SIGMA 3 (2007), 048, 13 pages      math.CV/0612108      https://doi.org/10.3842/SIGMA.2007.048
Contribution to the Vadim Kuznetsov Memorial Issue

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Pavel Etingof and Xiaoguang Ma
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 USA

Received December 05, 2006, in final form March 03, 2007; Published online March 14, 2007

Abstract
Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is βz², this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to *-representations of the deformed preprojective algebra of the affine quiver of type Â_m-1. We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.

Key words: Hele-Shaw flow; equilibrium measure; random normal matrices.

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References

1. Chau L.-L., Zaboronsky O., On the structure of correlation functions in the normal matrix model, Comm. Math. Phys. 196 (1998), 203-247, hep-th/9711091.
2. Crawley-Boevey W., Holland M.P., Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), 605-635.
3. Elbau P., Felder G., Density of eigenvalues of random normal matrices, Comm. Math. Phys. 259 (2005), 433-450, math.QA/0406604.
4. Entov V.M., Etingof P.I., Viscous flows with time-dependent free boundaries in a non-planar Hele-Shaw cell, Euro. J. Appl. Math. 8 (1997), 23-35.
5. Hedenmalm H., Makarov N., Quantum Hele-Shaw flow, math.PR/0411437.
6. Kostov I.K., Krichever I., Mineev-Weinstein M., Wiegmann P.B., Zabrodin A., The t-function for analytic curves, in Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge Univ. Press, Cambridge, 2001, 285-299.
7. Krichever I., Marshakov A., Zabrodin A., Integrable structure of the Dirichlet boundary problem in multiply-connected domains, Comm. Math. Phys. 259 (2005), 1-44, hep-th/0309010.
8. Marshakov A., Wiegmann P.B., Zabrodin A., Integrable structure of the Dirichlet boundary problem in two dimensions, Comm. Math. Phys. 227 (2002), 131-153, hep-th/0109048.
9. Oas G., Universal cubic eigenvalue repulsion for random normal matrices, Phys. Rev. E 55 (1997), 205-211, cond-mat/9610073.
10. Varchenko A.N., Etingof P.I., Why the boundary of a round drop becomes a curve of order four, AMS, Providence, 1992.
11. Wiegmann P.B., Zabrodin A., Conformal maps and integrable hierarchies, Comm. Math. Phys. 213 (2000), 523-538, hep-th/9909147.
12. Wiegmann P.B., Zabrodin A., Large scale correlations in normal non-Hermitian matrix ensembles, J. Phys. A: Math. Gen. 36 (2003), 3411-3424, hep-th/0210159.