标题： 马尔可夫链的现代方面：熵、曲率和截止现象 显示英文标题

标题： Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon

摘要： 截断现象是某些马尔可夫过程在状态空间大小趋于无穷大的极限情况下，从非平衡到平衡的突然转变：它们的距离到平衡不是随时间逐渐衰减，而是在一段时间内保持在其最大值附近，当时间参数达到临界阈值时突然降至零。 四十年前在洗牌过程中发现这一令人惊讶的现象后，它已在各种模型中被观察到，从群或复杂网络上的随机游走，到相互作用的粒子系统。 现在人们认为，这是快速混合的高维过程中的普遍现象。 然而，目前的证明高度依赖于模型，确定触发截断的一般条件仍然是有限马尔可夫链定量分析中最大的挑战之一。 这些讲义的目的是提供对这一迷人问题的自包含介绍，并描述其与熵、曲率和集中性最近发现的关系。 显示英文摘要

摘要： The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of these lecture notes is to provide a self-contained introduction to this fascinating question, and to describe its recently-uncovered relations with entropy, curvature and concentration.