Conjecture on the limiting distribution of matrix elements for the quantum cat map

Let $A\in SL(2,\mathbf Z)$ be a linear hyperbolic map of the torus, let $N$ tend to infinity through primes, and let $\{\psi_j\}_{j=1}^N$ be a joint eigenbasis of the Hecke operators and the quantum map $U_N(A)$ on $\mathcal H_N=L^2(\mathbf Z/N\mathbf Z)$. For a smooth real-valued observable $f\in C^\infty(\mathbf T^2)$, define

$$F_j^{(N)}:=\sqrt{N}\left(\langle \operatorname{Op}_N(f)\psi_j,\psi_j\rangle-\int_{\mathbf T^2}f(x)\,dx\right).$$

Writing $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, set

$$Q(x,y)=cx^2+(d-a)xy-by^2$$

and, for an integer $\nu$, define

$$f^\#(\nu):=\sum_{\substack{n=(n_1,n_2)\in\mathbf Z^2\Q(n)=\nu}}(-1)^{n_1n_2}\widehat f(n).$$

Limiting-distribution conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is that of the random variable

$$X_f:=\sum_{\nu\neq 0}f^\#(\nu)\operatorname{tr}(U_\nu),$$

where the $U_\nu$ are independently chosen random matrices in $SU(2)$ endowed with Haar probability measure.

This conjecture predicts that the fluctuations of the Hecke-eigenfunction matrix elements for the quantum cat map are generally non-Gaussian, in contrast with the expected Gaussian fluctuation picture for general quantum systems. The source presents it as the main conjectural description; its resolution is not established in the supplied text.