Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions

TL;DR

This paper presents explicit algebraic formulas for the n-point functions of weighted double Hurwitz numbers, connecting spectral curve methods with graph sums, providing a polynomial expression in spectral data.

Contribution

It introduces a new explicit graph-sum formula for Orlov-Scherbin n-point functions using spectral curve variables, advancing the understanding of Hurwitz number correlations.

Findings

Derived explicit formulas for n-point functions

Connected spectral curve variables with graph sum representations

Provided polynomial expressions in spectral data

Abstract

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.