Matrix resolvent and the discrete KdV hierarchy

Abstract

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.

arXiv commentIt was added a remark after Theorem 1; added Appendix A; added references; two statements moved to Introduction; corrected typos. 26 pages

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Cited by in corpus (2)

• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method