UCLA

We consider a block random matrix model in which $D$ random systems on a circle, modeled by $N\times N$ Hermitian Wigner matrices, interact pairwise according to a deterministic matrix $A$. We measure the interaction strength using the Hilbert-Schmidt norm $\|A\|_{HS}$. Denoting the eigenvectors by $\mathbf{v}_k$ in increasing order of the eigenvalues, we prove two results on the localization and delocalization of the eigenvectors in the limit $N\to\infty$. First, if $\|A\|_{HS} \geq N^{1/3+\epsilon}k^{-1/3}$, where $k\leq DN/2$, the mass of the $k$th eigenvector is approximately equally distributed over the $D$ subsystems with probability $1 - o(1)$. On the other hand, if $k = O(1)$ and $\|A\|_{HS} \leq N^{1/6-\epsilon}k^{-1/6}$, we show that for Gaussian subsystems, the mass of the $k$th eigenvector is localized within a single subsystem with probability $1 - o(1)$. Similar results hold if $k > DN/2$. These results extend our previous work on localization and delocalization of eigenvectors in the bulk.