The spectrum of the random environment and localization of noise (Q5961959)

Consider a random walk in a mildly random environment on finite transitive \(d\)-regular graphs of increasing girth (the length of the shortest cycle). It is shown that after scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting situation occurs at \(d=2\): as the limit graph changes from a regular tree to the integers, the noise becomes localized.