  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.
