Edge statistics for random band matrices

Abstract

We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z}{)}^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to \infty$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d/6{)}_{+}}$. In the Hermitian case, we establish the Anderson transition for all dimensions $d<4$, and GUE edge universality when $d\ge 4$ under the condition $W\ge L^{1/3+\hat{I}\mu }$ for any $\hat{I}\mu >0$. In the symmetric case, we also establish parallel but more subtle transition phenomena after tadpole diagram renormalization. These findings extend Sodin's pioneering work [Ann. Math. 172, 2010], which was limited to the one-dimensional case and did not address the critical phenomena.

Page 88, figures 14, Added Section 5 on tadpole diagram renormalization enables our results as better for symmetric case to as for Hermitian; We remove power-law band matrices that will be treated in a separate paper