The paper calculates how a broad, mid-scale spectral statistic behaves for two kinds of random quantum graphs. The authors study a linear spectral statistic, which is a sum of a test function applied to the graph’s eigenvalues on a mesoscopic scale — a scale bigger than the typical spacing of eigenvalues but smaller than the whole spectrum. They prove that, in the large-graph limit and on polynomial mesoscopic windows, the variance of this statistic agrees with the variance predicted by the Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE) from random matrix theory.

The work treats two models. In Model 1 the underlying discrete graph is a random d-regular graph on N vertices, the edge lengths are independent samples from a compactly supported measure, and each vertex is given a fixed symmetric “equi‑transmitting” unitary matrix. In Model 2 the discrete graph is the complete graph on N+1 vertices and the vertex unitary matrices are chosen independently from the Haar (uniform) measure on the unitary group; edge lengths are again included. For a smooth test function h and a smoothing window of size η, the authors study Lλ,η,h, the sum of h evaluated at the graph’s eigenvalues after rescaling by λ and η.

Their main theorems give the leading-order variance and error bounds. For Model 1 the variance equals 4 ∫0^∞ x |ĥ(x)|^2 dx plus an error term that vanishes when N is large and η is in the allowed mesoscopic range. For Model 2 the leading term is 2 ∫0^∞ x |ĥ(x)|^2 dx, again with control on the remainder. These leading integrals are precisely the mesoscopic variances of the GOE (factor 4) and GUE (factor 2), so the new results show agreement with random-matrix predictions for these mesoscopic linear statistics.

The proofs start from an exact trace formula that rewrites spectral sums in terms of sums over closed walks on the discrete graph. Because the Fourier transform of the test function has compact support, the sums truncate at walk lengths about η^{-1}. In the random-graph model the authors use combinatorial counting of closed walks and the configuration model to show that most relevant walks are simple cycles when η^{-1} is much smaller than N^{1/2}. In the Haar‑unitary model they average over the random vertex unitaries and use bounds on edge lengths and Cesàro convergence assumptions to control nonleading contributions.