Skeletons and Spectra: Bernoulli graphings are relatively Ramanujan

Abstract: The aim of this paper is to investigate the spectral theory of unimodular random graphs and graphings representing them. We prove that Bernoulli graphings are relatively Ramanujan with respect to their skeleton Markov chain. That is, the part of their spectrum that comes from the random labels falls within the appropriate Alon-Boppana bound. This result complements an example due to Fr\k{a}czyk of an ergodic unimodular random graph with almost sure spectral gap but non-expanding Bernoulli graphing. We also highlight connections of our work with the theory of finite random graphs. Exploiting the result of Bordenave and Collins on random lifts being relatively almost Ramanujan, we prove a strengthening of our main theorem for unimodular quasi-transitive quasi-trees.