Classical integrability for beta-ensembles and general Fokker-Planck equations

Abstract

Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g. there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system -a closed system of two nonlinear PDEs of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.

Painlevé Representation of Tracy–Widom $${_\beta}$$ β Distribution for $${\beta}$$ β = 6

“…This article is a sequel to [36]. Current results are a further demonstration of classical integrable structure present for values of β beyond the three special ones where it was known or always expected.…”

“…Considered together with eq. (1.12), they possess κ explicit first integrals found in [36], which we do not reproduce here because we find a different more convenient form of them in what follows.…”

“…(1.1) with t and x rescaled as x → x/κ 1/3 , t → t/κ 2/3 , κ = β/2. In [36] we found explicit 2 × 2 matrix Lax pairs of the form…”

“…. , κ, satisfy equations of motion for particles with Calogero interaction and additional time-dependent cubic external force which would lead to classical Painlevé II equation without the interaction [36]. Considered together with eq.…”

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In [36], we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of β. Using this general result, the case β = 6 is further considered here.

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