math.PR — Probability

Mobility Edge for Sparse Complex Gaussian Matrices

Contributed by Patrick Lopatto

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 $\sqrt\pi/2$ as $p$ tends to infinity. Thus the mobility edge lies far in the tail of the spectrum, rather than near the edge $2$ of the limiting semicircle law, as had been predicted in the physics literature.