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In the framework of the random matrix approach, we apply the theory of Selberg's integral to problems of quantum transport in chaotic cavities.
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∫ 0 1 d T 1 ⋯ ∫ 0 1 d T n ∏ j < k | T j − T k | 2 c ∏ i = 1 n T i a − 1 ( 1 − T i ) b − 1 = ∏ j = 0 n − 1 Γ ( 1 + c + j c ) Γ ( a + j c ) Γ ( b + j c ) Γ ( 1 + c ) Γ ( a + b + ( n + j − 1 ) c ) \int_{0}^{1}\!\mathrm{d}T_{1}\!\cdots\!\int_{0}^{1}\!\mathrm{d}T_{n}\prod_{j<k}|T_{j}-T_{k}|^{2c}\prod_{i=1}^{n}T_{i}^{a-1}(1-T_{i})^{b-1}=\prod_{j=0}^{n-1}\frac{\Gamma(1+c+jc)\Gamma(a+jc)\Gamma(b+jc)}{\Gamma(1+c)\Gamma(a+b+(n+j-1)c)} , where Γ ( x ) \Gamma(x) is the gamma function, gives the definition of Selberg’s integral which is valid at integer n ≥ 1 n\geq 1 , complex a a and b b with positive real parts, and complex c c with Re c > − min [ 1 / n \mathrm{Re\,}c>-\mathrm{min}[1/n , Re a / ( n − 1 ) \mathrm{Re\,}a/(n-1) , Re b / ( n − 1 ) ] \mathrm{Re\,}b/(n-1)] . Chapter 17 of Ref. 3 contains an elementary introduction into the Selberg’s integral theory and a derivation of some useful relations. For a current status of this field, see a recent review by P. J. Forrester and S. Ole Warnaar, arXiv: 0710.3981 [math.CA]
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This follows from the symmetry of the integral kernel ( 2
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W. Wieczorek, D. V. Savin, and H.-J. Sommers, in preparation
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2007
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