In this talk, I will show results on the spectral properties of an infinitely renormalizable random network model. This model is an 'anti-benchmark’ for Random Matrix Theory techniques because many working assumptions fail, such as moment matching methods, the low rank hypothesis for the signal matrix, and the non-self-averaging nature of the ESD. Despite these challenges, I will show an ansatz for the eigenfunctions based on the Laplace transform of the weights distribution. This allows to solve the eigenvalue equation for the signal matrix, which is a good proxy for the largest eigenmodes of the network realisations. The leading eigenvalues are all of order √n, alternate in sign, and lie at the intersection between the real axis and a logarithmic spiral in the complex plane. I will also show haow to calculate the associated eigenvectors, which display complex-valued scaling exponents and log-periodicity, indicating discrete scale invariance.