Asymptotic expansions for the Gaussian unitary ensemble
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Asymptotic expansions for the Gaussian unitary ensemble. / Haagerup, Uffe; Thorbjørnsen, Steen.
I: Infinite Dimensional Analysis, Quantum Probability and Related Topics, Bind 15, 2012, s. 1250003.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Asymptotic expansions for the Gaussian unitary ensemble
AU - Haagerup, Uffe
AU - Thorbjørnsen, Steen
PY - 2012
Y1 - 2012
N2 - Let g : R ¿ C be a C8-function with all derivatives bounded and let trn denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value ¿{trn(g(Xn))} for a rather general class of random matrices Xn, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix Xn that where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients aj(g), j ¿ N, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Trn[f(Xn)], Trn[g(Xn)]}, where f is a function of the same kind as g, and Trn = n trn. Special focus is drawn to the case where and for ¿, µ in C\R. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the .
AB - Let g : R ¿ C be a C8-function with all derivatives bounded and let trn denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value ¿{trn(g(Xn))} for a rather general class of random matrices Xn, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix Xn that where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients aj(g), j ¿ N, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Trn[f(Xn)], Trn[g(Xn)]}, where f is a function of the same kind as g, and Trn = n trn. Special focus is drawn to the case where and for ¿, µ in C\R. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the .
M3 - Journal article
VL - 15
SP - 1250003
JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics
JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics
SN - 0219-0257
ER -
ID: 45181614