Hermitian One-matrix Model Research Articles

TWO-DIMENSIONAL QUANTUM GRAVITY, MATRIX MODELS AND STRING THEORY

We review some results of the recent progress in understanding two-dimensional quantum gravity and low-dimensional string theories based on the lattice approach. The possibility to solve the lattice models exactly comes from their equivalence to large N matrix models. We describe various matrix models and their continuum limits, and discuss in some detail the phase structure of Hermitian one-matrix models. For the one-dimensional matrix model we discuss its field theoretic formulation through a collective field method and summarize some perturbative results. We compare the results obtained from matrix models to the results in the continuum approach to string theory.

SEMICLASSICAL APPROACH TO FINITE-N MATRIX MODELS

We reformulate the zero-dimensional Hermitian one-matrix model as a (nonlocal) collective field theory, for finite N. The Jacobian arising as a result of changing variables from matrix eigenvalues to their density distribution is treated exactly. The semiclassical loop expansion turns out not to coincide with the (topological) [Formula: see text] expansion, because the classical background has a nontrivial N dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong nonperturbative behavior around N=∞. This leads to IR singularities in the large-N expansion, but UV divergencies appear as well, despite remarkable cancelations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal-polynomial results.

Multicut solutions of the matrix Kontsevich-Penner model

Multicut solutions of the Hermitian one-matrix model parametrized by the recently introduced matrix model [1] with external field and Lagrangian having the form tr(ΛXΛX)-αN(log(1+X)−X) are considered. A brief review of the model, which describes the discretized moduli space of Riemann surfaces, is given. The general structure of multicut solutions is investigated, and it is shown that there arises an additional symmetry and thats parameters remain free for the (s+1)-cut solution. A detailed analysis of the one-cut solution is made. Among other results, all solutions of Kazakov type are reproduced. We also discuss the general form for the two-cut solution which arises as generalization of the string equation to the case of two cuts. The entire treatment is given in the approximation of planar diagrams.

UNIVERSALITY AND NONPERTURBATIVE DEFINITIONS OF 2D QUANTUM GRAVITY FROM MATRIX MODELS

The universality of the nonperturbative definition of Hermitian one-matrix models following the quantum, stochastic, or d=1-like stabilization is discussed in comparison with other procedures. We also present another alternative definition, which illustrates the need of new physical input for d=0 matrix models to make contact with 2D quantum gravity at the nonperturbative level.

Higher genus correlators from the hermitian one-matrix model

We develop an iterative algorithm for the genus expansion of the hermitian N × N one-matrix model (is the Penner model in an external field). By introducing moments of the external field, we prove that the genus g contribution to the m-loop correlator depends only on 3 g−2+ m lower moments (3 g−2 for the partition function). We present the explicit results for the partition function and the one-loop correlator in genus one. We compare the correlators for the hermitian one-matrix model with those at zero momenta for c = 1 CFT and show an agreement of the one-loop correlators for genus zero.

CHAOS IN THE HERMITIAN ONE-MATRIX MODEL

The recursion coefficients Ri, which appear in the orthogonal polynomial method of solution for the Hermitian one-matrix model, are determined numerically for values of N up to a thousand. For some cases a chaotic behavior appears in some range, preventing a smooth flow from odd to even multicrititical models. This behavior is studied both for single-well and multiwell potentials. For multiwell potentials, the coefficients Ri generically tend toward more than one function in the N→∞ limit, and this structure is analyzed for small i using the Gaussian approximation.

A hint on the external field problem for matrix models

We reexamine the external field problem for N × N hermitian one-matrix models. We prove an equivalence of the models with the potentials tr[( 1 2 N)X 2 + logX − ΛX ] and Σ k=1 ∞ t k tr X k provided the matrix Λ is related to { t k } by t k = ( 1 k ) tr Λ −k − ( N 2 )δ k2 . Based on this equivalence we formulate a method for calculating the partition function by solving the Schwinger-Dyson equations order by order of genus expansion. Explicit calculations of thepartition function and of correlators of conformal operators with the puncture operator are presented in genus one. These results support the conjecture that our models are associated with the c = 1 case in the same sense as the Kontsevich model describes c = 0.

THE CUBIC MATRIX MODEL WITH IMAGINARY COUPLING AND 2D GRAVITY

We study the continuum properties of the Hermitian one-matrix model defined by a cubic potential with imaginary coupling, using the orthogonal polynomials and loop-equation approaches. The model is well defined mathematically and has a continuum limit which cannot be naively interpreted as pure gravity since each term of the sum over surfaces is not positive-definite. Nevertheless, the model may be considered as an analytic continuation of the standard matrix-model formulation of gravity. We study in detail the conditions under which the analytic continuation may be performed. In particular, the specific heat is found to obey a Painlevé equation. Although we find a solution that is compatible, as far as global stability is concerned, with the one proposed by David, it is not completely clear that the two solutions are the same.

PROPERTIES OF HERMITIAN MATRIX MODELS IN AN EXTERNAL FIELD

We consider a Hermitian one-matrix model in an (Hermitian) external field. We drive the Schwinger-Dyson equations and show that those can be represented as a set of Virasoro constraints which are imposed on the partition function. We prove that these Virasoro constraints are equivalent (at least at large N) to a single integral equation whose solution can be found. We use this solution to study properties of the Kontsevich model in genus zero.

PHASE DIAGRAM AND ORTHOGONAL POLYNOMIALS IN MULTIPLE-WELL MATRIX MODELS

We investigate a zero-dimensional Hermitian one-matrix model in a triple-well potential. Its tree-level phase structure is analyzed semiclassically as well as in the framework of orthogonal polynomials. Some multiple-arc eigenvalue distributions in the first method correspond to quasiperiodic large-N behavior of recursion coefficients for the second. We further establish this connection between the two approaches by finding three-arc saddle points from orthogonal polynomials. The latter require a modification for nondegenerate potential minima; we propose weighing the average over potential wells.

