Colored graphs, Gaussian integrals and stable graph polynomials

TL;DR

This paper develops a combinatorial framework using colored graphs to analyze Gaussian integrals, deriving a system of PDEs and explicit solutions for the genus expansion of generating functions, including stable graph polynomials.

Contribution

It introduces a general combinatorial approach to Gaussian integrals via colored graphs, proving the uniqueness of solutions and explicit formulas for higher genus terms.

Findings

Generating functions satisfy specific PDEs similar to Gaussian integrals.

Explicit solutions for higher genus terms involve universal polynomials and stable graph polynomials.

Recurrence relations for stable graph polynomials are established.

Abstract

Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored graphs. We prove that the generating power series for such graphs satisfy the same system of partial differential equations as the Gaussian integral and the formal power series solution of this system is unique. The solution is obtained as the genus expansion of the generating power series. The initial term of this expansion is the corresponding generating function for trees. The consequence equations for this term turns to be equivalent to the inversion problem for the gradient mapping defined by the initial condition. The equations for the higher terms of the genus expansion are linear. The solutions of these equations can be expressed explicitly by substitution of the initial conditions and the initial term (the tree expansion) into some universal polynomials (for g>1) which are generating functions for stable closed graphs. (For g=1 instead of polynomials appears logarithm.) The stable graph polynomials satisfy certain recurrence. In [1] some of these results were obtained for the one dimensional case by more or less direct solution of differential equations. Here we present purely combinatorial proofs.