Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Understand

We study the eigenvector mass distribution of an $N\times N$ Wigner matrix on a set of coordinates $I$ satisfying $| I | \ge c N$ for some constant $c >0$.

• For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the sum of the mass on these coordinates converges to a Gaussian in the $N \rightarrow \infty$ limit, after a suitable rescaling and centering.
• The proof proceeds by a two moment matching argument.
• We directly compare edge eigenvector observables of an arbitrary Wigner matrix to those of a Gaussian matrix, which may be computed explicitly.