Hard Edge for β-Ensembles and Painlevé III

Starting from the diffusion equation at the β random matrix hard edge obtained by Ramirez and Rider (2008), we study the question of its relation with Lax pairs for Painlevé III (PIII). With this example, we develop a general method of finding a correspondence between certain diffusion equations and Painlevé Lax pairs. Our considerations can be extended to the other Painlevé equations since the corresponding diffusions are in fact known as nonstationary (imaginary time) Schrödinger equations for quantum Painlevé Hamiltonians. The results for the hard edge are in many respects similar to the ones found for the soft edge by Bloemendal and Virag (2010). In particular, the values β=2 and 4 (but not β=1) allow for a simple connection with PIII solutions and Lax pairs. However, there is an additional surprise for a special relation of parameters where a simple solution of the diffusion equation can be obtained, which is a one-parameter generalization of the Gumbel distribution. We also track the hard-to-soft-edge limit transition in terms of our Lax pairs.