Skeletons and Spectra: Bernoulli graphings are relatively Ramanujan
Héctor Jardón-Sánchez, László Márton Tóth
TL;DR
The paper develops a spectral framework for unimodular random graphs and their Bernoulli graphings, introducing a random/structured decomposition of the spectrum and a skeleton Markov chain that captures the structured part. It proves that Bernoulli graphings are Ramanujan relative to their skeleton, i.e., the random part of the spectrum adheres to the Alon–Boppana bound, and shows spectrum containment $\sigma(G,o)\subset\sigma_R(\mathcal{G})$; it also proves an equality result for unimodular quasi-transitive quasi-trees by leveraging recent random-lifts results. By connecting to finite random lifts (Friedman-type results and Bordenave–Collins), the work provides a finite-to-infinite strengthening and clarifies how finite eigenvalue phenomena translate to measurable graphings. The findings illuminate obstructions to relative Ramanujan behavior (Frączyk’s example) and identify conditions under which full equality holds, offering a robust bridge between finite graph theory and measurable limit objects with potential to inform random-graph models and limit constructions.
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 Frączyk 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.