Sparse block matrices: local laws and eigenvector delocalization

Ioana Dumitriu
University of Washington

We prove a local law in the bulk for general Wigner-type matrices, down to polylog(n)/n interval length, and use it to show optimal eigenvector delocalization. We also use these to deduce local laws and delocalization for sparsified versions of the ensembles (down to an average of $\omega(\log n)$ entries per row), including the well-known stochastic block model. Finally, we show that in the case of block models, the empirical spectral distributions converge, even in certain cases when the number of blocks is unbounded. This is joint work with Yizhe Zhu.

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