Rogers-Ramanujan identities in Statistical Mechanics

TL;DR

This paper explores the historical and mathematical development of Rogers-Ramanujan identities, highlighting their surprising entry into statistical mechanics and their ongoing influence in physics and mathematics over the past 44 years.

Contribution

It provides a comprehensive narrative linking Rogers-Ramanujan identities with developments in statistical mechanics and introduces new potential crossovers involving elliptic functions and vector partition functions.

Findings

Connection between Rogers-Ramanujan identities and the Hard Hexagon Model

Proofs by George E Andrews of related Baxter identities

Potential new crossovers with elliptic q-gamma functions and vector partition equations

Abstract

We describe the story of the Rogers-Ramanujan identities; being known for 85 years and having about 130 pure mathematics proofs, suddenly entering physics when Rodney Baxter solved the Hard Hexagon Model in Statistical Mechanics in 1980. We next cover the accompanying proofs by George E Andrews of other related Baxter identities arisen of Rogers-Ramanujan type, leading into a new flourishing partnership of Physics and Mathematics. Our narrative goes into the subsequent 44 years, explaining the progress in physics and mathematical analysis. Finally we show some related crossovers with regard to the Elliptic q-gamma function and some Vector Partition generating functional equations; the latter of which may be new. The present paper is essentially chapter 11 of a 32 chapter book to appear in June 2024.