Feingold–Peres diagonal variance conjecture

Let $\Omega$ be a billiard domain and let $\hat{A}$ be multiplication by a test function $A$. Define the local diagonal variance $V_A(E)$ as the mean square of $\langle\phi_n,\hat{A}\phi_n\rangle-\overline{A}$ over an energy window, and let $\widetilde{C}_A(0)$ denote the zero-frequency classical autocorrelation transform of $A$. Feingold–Peres diagonal variance conjecture. For ergodic flow with no symmetries other than time reversal,

$$V_A(E)\sim\frac{g\widetilde{C}_A(0)}{\operatorname{vol}(\Omega)}E^{-1/2},$$

where the symmetry factor is $g=2$. This refines the power-law conjecture by predicting both the exponent and prefactor. It is a physics heuristic tested numerically in the paper, while arithmetic systems can have different prefactors because of additional symmetries.