Normal Matrix Model Research Articles

Orthogonal polynomials in the normal matrix model with a cubic potential

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder.The main goal of this paper is to develop the Riemann–Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures (μ1,μ2) on the complex plane. We then formulate a 3×3 matrix valued RH problem for the orthogonal polynomials in hand, and we apply the nonlinear steepest descent method of Deift–Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure (μ1,μ2), and the construction of a global parametrix.The main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure μ1, the first component of the equilibrium measure. We also obtain analytical results for the measure μ1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve.

Random normal matrices and polynomial curves

We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the orthogonal polynomials corresponding to this matrix model and show that, under certain assumptions, the density of the zeros of the highest relevant orthogonal polynomial in the large matrix limit is (up to some constant factor) given by the discontinuity of the Schwarz function of this polynomial curve.

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

Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Maka- rov, 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 |z| 2 , 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 ˆ Am 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.

Genus-zero Whitham hierarchies in conformal-map dynamics☆☆Partially supported by MEC project FIS2005-00319.

A scheme for solving quasiclassical string equations is developped to prove that genus-zero Whitham hierarchies describe the deformations of planar domains determined by rational conformal maps. This property is applied in normal matrix models to show that deformations of simply-connected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies.

Noncritical string correlators, finite-N matrix models and the vortex condensate

We carry out a systematic study of correlation functions of momentum modes in the Euclidean c=1 string, as a function of the radius and to all orders in perturbation theory. We obtain simple explicit expressions for several classes of correlators in terms of special functions. The Normal Matrix Model is found to be a powerful calculational tool that computes c=1 string correlators even at finite N. This enables us to obtain a simple combinatoric formula for the 2n-point function of unit momentum modes, which after T-duality determines the vortex condensate. We comment on possible applications of our results to T-duality at c=1 and to the 2d black hole/vortex condensate problem.

Wilson loops in Script N = 4 SYM and fermion droplets

The matrix models which are conjectured to compute the circle Wilson loop and its correlator with chiral primary operators are mapped onto normal matrix models. A fermion droplet picture analogous to the well-known one for chiral primary operators is shown to emerge in the large N limit. Several examples are computed. We find an interesting selection rule for the correlator of a single trace Wilson loop with a chiral primary operator. It can be non-zero only if the chiral primary is in a representation with a single hook. We show that the expectation value of the Wilson loop in a large representation labelled by a Young diagram with a single row has a first order phase transition between a regime where it is identical to a large column representation and a regime where it is a large wrapping number single trace Wilson loop.

C= 1 Matrix models: equivalences and open-closed string duality

We give an explicit demonstration of the equivalence between the Normal Matrix Model (NMM) of c=1 string theory at selfdual radius and the Kontsevich-Penner (KP) model for the same string theory. We relate macroscopic loop expectation values in the NMM to condensates of the closed string tachyon, and discuss the implications for open-closed duality. As in c<1, the Kontsevich-Miwa transform between the parameters of the two theories appears to encode open-closed string duality, though our results also exhibit some interesting differences with the c<1 case. We also briefly comment on two different ways in which the Kontsevich model originates.

Schur function expansion for normal matrix model and associated discrete matrix models

We consider Schur function expansion for the partition function of the model of normal matrices. This expansion coincides with Takasaki's expansion for tau functions of Toda lattice hierarchy. We show that the partition function of the model of normal matrices is, at the same time, a partition function of certain discrete models, which can be solved by the method of orthogonal polynomials. We obtain discrete versions of various known matrix models: models of non-negative matrices, unitary matrices, normal matrices. We also introduce Hermitian and unitary two-matrix models with generalized interaction terms in continuous and discrete versions.

New Applications of Non-Hermitian Random Matrices

We discuss recently discovered links of the statistical models of normal random matrices to some important physical problems of pattern formation and to the quantum Hall effect. Specifically, the large N limit of the normal matrix model with a general statistical weight describes dynamics of the interface between two incompressible fluids with different viscosities in a thin plane cell (the Saffman-Taylor problem). The latter appears to be mathematically equivalent to the growth of semiclassical 2D electronic droplets in a strong uniform magnetic field with localized magnetic impurities (fluxes), as the number of electrons increases. The equivalence is most easily seen by relating both problems to the matrix model.

String field theory vertices, integrability and boundary states

We study Neumann coefficients of the various vertices in the Witten's open string field theory (SFT). We show that they are not independent, but satisfy an infinite set of algebraic relations. These relations are identified as so-called Hirota identities. Therefore, Neumann coefficients are equal to the second derivatives of tau-function of dispersionless Toda Lattice hierarchy (this tau-function is just the partition sum of normal matrix model). As a result, certain two-vertices of SFT are identified with the boundary states, corresponding to boundary conditions on an arbitrary curve. Such two-vertices can be obtained by the contraction of special surface states with Witten's three vertex. We analyze a class of SFT surface states,which give rise to boundary states under this procedure. We conjecture that these special states can be considered as describing D-branes and other non-perturbative objects as "solitons" in SFT. We consider some explicit examples, one of them is a surface states corresponding to orientifold.

2D string theory as normal matrix model

We show that the c=1 bosonic string theory at finite temperature has two matrix-model realizations related by a kind of duality transformation. The first realization is the standard one given by the compactified matrix quantum mechanics in the inverted oscillator potential. The second realization, which we derive here, is given by the normal matrix model. Both matrix models exhibit the Toda integrable structure and are associated with two dual cycles (a compact and a non-compact one) of a complex curve with the topology of a sphere with two punctures. The equivalence of the two matrix models holds for an arbitrary tachyon perturbation and in all orders in the string coupling constant.

On the Structure of Correlation Functions in the Normal Matrix Model

We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with axially symmetric potentials can be expressed in terms of holomorphic functions of one variable. This observation is used to demonstrate the exact solvability of the model. The two-point correlation function is calculated in the scaling limit by solving the BBGKY chain of equations. The answer is shown to be universal (i.e. potential independent up to a change of the scale). We then develop a two-dimensional free fermion formalism and construct a family of completely integrable hierarchies (which we call the extended-KP(N) hierarchies) of non-linear differential equations. The well-known KP hierarchy is a lower-dimensional reduction of this family. The extended-KP(1) hierarchy contains the (2+1)-dimensional Burgers equations. The partition function of the N*N NMM is the tau function of the extended-KP(N) hierarchy invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane.

Universal cubic eigenvalue repulsion for random normal matrices

Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices. The density of eigenvalues, all correlation functions, and level spacing statistics are calculated. Normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a Gaussian distribution are solvable. The statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen.

Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects

We formulate matrix models with random interactions from normal matrices and construct the corresponding unitary polynomials. The unitary polynomials can be viewed as a set of one-body wave functions of a two-dimensional repulsive Coulomb gas of particles. We then construct the N-body ground state wave functions from these unitary polynomials according to symmetry and statistics requirements. Interestingly, in the case of the zero matrix-model random interaction we precisely obtain the Laughlin wave functions for the fractional quantum Hall effect. In the case of the nonzero matrix-model random interactions, we demonstrate that the N-body wave functions obtained from our normal matrix model describe electrons on a plane under the influence of nonuniform external electromagnetic fields. Thus, our solutions from the normal matrix model show that fractional quantum Hall effects do exist in complex situations of external electromagnetic fields. It will be interesting to observe such fractional quantum Hall effects experimentally.