Quantitative convergence conjecture for Green's functions on the percolation cluster

Let $d$ be the dimension, let $obreak\mathfrak{p}$ and $obreak\lambda$ be the model parameters, and let $p(t,x,y)$ and $g(x,y)$ denote the parabolic and elliptic Green's functions on the infinite percolation cluster $obreak\mathscr{C}_\infty$. Let $\theta(\mathfrak{p})$ be the density of the infinite cluster, and let $\overline p$ and $\overline g$ be the corresponding homogenized Green's functions. For stochastic integrability, write $\mathcal{O}_s$ as in the source. Fix $s \in \left(0,\frac{2(d-1)}{d}\right)$.

Quantitative Green's-function convergence conjecture. There exists a positive constant $C<\infty$, depending on $d$, $\mathfrak{p}$, $\lambda$ and $s$, such that for every $t>0$ and every $x,y\in\mathbb{Z}^d$ with $|x-y|\leq t$, conditionally on $\{x,y\in\mathscr{C}_\infty\}$,

$$\left|p(t,x,y)-\theta(\mathfrak{p})^{-1}\overline p(t,x-y)\right|\leq\begin{cases}\mathcal{O}_s\left(Ct^{-\frac d2-\frac12}\exp\left(-\frac{|x-y|^2}{Ct}\right)\right),&d\geq3,\\ \mathcal{O}_s\left(C\log^{\frac12}(1+t)t^{-\frac32}\exp\left(-\frac{|x-y|^2}{Ct}\right)\right),&d=2.\end{cases}$$

For the elliptic Green's function, in dimension $d\geq3$, for every $x,y\in\mathbb{Z}^d$, conditionally on $\{x,y\in\mathscr{C}_\infty\}$,

$$\left|g(x,y)-\overline g(x-y)\right|\leq\mathcal{O}_s\left(C|x-y|^{1-d}\right).$$

In dimension $2$, for every $y\in\mathbb{Z}^2$, conditionally on $\{y\in\mathscr{C}_\infty\}$, the limit

$$K(y):=\lim_{x\to\infty}\bigl(g(x,y)-\overline g(x-y)\bigr)$$

exists, is finite almost surely, and satisfies $|K(y)|\leq\mathcal{O}_s(C)$. Moreover, for every $x\in\mathbb{Z}^2$, conditionally on $\{x,y\in\mathscr{C}_\infty\}$,

$$\left|g(x,y)-\overline g(x-y)-K(y)\right|\leq\mathcal{O}_s\left(C\log^{\frac12}(1+|x-y|)|x-y|^{-1}\right).$$

The conjecture predicts optimal quantitative homogenization rates for both parabolic and elliptic Green's functions in degenerate percolation environments; the paper explicitly expects these rates to improve the exponents obtained in its theorems. The parser supplies no resolution evidence, so the conjecture is recorded as open.