Quantitative dynamical exclusion-sensitivity threshold conjecture for the voter model

Let $(G_n)_{n\in\mathbb N}$ be a sequence of connected graphs, with $n$ vertices, edge set $E_n$, and maximum degree $\Delta_n$. Let $t_n$ be a non-negative sequence of times, let $f_n(\eta_0,\omega)$ denote the consensus opinion of the voter model, and let $\omega_{t_n}$ denote the dynamically exclusion-perturbed configuration at time $t_n$. Quantitative dynamical exclusion-sensitivity threshold conjecture.

$$\lim_{n\to\infty}\frac{t_n n\Delta_n}{|E_n|}=0 \quad\Longrightarrow\quad \liminf_{n\to\infty}\operatorname{Cov}\big(f_n(\eta_0,\omega),f_n(\eta_0,\omega_{t_n})\big)>0,$$

and

$$\lim_{n\to\infty}\frac{t_n n\Delta_n}{|E_n|}=+\infty \quad\Longrightarrow\quad \liminf_{n\to\infty}\operatorname{Cov}\big(f_n(\eta_0,\omega),f_n(\eta_0,\omega_{t_n})\big)=0.$$

The first implication is already given by the preceding exclusion-stability corollary, while the second would provide a quantitative description of the conjectured sensitivity regime. The intermediate-scale behavior is not specified.