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Mobility Edge for Sparse Complex Gaussian Matrices

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Patrick Lopatto ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

We consider the sparse Hermitian random matrix supported on the Erdős–Rényi graph G(n,p/n)G(n,p/n), whose nonzero entries are independent complex Gaussian variables of variance 1/p1/p, together with its local limit, a weighted Poisson–Galton–Watson tree. We prove that for every fixed and sufficiently large pp, the limiting operator on the tree conditioned on survival has exactly one positive mobility edge Ec(p)E_c(p): almost surely, its spectrum is purely absolutely continuous on (−Ec(p),0)∪(0,Ec(p))(-E_c(p),0)\cup(0,E_c(p)) and pure point outside [−Ec(p),Ec(p)][-E_c(p),E_c(p)]. Moreover, Ec(p)/pE_c(p)/\sqrt p converges to π/2\sqrt\pi/2 as pp tends to infinity. Thus the mobility edge lies far in the tail of the spectrum, rather than near the edge 22 of the limiting semicircle law, as had been predicted in the physics literature.

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  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 08, 2026