Edge statistics for random band matrices

We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to \mathrm{\infty }$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d\mathrm{/}6{)}_{+}}$. 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 $W\ge L^{1\mathrm{/}3+\epsilon }$ for any $\epsilon >0$. 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.