Applied Mathematicsematics

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32). 37). 2 in the case β = 2 gives that the distribution of A is proportional to (det Y )(n1 −n2 ) (det(1n2 − Y ))(N −n1 −n2 ) . 13) gives the stated result. f. 30) in the case β = 2 by setting n1 = n2 = m, N = n + m and changing variables 1 − yj = λ2j . May 5, 2009 Master Review Vol. 9. Classical random matrix ensembles Let β = 1, 2 or 4 according to the elements being real, complex or quaternion real respectively. ’s derived above all have the general form 1 C N g(xl ) l=1 1≤j

Of Bn = 1/2 1/2 (1/N )Tn Xn Xn∗ Tn corresponding to limiting ratio d and limiting F Tn G. 2. ([2]). 1, Let f1 , . . , fr be C 1 functions on R with bounded derivatives, and analytic on an open interval containing √ √ n n I(0,1) (c)(1 − c)2 , lim sup λTmax (1 + c)2 ]. [lim inf λTmin n n May 5, 2009 Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS) 01-Silverstein 23 Stieltjes Transform and Random Matrices Let m = mFˆ . Then (1) the random vector f1 (x) dGn (x), . . 1) forms a tight sequence in n.

For scattering matrices in quantum mechanics, or more generally evo- May 5, 2009 Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS) 46 02-Forrester P. J. Forrester lution operators, time reversal symmetry requires that T −1 ST = S † . 5) D S = Z2N S T Z−1 2N =: S . 6). For random scattering matrices it is hypothesized that the statistical properties of S are determined soley by the global time reversal symmetry, and are invariant under the same conjugations as the corresponding Hamiltonians.