标题： 带有一般初始条件的噪声选民模型 显示英文标题

标题： The noisy voter model with general initial conditions

摘要： 我们研究具有$q\geq 2$个状态和噪声概率$\theta$的噪声投票模型，在任意有界度的$n$顶点图$G$上，其球体的生长具有次指数性质（例如，$\mathbb{Z}^d$的有限子集）。Cox, Peres 和 Steif（2016）对于二元情况$q=2$（以及更广泛的链类）表明，当从最坏情况的初始状态开始时，这个马尔可夫链在$t_n=\frac1{2\theta}\log n$处具有总变分截止。 第二作者和Sly (2021) 分析了1D Ising模型Glauber动力学的更快初始条件，这是$q=2$和$G=\mathbb{Z}/n\mathbb{Z}$的噪声投票者。 他们证明了如果$\theta\geq \frac23$，“交替”初始状态是最快的，并推测这在所有噪声$\theta$的值下都成立。 此处我们证明，对于如上所述的每个图$G$以及所有$\theta,q$和初始状态$x_0$，噪声投票模型在以$x_0$开始的模型的自相关函数的显式函数处表现出截止现象。 因此，对于$G=\mathbb{Z}/n\mathbb{Z}$和$q=2$（1D Ising 模型的 Glauber 动力学），我们确认了 [LS21] 的猜想，即交替初始条件对于所有$\theta$来说在渐近意义上是最快速的。 类似的结论适用于$\mathbb{Z}_n^d$对于$q=2$和所有$d\geq 1$（“棋盘”初始条件最快）以及对于$d=1$和所有$q\geq 2$（“彩虹”初始条件最快）。 显示英文摘要

摘要： We study the noisy voter model with $q\geq 2$ states and noise probability $\theta$ on arbitrary bounded-degree $n$-vertex graphs $G$ with subexponential growth of balls (e.g., finite subsets of $\mathbb{Z}^d$). Cox, Peres and Steif (2016) showed for the binary case $q=2$ (and a wider class of chains) that, when starting from a worst-case initial state, this Markov chain has total variation cutoff at $t_n=\frac1{2\theta}\log n$. The second author and Sly (2021) analyzed faster initial conditions for Glauber dynamics for the 1D Ising model, which is the noisy voter for $q=2$ and $G=\mathbb{Z}/n\mathbb{Z}$. They showed that the ``alternating'' initial state is the fastest one if $\theta\geq \frac23$, and conjectured that this holds for all values of the noise $\theta$. Here we show that for every graph $G$ as above and all $\theta,q$ and initial states $x_0$, the noisy voter model exhibits cutoff at an explicit function of the autocorrelation of the model started at $x_0$. Consequently, for $G=\mathbb{Z}/n\mathbb{Z}$ and $q=2$ (Glauber dynamics for the 1D Ising model), we confirm the conjecture of [LS21] that the alternating initial condition is asymptotically fastest for all $\theta$. Analogous results hold in $\mathbb{Z}_n^d$ for $q=2$ and all $d\geq 1$ (``checkerboard'' initial conditions are fastest) as well as for $d=1$ and all $q\geq 2$ (``rainbow'' initial conditions are fastest).