Limiting empirical spectral distribution for the non-backtracking matrix of an Erdos-Renyi random graph

In this note, we give a precise description of the limiting empirical spectral distribution for the non-backtracking matrices for an Erdos-Renyi graph G(n, p) assuming np / log n tends to infinity. We show that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably, and then, we use Tao and Vu's replacement principle and the Bauer-Fike theorem to show that the partly derandomized spectrum is, in fact, very close to the original spectrum.