In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature $\beta$ ranges from 0 to $\beta_0=\frac12\mathrm{arctanh}(\frac13)$. Moreover, the dominant test function depends on the temperature: at $\beta \beta_0$ it is the Hamiltonian.

中文翻译：

---

芒硝动力学的快速初始条件

在马尔可夫链混合时间的研究中，分析集中在最坏情况起始状态的性能上。在这里，在用于一维 Ising 模型的 Glauber 动力学的背景下，我们展示了如何使用来自信息渗透的新想法从其他起始状态建立混合时间。在高温下，我们表明交替初始条件是渐近最快的，并且令人惊讶的是，它的混合时间比无限温度下更快，随着逆温度 $\beta$ 范围从 0 到 $\beta_0=\frac12 而加速\mathrm{arctanh}(\frac13)$。此外，主要的测试函数取决于温度：在 $\beta \beta_0$ 处，它是哈密顿量。