可逆扩散过程的不可逆提升及弛豫时间

Cutoff for random lifts of weighted graphs

We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a constant associated to $\mathcal{G}$, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all $n$-lifts of $\mathcal{G}$. In the particular case where the base graph is a vertex with $d/2$ loops, $d$ even, we obtain a cutoff for a $d$-regular random graph (as did Lubetzky and Sly in \cite{cutoffregular} with a …

驱动系统中的非对称热弛豫：旋转以相反方向发生

Asymmetric Thermal Relaxation in Driven Systems: Rotations go Opposite Ways

本文证明一般线性漂移驱动的非平衡系统仍存在热弛豫不对称，并发现升温和降温过程中旋转方向相反，说明含噪系统弛豫不必经过局部平衡态。

Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime

We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation dX_t=-\nabla F(X_t) dt+\sqrt2εdW_t,\qquad ε>0, and the spectrum near zero of its generator -L_ε\equiv εΔ-\nabla F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. For generic F to each local minimum of F there corresponds a metastable state. We prove that the distribution of its rescaled relaxation time converges to the exponential distribution as ε\downarrow 0 with optimal and uniform error estimates. Each metastable state can be vie…

松弛时间是大型集合的平稳命中时间

Relaxation times are stationary hitting times of large sets

本文通过大型集合的平稳期望命中时间刻画了松弛时间（精确到绝对常数因子），解决了 Aldous 和 Fill 的猜想，并推广到谱轮廓及非可逆情形。

Characterization of cutoff for reversible Markov chains

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least $α$, for some $α\in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $α$ was not hit by ti…

自 repellent 随机游走的收敛速率，其局部时间与事件链蒙特卡罗

Convergence rates of self-repellent random walks, their local time and Event Chain Monte Carlo

研究自 repellent 随机游走的收敛速率及其局部时间过程，证明其松弛时间下界为 $Ω(n^{3/2})$，并验证事件链蒙特卡罗算法在 MCMC 方法中的优势。

非可逆马尔可夫链的谱隙

Spectral gap of nonreversible Markov chains

本文以生成元第二小奇异值定义非可逆马尔可夫链的谱隙，证明其倒数可刻画经验平均收敛时间，并建立与 Cheeger 常数、混合时间及路径上界的联系。

非可逆马尔可夫链的非渐近混合时间分析

Non Asymptotic Mixing Time Analysis of Non-Reversible Markov Chains

本文提出了一个统一的算子理论框架，通过投影转移算子分析非可逆马尔可夫链的非渐近混合时间，并基于其谱特征建立了卡方散度界限，为动量采样器等非可逆链提供了精确的收敛分析。

Mixing time estimation in reversible Markov chains from a single sample path

The spectral gap $γ$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $γ$ from this data. Let $π$ be the stationary distribution of $P$, and $π_\star = \min_x π(x)$. We show that if $n = \tilde{O}\bigl(\frac{1}{γπ_\star}\bigr)$, then $γ$ can be estimated to within multiplicative constants with high probability. When $π$ is uniform on $d$ states, this matc…

定量谱刚性与可逆马尔可夫链中的有限时间谱热力学

Spectral Purification in Reversible Markov Chains: Hidden Parameters, Observable Equivalence, and Finite-Time Rigidity

本文研究了可逆马尔可夫链中有限时间谱刚性的本质，通过精确的谱弛豫动力学揭示了谱刚性的时间结构，并建立了谱分离比控制谱刚性出现的理论框架，同时提出了基于谱熵理论的精确误差身份和自适应停止准则。

Mixing and relaxation time for Random Walk on Wreath Product Graphs

Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z…