2021/07/31 by Fan Yang, Horng-Tzer Yau, Horng‐Tzer Yau +1 · 9 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Anomalous diffusion #Combinatorics #Delocalized electron #Density matrix renormalization group #Eigenvalues and eigenvectors #Hermitian matrix #Lattice (music) #Mathematical physics #Mathematics #Physics #Quantum mechanics #Quantum optics and atomic interactions #Random Matrices and Applications #Random matrix #Renormalization #Renormalization group #math.PR
paper · pdf · doi:10.1007/s00220-022-04474-y
published in Communications in Mathematical Physics 396(2), 527-622 (Springer Science+Business Media) · 74 pages. This is a sequel of arXiv:2104.12048. Accepted by Communications in Mathematical Physics
arxiv created 2022/07/08 · openalex publication_date 2022/08/17 · openalex created_date 2022/08/18 · arxiv updated 2022/11/23 · openalex updated_date 2026/08/01
We consider Green's functions G(z):=(H-z)-1 of Hermitian random band matrices H on the d-dimensional lattice (\mathbb Z/L\mathbb Z)d. The entries hxy= hyx of H are independent centered complex Gaussian random variables with variances sxy=\mathbb E|hxy|2. The variances satisfy a banded profile so that sxy is negligible if |x-y| exceeds the band width W. For any n∈ \mathbb N, we construct an expansion of the T-variable, Txy=|m|2 ∑αsxα|Gαy|2, with an error O(W-nd/2), and use it to prove a local law on the Green's function. This T-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions d≥ 8 in part I of this series.