Jiaoyang Huang

Eigenvector statistics of sparse random matrices

We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of Erdős-Renyi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combining with an isotropic local law for Greens function. As an auxiliary result, we prove that for the eigenvector flow of Dyson Brownian motion with general initial data, the eigenvectors are asymptotically jointly normal in the direction

Mesoscopic perturbations of large random matrices

q

Asymptotic Expansion of Spherical Integral

after time

Bulk eigenvalue statistics for random regular graphs

\eta_*\ll t\ll r

Bulk universality of sparse random matrices

, if in a window of size

Local Kesten--McKay law for random regular graphs

r

Laurent phenomenon sequences

, the initial density of states is bounded below and above down to the scale

Local law and mesoscopic fluctuations of Dyson Brownian motion for general

\eta_*

\beta

, and the initial eigenvectors are delocalized in the direction

and potential

q