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We derive a set of coupled non-linear algebraic equations for the asymptoticsof the Poisson kernel distribution describing the statistical properties of atwo-terminal double-barrier chaotic billiard (or ballistic quantum dot). Theequations are calculated from a diagrammatic technique for performing averagesover the unitary group, proposed by Brouwer and Beenakker [J. Math. Phys. 37,4904 (1996)]. We give strong analytical evidences that these equations areequivalent to a much simpler polynomial equation calculated from a recentextension of Nazarov's circuit theory [A. M. S. Macedo, Phys. Rev. B 66, 033306(2002)]. These results offer interesting perspectives for further developmentsin the field via the direct conversion of one approach into the other.Comment: 7 pages, to appear in Physical Review

Diagrammatic analysis of the unitary group for double-barrier ballistic cavities: Equivalence with circuit theory