Cutoff profile of ASEP on a segment

高密度和低密度相中TASEP的混合时间与截断现象

Mixing times and cutoff for the TASEP in the high and low density phase

本研究探讨开放边界TASEP在高密度和低密度相中的混合时间，发现体相内出现N阶截断，而共存线混合时间为N^2且无截断，并确定了共存线上的ε-混合时间，完善了该模型的混合时间图景。

Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion

In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of $N$ cards. We prove that around time $N^2\log N/(2π^2)$, the total variation distance to equilibrium of the deck distribution drops abruptly from $1$ to $0$, and that the separation distance has a similar behavior but with a transition occurring at time $(N^2\log N)/π^2$. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length $N$ with $k$ particles.

Mixing times for the simple exclusion process with open boundaries

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary paramete…

Mixing time and cutoff for one dimensional particle systems

We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results. A Markov chain is said to exhibit cutoff if on a certain time scale, the distance to equilibrium drops abruptly from $1$ to $0$. We also review a couple of techniques used to obtain these results by exposing and commenting some elements of proof.

Mixing time and cutoff for the weakly asymmetric simple exclusion process

We consider the simple exclusion process with $k$ particles on a segment of length $N$ performing random walks with transition $p>1/2$ to the right and $q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates $b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than $(\log k)/N$. We show that in the former case ($b \gg (\log k)/N$), the mixing time corresponds to the time needed to reach macroscopic e…

具有单调速率的非对称零程过程的截断现象

Cutoff phenomenon for asymmetric zero range process with monotone rates

该研究探究带非递减速率的线段非对称零程过程混合时间，证明凸通量完全非对称及强非对称凹通量情形下均存在截断现象，且混合发生于宏观系统平衡时，还提出了具独立价值的不规则初始数据流体动力学极限证明方法。

Hydrodynamic limit and cutoff for the biased adjacent walk on the simplex

We investigate the asymptotic in $N$ of the mixing times of a Markov dynamics on $N-1$ ordered particles in an interval. This dynamics consists in resampling at independent Poisson times each particle according to a probability measure on the segment formed by its nearest neighbours. In the setting where the resampling probability measures are symmetric, the asymptotic of the mixing times were obtained and a cutoff phenomenon holds. In the present work, we focus on an asymmetric version of the model and we establish a cutoff phenomenon. An important part of our analysis consists in the derivat…

Cutoff for the non reversible SSEP with reservoirs

We consider the Symmetric Simple Exclusion Process (SSEP) on the segment with two reservoirs of densities $p, q \in (0,1)$ at the two endpoints. We show that the system exhibits cutoff with a diffusive window, thus confirming a conjecture of Gantert, Nestoridi, and Schmid in \cite{Gantert2020}. In particular, our result covers the regime $p \neq q$, where the process is not reversible and there is no known explicit formula for the invariant measure. Our proof exploits the information percolation framework introduced by Lubetzky and Sly, the negative dependence of the system, and an anticoncent…

非均匀电导的简单排他过程的截断现象

Cutoff of the simple exclusion process with inhomogeneous conductances

本文研究了非均匀电导下线段上简单排他过程的混合时间，证明了在特定假设下，其到平衡态的总变差距离在特定时间点附近会发生从1到0的骤降截断现象。