A Probabilistic Proof of the Rogers Ramanujan Identities

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.Comment: Final version, to appear in Bull LMS. The one math change is to fix a typo in the limit in Corollary 2. We also make two historical correction