Matrix models with boundary terms and the generalized Painlevé II equation

We derive the specific heat for unitary and hermitian one-matrix models with boundary terms. We show that for the lowest critical point, the specific heat is a simple polynomial of a generalized Painlevé II transcendant. This suggests that topological gravity of surfaces with boundaries is intimately related to the Painlevé II equation. This is somewhat confirmed by calculating the operator products of puncture operators on the disk. Finally, we show that as the number of fermion flavors on the boundary diverges, the critical behavior of the unitary theory approaches that of the k=2 critical point of surfaces without boundaries. For the hermitian case, the critical behavior in this limit is also that of the k=2 critical point, except for an overall factor 1 2 in th specific heat.

A well-defined multicritical series in hermitian one-matrix models

A new multicritical series is described for hermitian one-matrix models with even potential U( φ) = Σ d i = 1 g 2 i φ 2 i . It is characterized by a negative g 2 and a positive g 2 d . The latter condition ensures that the matrix model is well-defined. The corresponding value of γ 0 is −1/( d − 1). The specific heat is F″ = 1 4 u 2 , where u satisfies a string equation of the mKdV type. For the case d=3, a numerical solution is presented which is pole free and completely determined by asymptotic boundary conditions. This new sequence, as well as the other two sequences previously found for hermitian one-matrix models with even potentials, are shown to be special cases of a two-parameter family of multicritical models characterized by an engenvalue distribution with n ⩽ 2 cuts.

MATRIX MODELS WITH NONEVEN POTENTIALS

We study examples of Hermitian one-matrix models with even and odd terms present in the potential. A definition of criticality is presented which in these cases leads to multicritical models falling into the same universality classes as those of the purely even potentials. We also show that (in our examples) for polynomial potentials ending in odd powers (unbounded), the coupling constants, in addition to their expected real critical values, also admit critical values which alternate between imaginary/real values in the odd/even terms. We find that, remarkably, the ensuing statistical models are insensitive to the real/imaginary nature of these critical values. This feature may be of relevance in the recently studied connection between matrix models and the moduli space of Riemann surfaces.

UNIVERSALITY OF THE MATRIX MODEL APPROACH TO TWO-DIMENSIONAL QUANTUM GRAVITY

Universality with respect to triangulations is investigated in the Hermitian one-matrix model approach to 2-D quantum gravity for a potential containing both even and odd terms, [Formula: see text]. With the use of analytical and numerical calculations, I find that the universality holds and the model describes pure gravity, which leads in the double scaling limit to coupled equations of Painlevé type.

LOGARITHMIC SCALING VIOLATION AND BOSE CONDENSATION IN ONE-MATRIX MODELS

In this paper, we are interested in studying critical behavior of Hermitian one-matrix models where the topological expansion for the free-energy develops a logarithmic singularity, Γ∝μ2 log μ+…. We consider models where the potential is of the type U(M)= t0V0(M)−tf log Vf(M). We show that, for tf>0, the model has an underlying fermionic structure. Continuing into the bosonic region where tf<0 allows the possibility of Bose condensation, which is shown to be responsible for a logarithmic scaling violation for the free energy.

TACHYONIC HERMITIAN ONE-MATRIX MODELS

We treat the hermitian one-matrix model in the case where the quadratic term of the matrix potential has a negative coefficient (negative mass). In that case two scaling functions are necessary in the continuum limit. We analyze the problem both in the spherical and the double scaling limits. We give the matrix potentials leading to the general multi-critical points, and find two sequences of critical behaviors, for potentials of degree 4m and 4m+2 respectively (m=1, 2, …). The two scaling functions satisfy coupled nonlinear differential equations. It is shown that the latter can be decoupled into equations of the KdV and the mKdV hierarchies. For potentials of degree 4m+2, the genus expansion of the specific heat coincides with that obtained in the positive mass case at degree 2m. For potentials of degree 4m, the genus expansion is different.

Instability of even m multicritical matrix models of 2D gravity

The hermitian one-matrix model of two-dimensional gravity is studied in the large N limit by the saddle-point method for a general even matrix potential giving a well defined partition function. A simple demonstration of the connection between instability of m even models and the existence of multi-arc phases is given.

Multiband structure and critical behavior of matrix models.

We discuss, perturbatively and nonperturbatively, the multiband phase structure that arises in Hermitian one-matrix models with potentials having several local minima. The tree-level phase diagram for the {phi}{sup 6} potential including critical exponents at the phase boundaries is presented. The multiband structure is then studied from the viewpoint of the orthogonal polynomial recursion coefficients {ital R}{sub {ital n}}, using the operator formalism to relate them to the large-{ital N} limit of the generating function {ital F}({ital z}){identical to}(1/{ital N}){l angle}tr1/({ital z}{minus}{Phi}){r angle}. We show how a periodicity structure in the sequence of the {ital R}{sub {ital n}} coefficients naturally leads to multiband structure, and in particular, provide an explicit example of a three-band phase. Numerical evidence for the periodicity structure among the recursion coefficients is given. We then present examples where we identify the double-scaling limit from a multiband phase. In particular, a ({ital k}=2)-type multicritical nonperturbative solution from the two-band phase in the {phi}{sup 8} potential, and a ({ital k}=1)-type nonperturbative solution from the three-band phase in the {phi}{sup 6} potential is found. Both solutions are described by differential equations related to the modified Korteweg--de Vries hierarchy. Finally, we comment on the other phases that coexist with the {italmore » k}=2 pure gravity solution.« less