The inverse-dimension conjecture for average resistance of toroidal grids

Let $T_{M^d}$ be the $d$-dimensional toroidal grid with all side lengths equal to $M$, and let $R_{\rm ave}(T_{M^d})$ denote its average effective resistance. More generally, let $T_{M_1,\ldots,M_d}$ be a toroidal grid with side lengths $M_1,\ldots,M_d$ and total size $N=M_1\times\cdots\times M_d$. Inverse-dimension conjecture. For fixed $M$, the average effective resistance satisfies

$$R_{\rm ave}(T_{M^d})=\Theta\left(\frac{1}{d}\right)\quad\text{for }d\to+\infty.$$

Equivalently, for grids with $d\geq 3$ and equal side lengths, the average effective resistance is inversely proportional to the dimension, with the order $1/d$ independent of the finite side length. The paper proves the corresponding order after the side lengths tend to infinity, and numerical experiments support the conjectured finite-side-length statement; the general finite-$M$ claim remains open.