Fluctuations in Quantum Unique Ergodicity at the Spectral Edge

Abstract

We study the eigenvector mass distribution of an $N\times N$ Wigner matrix on a set of coordinates $I$ satisfying $|I|\ge cN$ 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 $N\to \infty$ 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.

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