Diagrammatic Analysis of the Unitary Group for Double Barrier Ballistic Cavities: Equivalence with Circuit Theory

We derive a set of coupled non-linear algebraic equations for the asymptotics of the Poisson kernel distribution describing the statistical properties of a two-terminal double-barrier chaotic billiard (or ballistic quantum dot). The equations are calculated from a diagrammatic technique for performing averages over the unitary group, proposed by Brouwer and Beenakker [J. Math. Phys. 37, 4904 (1996)]. We give strong analytical evidences that these equations are equivalent to a much simpler polynomial equation calculated from a recent extension of Nazarov's circuit theory [A. M. S. Macedo, Phys. Rev. B 66, 033306 (2002)]. These results offer interesting perspectives for further developments in the field via the direct conversion of one approach into the other.