The glow of Fourier matrices: universality and fluctuations

The glow of an Hadamard matrix $H\in M_N(\mathbb C)$ is the probability measure $\mu\in\mathcal P(\mathbb C)$ describing the distribution of $\varphi(a,b)=<a,Hb>$, where $a,b\in\mathbb T^N$ are random. We prove that $\varphi/N$ becomes complex Gaussian with $N\to\infty$, and that the universality holds as well at order 2. In the case of a Fourier matrix, $F_G\in M_N(\mathbb C)$ with $|G|=N$, the universality holds up to order 4, and the fluctuations are encoded by certain subtle integrals, which appear in connection with several Hadamard-related questions. In the Walsh matrix case, $G=\mathbb Z_2^n$, we conjecture that the glow is polynomial in $N=2^n$.