The joint statistical independence conjecture for Hecke–Maass forms and Eisenstein series

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms and let $\{\widetilde{E_{T_j}}\}_{j=1}^J$ be an orthogonal family of normalized real Eisenstein series with $|T_j-T_{j'}|\ge 1$ for $j\ne j'$, where

$$\widetilde{E_T}(z):=\sqrt{\frac{\operatorname{vol}(\mathbb{X})}{\log(\frac14+T^2)}}\frac{\xi(1+2iT)}{|\xi(1+2iT)|}E_T(z).$$

Let $C_n$ be the $n$th moment of a standard real Gaussian. Joint statistical independence conjecture. For positive integers $a_i,b_j$, the families $\{f_i^{a_i}\}_{i=1}^I$ and $\{\widetilde{E_{T_j}}\}_{j=1}^J$ are statistically independent: for every $\psi\in C_c^\infty(\mathbb{X})$,

$$\int_{\mathbb{X}}\psi(z)\prod_{i=1}^I f_i^{a_i}(z)\prod_{j=1}^J\widetilde{E_{T_j}}^{b_j}(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}\sim\prod_{i=1}^I C_{a_i}\prod_{j=1}^J C_{b_j}\int_{\mathbb{X}}\psi(z)\frac{\mathrm{d}x\,\mathrm{d}y}{y^2}$$

as $\min\{t_{f_1},\ldots,t_{f_I},T_1,\ldots,T_J\}$ tends to infinity. This combines the conjectural independent random-wave behavior of cusp forms and normalized Eisenstein series.