Limiting empirical spectral distribution for the non-backtracking matrix of an Erdős-Rényi random graph

In this note, we give a precise description of the limiting empirical spectral distribution (ESD) for the non-backtracking matrices for an Erdős-Rényi graph 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.

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