The long-range conductance IIP conjecture

Let $d\ge2$ and let $p,q>d/2$ satisfy $1/p+1/q<2/d$. Consider symmetric long-range conductance configurations $\{c(x,y)=c(y,x):x,y\in\mathbb Z^d\}$ that are averaging and ergodic in the sense of the paper. Assume

$$\sup_{n\ge1}\frac1{|\Lambda_n|}\sum_{x\in\Lambda_n}\left(\sum_{y\in\mathbb Z^d}c(x,y)|y-x|^2\right)^p<\infty$$

and

$$\sup_{n\ge1}\frac1{|\Lambda_n|}\sum_{(x,y)\in E(\Lambda_n)}c(x,y)^{-q}<\infty.$$

Here $E(\Lambda)$ is the set of nearest-neighbor edges incident with $\Lambda$, and nearest-neighbor conductances are required to be strictly positive. The long-range conductance IIP conjecture. An IIP holds for the random walk among these long-range conductance configurations. The setting includes long-range percolation with zero-density modifications and spatially inhomogeneous truncations, and may help address the lack of everywhere sublinearity of the corrector.