Mesoscopic Linear Statistics for Two Ensembles of Quantum Graphs

We study mesoscopic linear spectral statistics for two ensembles of random quantum graphs.

These are defined by a discrete graph $G$ and a unitary-matrix-valued function $U(k)$ indexed by directed edges of $G$.

The matrix function $U(k)$ is constructed from unitary matrices $U^{(v)}$ indexed by the neighbours of each vertex $v$.

The first ensemble is obtained by sampling the underlying discrete graph uniformly from the set of $d$-regular graphs.

The second ensemble is obtained by sampling $U^{(v)}$ uniformly from the Haar measure, independently for each vertex.

We prove that the variance of a linear spectral statistic in the large graph limit on polynomial mesoscopic scales coincides with that of the Gaussian Orthogonal/Unitary Ensemble.