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Random matrices: Universality of local spectral statistics of non-Hermitian matrices
Random matrices non-Hermitian matrices Probability
2012/6/30
It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n \times n$ gaussian matrix with independent entries of mean zero and unit variance are asymp...
Lehmer's conjecture for Hermitian matrices over the Eisenstein and Gaussian integers
Lehmer's conjecture Hermitian matrices Eisenstein and Gaussian integers Number Theory
2012/6/29
We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least ...
Spectra of Random Hermitian Matrices with a Small-Rank External Source: The critical and near-critical regimes
Spectra of Random Hermitian Matrices Small-Rank External Source
2010/11/24
Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, whi...
Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
Spectra of random Hermitian matrices small-rank external source: supercritical subcritical regimes
2010/12/9
Random Hermitian matrices with a source term arise, for instance, in the study of non-ntersecting Brownian walkers [1, 20] and sample covariance matrices [4]. We consider the case when the n × n exter...
Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution
division ring with involution hermitian matrix adjacency geometry of matrices
2007/12/11
Let $D$ be any division ring with an involution, ${\mathscr H}_n(D)$ be the space of all $n\times n$ hermitian matrices over $D$. Two hermitian matrices $A$ and $B$ are said to be adjacent if ${\rm ra...