All orders asymptotic expansion of large partitions

TL;DR

This paper develops a comprehensive asymptotic expansion for large partitions using matrix integrals, with applications in physics and algebraic geometry, including Gromov-Witten invariants and mirror symmetry.

Contribution

It provides the first all-orders asymptotic expansion for partitions under the Plancherel measure and links generating functions to matrix models and symplectic invariants.

Findings

Computed Gromov-Witten invariants of X_p Calabi-Yau 3-fold.

Proved Marino's conjecture relating generating functions to matrix models.

Established the connection between asymptotic expansions and mirror symmetry.

Abstract

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.