All orders asymptotic expansion of large partitions

Bertrand Eynard (1)

1 SPhT - Service de Physique Théorique

The generating function which counts partitions with the Plancherel measure (and its q-deformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the Gromov-Witten invariants of the X_p Calabi-Yau 3-fold, and we prove a conjecture of M. Marino, that the generating functions F_g of Gromov--Witten invariants of X_p, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.

Mots clés: Plancherel; partitions; Gromov Witten; random matrix; spectral curve; topoloical expansion

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]; Mécanique statistique [cond-mat.stat-mech]

Bertrand Eynard. All orders asymptotic expansion of large partitions. 2008. ⟨hal-00269379v2⟩