The noisy voter model with general initial conditions

We study the noisy voter model with q≥ 2 states and noise probability $θ$ on arbitrary bounded-degree n-vertex graphs G with subexponential growth of balls (e.g., finite subsets of Zᵈ). 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 tₙ=12θlog 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=Z/nZ. They showed that the ``alternating'' initial state is the fastest one if θ≥ 23, and conjectured that this holds for all values of the noise $θ$. Here we show that for every graph G as above and all $θ,q$ and initial states x₀, the noisy voter model exhibits cutoff at an explicit function of the autocorrelation of the model started at x₀. Consequently, for G=Z/nZ 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 $θ$. Analogous results hold in Zₙᵈ for $q=2$ and all d≥ 1 (``checkerboard'' initial conditions are fastest) as well as for $d=1$ and all q≥ 2 (``rainbow'' initial conditions are fastest).