# Matrix resolvent and the discrete KdV hierarchy

## TL;DR

This paper introduces a matrix-resolvent approach to the discrete KdV hierarchy, defining tau-functions and deriving explicit formulas, with applications to ribbon graph enumeration and cubic Hodge integrals.

## Contribution

It develops a novel matrix-resolvent method for the discrete KdV hierarchy and connects tau-functions with combinatorial and geometric applications.

## Key findings

- Explicit formulas for tau-function derivatives are derived.
- Connections between tau-functions and ribbon graph enumeration are established.
- Applications to cubic Hodge integrals are demonstrated.

## 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.