Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of eigenvectors“, submitted to Random Matrices: Theory and Applications. This paper concerns an extension of our four moment theorem for eigenvalues. Roughly speaking, that four moment theorem asserts (under mild decay conditions on the coefficients of the random matrix) that the fine-scale structure of individual eigenvalues of a Wigner random matrix depend only on the first four moments of each of the entries.

In this paper, we extend this result from eigenvalues to eigenvectors, and specifically to the coefficients of, say, the ${i^{th}}$ eigenvector ${u_i(M_n)}$ of a Wigner random matrix ${M_n}$. Roughly speaking, the main result is that the distribution of these coefficients also only depends on the first four moments of each of the entries. In particular, as the distribution of coefficients eigenvectors of invariant ensembles such as GOE or GUE are known to be asymptotically gaussian real (in the GOE case) or gaussian complex (in the GUE case), the same asymptotic automatically holds for Wigner matrices whose coefficients match GOE or GUE to fourth order.

(A technical point here: strictly speaking, the eigenvectors ${u_i(M_n)}$ are only determined up to a phase, even when the eigenvalues are simple. So, to phrase the question properly, one has to perform some sort of normalisation, for instance by working with the coefficients of the spectral projection operators ${P_i(M_n) := u_i(M_n) u_i(M_n)^*}$ instead of the eigenvectors, or rotating each eigenvector by a random phase, or by fixing the first component of each eigenvector to be positive real. This is a fairly minor technical issue here, though, and will not be discussed further.)

This theorem strengthens a four moment theorem for eigenvectors recently established by Knowles and Yin (by a somewhat different method), in that the hypotheses are weaker (no level repulsion assumption is required, and the matrix entries only need to obey a finite moment condition rather than an exponential decay condition), and a slightly stronger conclusion (less regularity is needed on the test function, and one can handle the joint distribution of polynomially many coefficients, rather than boundedly many coefficients). On the other hand, the Knowles-Yin paper can also handle generalised Wigner ensembles in which the variances of the entries are allowed to fluctuate somewhat.

The method used here is a variation of that in our original paper (incorporating the subsequent improvements to extend the four moment theorem from the bulk to the edge, and to replace exponential decay by a finite moment condition). That method was ultimately based on the observation that if one swapped a single entry (and its adjoint) in a Wigner random matrix, then an individual eigenvalue ${\lambda_i(M_n)}$ would not fluctuate much as a consequence (as long as one had already truncated away the event of an unexpectedly small eigenvalue gap). The same analysis shows that the projection matrices ${P_i(M_n)}$ obeys the same stability property.

As an application of the eigenvalue four moment theorem, we establish a four moment theorem for the coefficients of resolvent matrices ${(\frac{1}{\sqrt{n}} M_n - zI)^{-1}}$, even when ${z}$ is on the real axis (though in that case we need to make a level repulsion hypothesis, which has been already verified in many important special cases and is likely to be true in general). This improves on an earlier four moment theorem for resolvents of Erdos, Yau, and Yin, which required ${z}$ to stay some distance away from the real axis (specifically, that ${\hbox{Im}(z) \geq n^{-1-c}}$ for some small ${c>0}$).