All orders asymptotic expansion of large partitions

TL;DR

This work shows that sums over partitions with Plancherel (and q-deformed) weight can be rewritten as matrix integrals and expanded to all orders via topological recursion. The central object is a spectral curve whose symplectic invariants F_g encode the complete genus-g contributions to ln Z, tying combinatorics to the geometry of the mirror curve and to topological strings. The method yields explicit F_g for examples such as X_p Calabi–Yau threefolds and proves Marino’s conjecture that Gromov–Witten invariants come from the matrix-model symplectic invariants of the mirror curve. The results bridge partition combinatorics, random-matrix theory, and algebraic geometry, with applications to LIS, crystal growth, TASEP, and GW theory of P^1 and CY manifolds.

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.