Localization-delocalization transition for a random block matrix model at the edge

Consider a random block matrix model consisting of $D$ random systems arranged along a circle, where each system is modeled by an independent $N\times N$ complex Hermitian Wigner matrix. Neighboring systems interact via an arbitrary deterministic $N\times N$ matrix $A$. In this paper, we extend the localization-delocalization transition previously established in arxiv:2312.07297 for the bulk eigenvalue spectrum to the entire spectrum, including the spectral edges. Let $[E^-,E^+]$ denote the support of the limiting spectral density, and define $\kappa_E:=|E-E^+|\wedge |E-E^-|$ as the distance from a given energy $E \in [E^-, E^+]$ to the spectral edges. We show that for eigenvalues near $E$, the corresponding eigenvectors undergo a localization-delocalization transition when $\|A\|_{\mathrm{HS}}$ crosses the critical threshold $(\kappa_E + N^{-2/3})^{-1/2}$. In the delocalized phase, the extreme eigenvalues asymptotically follow the Tracy-Widom distribution, while in the localized phase, the edge eigenvalue statistics asymptotically match those of $D$ independent GUE ensembles, up to a deterministic shift. Our results recover those of arxiv:2312.07297 in the bulk regime, where $\kappa_E \asymp 1$, and further reveal the presence of mobility edges near $E^\pm$ when $1 \ll \|A\|_{\mathrm{HS}} \ll N^{1/3}$. Specifically, bulk eigenvectors corresponding to energies $E$ with $\kappa_E \gg \|A\|_{\mathrm{HS}}^{-2}$ are delocalized, while those with $\kappa_E \ll \|A\|_{\mathrm{HS}}^{-2}$ are localized.