Inhomogeneous circular law for correlated matrices
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Inhomogeneous circular law for correlated matrices. / Alt, Johannes; Krüger, Torben.
In: Journal of Functional Analysis, Vol. 281, No. 7, 109120, 2021.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Inhomogeneous circular law for correlated matrices
AU - Alt, Johannes
AU - Krüger, Torben
N1 - Publisher Copyright: © 2021
PY - 2021
Y1 - 2021
N2 - We consider non-Hermitian random matrices X∈Cn×n with general decaying correlations between their entries. For large n, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear n×n matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of X. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.
AB - We consider non-Hermitian random matrices X∈Cn×n with general decaying correlations between their entries. For large n, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear n×n matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of X. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.
KW - Brown measure
KW - Delocalisation
KW - Local law
KW - Non-Hermitian random matrix
U2 - 10.1016/j.jfa.2021.109120
DO - 10.1016/j.jfa.2021.109120
M3 - Journal article
AN - SCOPUS:85108151116
VL - 281
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 7
M1 - 109120
ER -
ID: 306674188