Noise sensitivity of crossing events under nearest-neighbor dynamics

Let $\{\eta_t\}$ denote the nearest-neighbor dynamics on the triangular lattice at critical probability $p_c=1/2$, and let $f_n$ be the Boolean function considered in the paper. Let $\alpha_4(n)$ denote the critical four-arm probability to distance $n$. The conjectured statement is about time scales $t_n$ satisfying

$$t_n \gg (n^2\alpha_4(n))^{-1} \approx n^{-3/4}.$$

Nearest-neighbor noise-sensitivity conjecture. If $t_n \gg (n^2\alpha_4(n))^{-1} \approx n^{-3/4}$, then

$$\lim_{n\to\infty}\mathbb{E}[f_n(\eta_0)f_n(\eta_{t_n})]-\mathbb{E}[f_n(\eta_0)]^2=0.$$

This predicts decorrelation of the Boolean observable under nearest-neighbor dynamics once the time exceeds the inverse pivotality scale. The source presents it as a result that should hold; no resolution is supplied here.