Spectral non-concentration near the top for unimodular random graphs

In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the top of the spectrum. We extend this result to infinite unimodular random graphs. As a corollary, the spectral distribution of the adjacency operator cannot have an atom at the top. For an infinite regular expander, we deduce that the singularity of the spectral measure at the top satisfies $\mu_G[(1-\theta)\rho,\rho] \lesssim \theta^c$ for some constant $c>0$, where $\rho$ is the spectral radius of the adjacency operator of the graph. This implies new general estimates on the return probabilities of random walks.