Matrix Models for Random Partitions

Abstract

We derive exact matrix integral representations for different sums over partitions. The characteristic feature of all obtained matrix models is the presence of logarithmic (or, vice versa, exponential) terms in the potential. Our derivation is based on the application of the higher Casimir operators. The Toda lattice integrability of the basic sums over partitions can be easily derived from the matrix model representation.

26 pages, presentation improved, references corrected

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• Proving AGT conjecture as HS duality: extension to five dimensions
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• On KP-integrable Hurwitz functions
• A direct proof of AGT conjecture at beta = 1
• M5-branes, toric diagrams and gauge theory duality
• Free fermions and tau-functions
• Correlators in tensor models from character calculus
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Toda hierarchies and their applications
• Cut and join operator ring in Aristotelian tensor model
• A slow review of the AGT correspondence
• Towards matrix model representation of HOMFLY polynomials
• Weighted Hurwitz numbers and topological recursion: an overview
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Weighted Hurwitz numbers and topological recursion
• On determinant representation and integrability of Nekrasov functions
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Loop equations and a proof of Zvonkine's $qr$-ELSV formula
• Complex matrix model duality
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Refined open intersection numbers and the Kontsevich-Penner matrix model
• On W-operators and superintegrability for dessins d'enfant
• On Products of Random Matrices
• Challenges of beta-deformation
• Connection between cut-and-join and Casimir operators
• Laplacian growth in a channel and Hurwitz numbers
• From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
• Hurwitz numbers from Feynman diagrams
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Special cases of the orbifold version of Zvonkine's $r$-ELSV formula
• Faces of matrix models
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Hurwitz numbers and integrable hierarchy of Volterra type
• Topological recursion for monotone orbifold Hurwitz numbers: a proof of the Do-Karev conjecture
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Integrals of tau functions
• Matrix model for the stationary sector of Gromov-Witten theory of ${\mathbf{P}}^1$
• Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
• Tau functions of the UC hierarchy as partition functions of matrix models