Asymptotic properties of large random matrices with independent entries
- B. I. Verkin Institute for Low Temperature Physics, Kharkov, 310164 (Ukraine)
- 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, Vol. 37, Issue 10; Other Information: PBD: Oct 1996
- Country of Publication:
- United States
- Language:
- English
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