We continue the analysis of the spectral curve of the normal random matrix ensemble, introduced in an earlier paper. Evolution of the full quantum curve is given in terms of compatibility equations of independent flows. The semiclassical limit of these flows is expressed through canonical differential forms of the spectral curve. We also prove that the semiclassical limit of the evolution equations is equivalent to Whitham hierarchy.
Bibliographical noteFunding Information:
We are indebted to A. Kapaev, V. Kazakov, I. Krichever, I. Kostov, A. Marshakov and M. Mineev-Weinstein for useful discussions, interest in the subject and help. P.W. and R.T. were supported by the NSF MRSEC Program under DMR-0213745, NSF DMR-0220198 and by the Humboldt foundation. A.Z. and P.W. acknowledge support by the LDRD project 20020006ER “Unstable Fluid/Fluid Interfaces” at Los Alamos National Laboratory and M. Mineev-Weinstein for the hospitality in Los Alamos. A.Z. was also supported in part by RFBR grant 03-02-17373,w and by the grant for support of scientific schools NSh-1999.2003.2. P.W. is grateful to K.B. Efetov for the hospitality in Ruhr-Universitaet Bochum and to A. Cappelli for the hospitality in the University of Florence.
- Integrable systems
- Random matrix theory