For inhomogeneous random graphs with a graphon variance profile, the traces of powers of the adjacency matrix, suitably rescaled, converge to Gaussian processes with covariances expressible as graphon homomorphism densities.
Mesoscopic Spectral CLT for Block Correlated Random Matrices
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For random matrices with block correlation structure we show that the fluctuations of linear eigenvalue statistics are Gaussian on all mesoscopic scales with universal variance which coincides with that of the Gaussian unitary or Gaussian orthogonal ensemble, depending on the symmetry class of the model. The main tool used for determining this variance is a two-point version of the matrix-valued Dyson equation, that encodes the asymptotic behavior of the product of resolvents at different spectral parameters.
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Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits
For inhomogeneous random graphs with a graphon variance profile, the traces of powers of the adjacency matrix, suitably rescaled, converge to Gaussian processes with covariances expressible as graphon homomorphism densities.