The noisy voter model with general initial conditions

Abstract

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).