# 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 partial differential equations (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.

- Authors:

- Department of Applied Mathematics, CU Boulder, Boulder, Colorado 80309 (United States)

- Publication Date:

- OSTI Identifier:
- 22403090

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 1; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVECTORS; FOKKER-PLANCK EQUATION; HAMILTONIANS; INTEGRAL CALCULUS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; TIME DEPENDENCE

### Citation Formats

```
Rumanov, Igor, E-mail: igor.rumanov@colorado.edu.
```*Classical integrability for beta-ensembles and general Fokker-Planck equations*. United States: N. p., 2015.
Web. doi:10.1063/1.4906067.

```
Rumanov, Igor, E-mail: igor.rumanov@colorado.edu.
```*Classical integrability for beta-ensembles and general Fokker-Planck equations*. United States. doi:10.1063/1.4906067.

```
Rumanov, Igor, E-mail: igor.rumanov@colorado.edu. Thu .
"Classical integrability for beta-ensembles and general Fokker-Planck equations". United States.
doi:10.1063/1.4906067.
```

```
@article{osti_22403090,
```

title = {Classical integrability for beta-ensembles and general Fokker-Planck equations},

author = {Rumanov, Igor, E-mail: igor.rumanov@colorado.edu},

abstractNote = {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 partial differential equations (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.},

doi = {10.1063/1.4906067},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 56,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}