Asymptotic properties of large random matrices with independent entries

Journal Article · Tue Oct 01 04:00:00 EDT 1996 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.531589· OSTI ID:434702

Khorunzhy, A M ^[1]; Khoruzhenko, B A ^[2]; Pastur, L A ^[1]

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1. B. I. Verkin Institute for Low Temperature Physics, Kharkov, 310164 (Ukraine)
2. Queen Mary & Westfield College, University of London, London, E1 4NS (United Kingdom)

We study the normalized trace {ital g}{sub {ital n}}({ital z})={ital n}{sup {minus}1}tr({ital H}{minus}{ital zI}){sup {minus}1} of the resolvent of {ital n}{times}{ital n} real symmetric matrices {ital H}=[(1+{delta}{sub {ital jk}}){ital W}{sub {ital jk}}{radical}{ital n}]{sub {ital j},{ital k}=1}{sup {ital n}} assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of {ital g}{sub {ital n}}({ital z}) for {vert_bar} {ital Iz}{vert_bar}{ge}{eta}{sub 0} where {eta}{sub 0} is determined by the second moment of {ital W}{sub {ital jk}}. By using this method we find the asymptotic form of the expectation {bold E}{l_brace}{ital g}{sub {ital n}}({ital z}){r_brace} and of the connected correlator {bold E}{l_brace}{ital g}{sub {ital n}}({ital z}{sub 1}){ital g}{sub {ital n}}({ital z}{sub 2}){r_brace}{minus}{bold E}{l_brace}{ital g}{sub {ital n}}({ital z}{sub 1}){r_brace}{bold E}{l_brace}{ital g}{sub {ital n}} ({ital z}{sub 2}){r_brace}. We also prove that the centralized trace {ital ng}{sub {ital n}}({ital z}){minus}{bold E}{l_brace}{ital ng}{sub {ital n}}({ital z}){r_brace} has the Gaussian distribution in the limit {ital n}={infinity}. Based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles. {copyright} {ital 1996 American Institute of Physics.}

• OSTI ID: 434702
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 37; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ASYMPTOTIC SOLUTIONS
CORRELATION FUNCTIONS
EIGENVALUES
EXPECTATION VALUE
GAUSSIAN PROCESSES
MATRICES
STOCHASTIC PROCESSES