Asymptotic expansions for the Gaussian unitary ensemble

TY - JOUR

T1 - Asymptotic expansions for the Gaussian unitary ensemble

AU - Haagerup, Uffe

AU - Thorbjørnsen, Steen

PY - 2012/3

Y1 - 2012/3

N2 - Let g : R → C be a C ∞-function with all derivatives bounded and let tr n denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr n(g(X n))} for a rather general class of random matrices X n, 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 GUE(n, 1\n)random matrix X n that (Chemical Equation Presented?) where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients α j(g), j ∈ N , as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr n[f(X n)], Tr n[g(X n)]}, where f is a function of the same kind as g, and Tr n = n tr n. Special focus is drawn to the case where g(x) = 1/4{1}{λ-x} and f(x) = 1/4{1}{μ-x} for λ, μ in . In this case the mean and covariance considered above correspond to, respectively, the one-and two-dimensional Cauchy (or Stieltjes) transform of the {rm GUE}(n, frac{1}{n}).

AB - Let g : R → C be a C ∞-function with all derivatives bounded and let tr n denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr n(g(X n))} for a rather general class of random matrices X n, 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 GUE(n, 1\n)random matrix X n that (Chemical Equation Presented?) where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients α j(g), j ∈ N , as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr n[f(X n)], Tr n[g(X n)]}, where f is a function of the same kind as g, and Tr n = n tr n. Special focus is drawn to the case where g(x) = 1/4{1}{λ-x} and f(x) = 1/4{1}{μ-x} for λ, μ in . In this case the mean and covariance considered above correspond to, respectively, the one-and two-dimensional Cauchy (or Stieltjes) transform of the {rm GUE}(n, frac{1}{n}).

M3 - Journal article

SN - 0219-0257

VL - 15

SP - 1250003

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

ER -