The Hierarchical Laplacian, proposed in Dyson's theory of one-dimensional ferromagnetic phase transitions, has a discrete spectrum with each isolated eigenvalue having infinite multiplicity. As a result, the integrated density of states is piecewise constant and the density of states is a sum of point-masses located on its spectrum. To correct these "defects," we modify the Hierarchical Laplacian by allowing its deterministic coefficients to instead vary randomly, but without changing the eigenfunctions. The resulting spectrum is deterministic but the eigenvalues are now random with finite multiplicity and we obtain an absolutely continuous density of states. Examining the eigenvalue statistics near an individual point of the spectrum, we find that, locally, the spectrum is approximately a Poisson point process.
