Mobility Edge for Sparse Complex Gaussian Matrices
We consider the sparse Hermitian random matrix supported on the Erdős–Rényi graph $G(n,p/n)$, whose nonzero entries are independent complex Gaussian variables of variance $1/p$, together with its local limit, a weighted Poisson–Galton–Watson tree. We prove that for every fixed and sufficiently large $p$, the limiting operator on the tree conditioned on survival has exactly one positive mobility edge $E_c(p)$: almost surely, its spectrum is purely absolutely continuous on $(-E_c(p),0)\cup(0,E_c(p))$ and pure point outside $[-E_c(p),E_c(p)]$. Moreover, $E_c(p)/\sqrt p$ converges to…
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