Abstract
We consider the 1/N-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting N scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices β = 1 and β = 2), we obtain a recursion relation, which efficiently generates the coefficients of the 1/N-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.
Highlights
Formula (5a) behaves better than (5b); the number of terms in the sum is unbounded for large N, it is a sum of positive terms and the only asymptotic analysis one needs is the complete asymptotic series for the ratio of two Gamma functions
We considered the average of power traces E TrQk of the time-delay matrix for ballistic chaotic cavities
The large-N expansion of these averages has been computed for systems with and without broken time reversal symmetry ( β = 2 and β = 1, respectively); we suggest that the
Summary
The Wigner-Smith[11,37,40] time-delay matrix Q plays a central role in the theory of quantum transport.[14,38] It is defined in terms of the N-channel scattering matrix S via the relation. For complex Gaussian Hermitian matrices (Gaussian unitary ensemble (GUE)), the large-N expansion of the moments enumerates maps of given genus.[12,41] we cannot prove that our expansion is related to the enumeration of maps, we have strong evidence of an underlying enumeration problem for the moments of the ensemble modelling the Wigner-Smith time-delay matrix. Formula (5a) behaves better than (5b); the number of terms in the sum is unbounded for large N, it is a sum of positive terms and the only asymptotic analysis one needs is the complete asymptotic series for the ratio of two Gamma functions Along these lines of reasoning, the first three terms τk(2,g) (g = 0, 2, 4) of the large-N expansion of τk(2) have been obtained in Ref. 27.
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