Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where $A$ is the adjacency matrix of the graph and $D$ is diagonal with $D\_{ii} = $ degree of node $i$). Let finally $0 = \lambda\_0 \leq \lambda\_1 \leq \lambda\_2 \leq \cdots$ be the eigenvalues of $L$. Is an analytic expression known for the limiting empirical distribution of these eigenvalues, i.e. for


$\lim\_{n \to \infty} \frac{1}{n} \#\{ 1 \leq j \leq n : \lambda\_j \leq t \}, t \in \mathbb{R}?$



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."


