The Fuchsian differential equation of the square lattice Ising model $\chi(3)$ susceptibility

TL;DR

This paper derives and analyzes a seventh-order Fuchsian differential equation for the three-particle susceptibility contribution in the square lattice Ising model, providing insights into its mathematical structure and properties.

Contribution

It introduces a novel polynomial-time expansion method to generate series coefficients and explicitly derives the differential equation governing the three-particle susceptibility.

Findings

Derived a seventh-order Fuchsian differential equation for χ^{(3)}

Generated a long series of coefficients using an efficient algorithm

Analyzed the properties of the differential equation

Abstract

Using an expansion method in the variables $ x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the 3-particle contribution $\chi^{(3)}$, using a $ N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $\chi^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.