The spectrum of the random environment and localization of noise

We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized. [three figures omitted] The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.