The noisy voter model with general initial conditions

TL;DR

This paper investigates the mixing times and cutoff phenomena of the noisy voter model with multiple states on various graphs, confirming conjectures about optimal initial conditions and providing explicit cutoff criteria based on autocorrelation.

Contribution

It establishes a general cutoff criterion for the noisy voter model on arbitrary graphs and confirms conjectures about fastest initial conditions in specific cases.

Findings

The noisy voter model exhibits cutoff at a time depending on initial autocorrelation.

Alternating initial states are fastest for the 1D Ising model on cyclic graphs.

Checkerboard and rainbow initial conditions are fastest in higher dimensions and for multiple states.

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).