Universal entrywise eigenvector fluctuations in delocalized spiked matrix models and asymptotics of rounded spectral algorithms

We consider the distribution of the top eigenvector $\widehat{v}$ of a spiked matrix model of the form $H = θvv^* + W$, in the supercritical regime where $H$ has an outlier eigenvalue of comparable magnitude to $\|W\|$. We show that, if $v$ is sufficiently delocalized, then the distribution of the individual entries of $\widehat{v}$ (not, we emphasize, merely the inner product $\langle \widehat{v}, v\rangle$) is universal over a large class of generalized Wigner matrices $W$ having independent entries, depending only on the first two moments of the distributions of the entries of $W$. This complements the observation of Capitaine and Donati-Martin (2018) that these distributions are not universal when $v$ is instead sufficiently localized. Further, for $W$ having entrywise variances close to constant and thus resembling a Wigner matrix, we show by comparing to the case of $W$ drawn from the Gaussian orthogonal or unitary ensembles that averages of entrywise functions of $\widehat{v}$ behave as they would if $\widehat{v}$ had Gaussian fluctuations around a suitable multiple of $v$. We apply these results to study spectral algorithms followed by rounding procedures in dense stochastic block models and synchronization problems over the cyclic and circle groups, obtaining the first precise asymptotic characterizations of the error rates of such algorithms.