All order asymptotic expansion of large partitions

2008/04/08 by Bertrand Eynard, B Eynard · 6 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Mathematical functions and polynomials #Random Matrices and Applications #cond-mat.stat-mech #hep-th #math-ph #math.MP

paper · pdf · doi:10.1088/1742-5468/2008/07/p07023

published as J.Stat.Mech.0807:P07023,2008 · 37 pages, latex, 10 figures, reference and example added, few misprints corrected

arxiv created 2008/04/08 · openalex publication_date 2008/07/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30

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 one to compute its asymptotic expansion to all orders. There are applications in the statistical physics of growing/melting crystals, TASEP (totally asymmetric exclusion processes), and also algebraic geometry. In particular we compute the Gromov–Witten invariants of the Calabi–Yau threefold, and we prove a conjecture of Mariño, 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.

Citations

• Breakdown of universality in multi-cut matrix models
• Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues
• A planar diagram theory for strong interactions

Cited by

• Matrix models for2*theories
• Matrix models for β-ensembles from Nekrasov partition functions
• Open Topological Strings and Integrable Hierarchies: Remodeling the A-Model
• Generalization of the Tavis–Cummings model for multi-level anharmonic systems
• Загадки β-деформации
• A-polynomial, B-model, and quantization

Related

• Critical dimensions and small cycle dominance from all-orders asymptotics of d-matrix theory
• The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion
• Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration
• Random partitions and asymptotic theory of symmetric groups, Hecke algebras and finite Chevalley groups
• The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants