Local Kesten–McKay Law for Random Regular Graphs

Abstract

We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum [−2down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten-McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs.

$$L^p$$ Norms and Support of Eigenfunctions on Graphs

“…This very simple result somehow complements the bounds ψ λ ∞ log |G| |G| 1/2 proved for random graphs in [14] for λ in the bulk of the spectrum (this said, (1.4) is of course much easier to prove). In our case, we know we can still take ℓ G = 1…”

“…• The fact that eigenfunction delocalization depends on the behavior of the Green function was not apparent in [21], but is actually well-known, see [24,14,17,9] to name a few references. For example, note that if ψ j is an eigenfunction of H G on G with corresponding eigenvalue λ j , then for any η > 0,…”

“…If |G| = N and if one can take η ≈ 1 N , and guarantee that | Im g λ+ 1 N (x, x)| stays bounded, this implies that |ψ j (x)| 2 1 N , which is the ideal delocalization one can hope to achieve. This idea is implemented in [14] in the framework of random regular graphs, so that most of the proof is devoted to controlling the Green function g γ and show that it is close to the Green function G γ of a tree-like graph, which is well-behaved.…”

“…• Our upper bound is in terms of ℓ G . It is natural to ask if we can go beyond this and prove strong delocalization bounds as in [14]. In our framework of deterministic graphs, one cannot hope for an upper bound involving |G| directly.…”

“…All the preceding results apply to deterministic graphs. In case of random regular graphs of high degree, much better delocalization properties were obtained in [14]. It is shown essentially that with high probability, all the ideal properties of N -cycles remain true, modulo logarithmic corrections.…”