If G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of ...

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix orthogonal matrix eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing stochastic eigenstate equation (SEE) (Bourgade and Yau Comm Math Phys, 2013) which describes Lie group valued flow induced by Brownian m...