on Mixing Times: March 17-21, 2025

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Organized by:

  • Evita Nestoridi ( Stony Brook University)
  • Dominik Schmid ( Bonn University)

Markov chains serve as indispensable tools for generating random structures, such as graph colorings, vector space bases, and polygon triangulations. Mixing times capture the temporal evolution towards equilibrium. Of particular interest is the abrupt transition from unmixed to mixed – the cutoff phenomenon. The cutoff phenomenon was first discover by Diaconis and Shashahani in the context of card shuffling, and called received its name in . Many techniques from representation theory, combinatorics and probability, such as comparison theory, Nash inequalities, evolving sets, distinguishing statistics, and many more have been developed to understand mixing times. For a standard reference on these classical approaches, we refer to. The question whether cutoff occurs was solved over the years for many models, for example random walks on random graphs, the card shuffles,or the east process. In recent years, many new techniques to verify the occurrence or absence of cutoff, as well as even more refined results on the convergence towards the stationary distribution, were established.