limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Maybe relevant:

"Lifshitz tails for spectra of Erdős–Rényi random graphs"

Oleksiy Khorunzhiy, Werner Kirsch, and Peter Müller

"We consider the discrete Laplace operator (the graph Laplacian) on Erdos–Rényi random graphs and show in Theorem 2.5 that the asymptotic behavior of its limiting integrated density of states at the lower spectral edge is given by a Lifshitz tail with Lifshitz exponent 1/2."

As far as I know, the spectra of Laplacian of a graph follows a distribution that is the free additive convolution of a Gaussian and the semicircle law.

Take a look, for example at the article:

"Spectra of random graphs with arbitrary expected degrees" Raj Rao Nadakuditi, M. E. J. Newman

https://arxiv.org/abs/1208.1275

In your case the arbitrary expected degree is constant, therefore the computation should result simpler than in the more general case.