
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 WiegmannZabrodin, ElbauFelder,
HedenmalmMakarov, 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 HeleShaw flows on curved surfaces,
considered by Entov and the first author in 1990s. 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 βz^{2}, this is done by WiegmannZabrodin and
ElbauFelder.
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_{m1}. 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:
HeleShaw flow; equilibrium measure; random normal matrices.
pdf (250 kb)
ps (179 kb)
tex (15 kb)
References
 Chau L.L., Zaboronsky O.,
On the structure of correlation functions in the normal matrix model,
Comm. Math. Phys. 196 (1998), 203247, hepth/9711091.
 CrawleyBoevey W., Holland M.P.,
Noncommutative deformations of Kleinian singularities,
Duke Math. J. 92 (1998), 605635.
 Elbau P., Felder G.,
Density of eigenvalues of random normal matrices,
Comm. Math. Phys. 259 (2005), 433450, math.QA/0406604.
 Entov V.M., Etingof P.I.,
Viscous flows with timedependent free boundaries in a nonplanar HeleShaw cell,
Euro. J. Appl. Math. 8 (1997), 2335.
 Hedenmalm H., Makarov N.,
Quantum HeleShaw flow,
math.PR/0411437.
 Kostov I.K., Krichever I., MineevWeinstein M., Wiegmann P.B., Zabrodin A.,
The tfunction for analytic curves, in Random Matrix Models and Their Applications,
Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge Univ. Press, Cambridge, 2001, 285299.
 Krichever I., Marshakov A., Zabrodin A.,
Integrable structure of the Dirichlet boundary problem in multiplyconnected domains, Comm. Math. Phys. 259 (2005), 144, hepth/0309010.
 Marshakov A., Wiegmann P.B., Zabrodin A.,
Integrable structure of the Dirichlet boundary problem in two dimensions,
Comm. Math. Phys. 227 (2002), 131153, hepth/0109048.
 Oas G.,
Universal cubic eigenvalue repulsion for random normal matrices,
Phys. Rev. E 55 (1997), 205211, condmat/9610073.
 Varchenko A.N., Etingof P.I.,
Why the boundary of a round drop becomes a curve of order four,
AMS, Providence, 1992.
 Wiegmann P.B., Zabrodin A.,
Conformal maps and integrable hierarchies,
Comm. Math. Phys. 213 (2000), 523538, hepth/9909147.
 Wiegmann P.B., Zabrodin A.,
Large scale correlations in normal nonHermitian matrix ensembles,
J. Phys. A: Math. Gen. 36 (2003), 34113424,
hepth/0210159.

