A Probabilistic Proof of the Rogers Ramanujan Identities

TL;DR

This paper provides a probabilistic approach using Markov chains to prove the Rogers-Ramanujan identities, connecting asymptotic measures on partitions with algebraic identities and extending to quivers.

Contribution

It introduces a simple probabilistic method via Markov chains to prove Rogers-Ramanujan identities and interprets Bailey's lemma in terms of transition matrix eigenvectors.

Findings

Elementary probabilistic proofs of Rogers-Ramanujan identities

Interpretation of Bailey's lemma through Markov chain eigenvectors

Extension of Markov chain approach to quivers

Abstract

The asymptotic probability theory of conjugacy classes of the finite general linear and unitary groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given and compared with work on the uniform measure. Elementary probabilistic proofs of the Rogers-Ramanujan identities follow. As a corollary, the main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of the Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.