We consider randomized dynamics over the $n$-simplex, where at each step a random set, or block, of coordinates is evenly averaged. When all blocks have size 2, this reduces to the repeated averages studied in [CDSZ22], a version of the averaging process on a graph [AL12]. We study the convergence to equilibrium of this process as a function of the distribution of the block size, and provide sharp conditions for the emergence of the cutoff phenomenon. Moreover, we characterize the size of the cutoff window and provide an explicit Gaussian cutoff profile. To complete the analysis, we study in detail the simplified case where the block size is not random. We show that the absence of a cutoff is equivalent to having blocks of size $n^{Ω(1)}$, in which case we provide a convergence in distribution for the total variation distance at any given time, showing that, on the proper time scale, it remains constantly 1 up to an exponentially distributed random time, after which it decays following a Poissonian profile.
Mixing time for the Repeated Balls into Bins dynamics
本文估计了一类不可逆有限马尔可夫链--重复球入仓（RBB）过程的混合时间。该过程是一个具有并行更新的离散时间守恒相互作用粒子系统。最初放置在$L$箱中的$rL$球中，其中$r$为固定的正常数。在每个时间步骤，从每个非空的箱子中取出一个球。然后将\emph{all the balls}统一重新分配到bin中。证明了RBB过程的混合时间与初态的最大球占有数成线性关系。因此，如果初始配置使得球的最大占用数量为$L$阶，则混合时间为相同的正确阶。而如果初始构型更稀，则在$(\log L)^c$阶时间内达到平衡。
Cutoff for mixtures of permuted Markov chains: reversible case
Département d’Informatique de l’ENS, École Normale Supérieure（巴黎高等师范学院计算机科学系）；CNRS（法国国家科学研究中心）；PSL Research University（巴黎文理研究大学）；INRIA Paris（法国国家信息与自动化研究所巴黎分部）；CNRS, I2M, Aix-Marseille Université（法国国家科学研究中心、马赛数学研究所、艾克斯-马赛大学）
