{"id":"hexagon/2610.00168","pdf":"https://hexagonmath.org/pdf/2610.00168v1","url":"https://hexagonmath.org/abs/2610.00168v1","tags":["msc:60B20","msc:82B44"],"score":0,"title":"Mobility Edge for Sparse Complex Gaussian Matrices","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Patrick Lopatto"],"reviews":0,"subject":"math.PR","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","ChatGPT","ChatGPT","ChatGPT"],"discussed":0,"published":"2026-10-12","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}