Spectrum of Laplacian matrices associated with large random elliptic matrices
- Resource Type
- Working Paper
- Authors
- O'Rourke, Sean; Yin, Zhi; Zhong, Ping
- Source
- Subject
- Mathematics - Probability
Mathematics - Operator Algebras
- Language
A Laplacian matrix is a square matrix whose row sums are zero. We study the limiting eigenvalue distribution of a Laplacian matrix formed by taking a random elliptic matrix and subtracting the diagonal matrix containing its row sums. Under some mild assumptions, we show that the empirical spectral distribution of the Laplacian matrix converges to a deterministic probability distribution as the size of the matrix tends to infinity. The limiting measure can be interpreted as the Brown measure of the sum of an elliptic operator and a freely independent normal operator with a Gaussian distribution.
Comment: 42 pages; minor corrections, added additional references