Penner Matrix Model Research Articles

Phase space and phase transitions in the Penner matrix model with negative coupling constant

The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with ’t Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars–McLaughlin sequences instead of ’t Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation.

Topological recursion for Gaussian means and cohomological field theories

We introduce explicit relations between genus-filtrated s-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M_(g,s)^(disc) (discrete volumes). Using these relations, we express Gaussian means in all orders of the genus expansion as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate the topological recursion of the Gaussian model into recurrence relations for the coefficients of this expansion, which allows proving that they are integers and positive. We find the coefficients in the first subleading order for M_(g,1) for all g in three ways: using the refined Harer–Zagier recursion, using the Givental-type decomposition of the KPMM, and counting diagrams explicitly.

Models of discretized moduli spaces, cohomological field theories, and Gaussian means

We prove combinatorially the explicit relation between genus filtrated s -loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces M g , s disc given by N g , s ( P 1 , … , P s ) for ( P 1 , … , P s ) ∈ Z + s . This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretization of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for M g , 1 in two ways: by using the refined Harer–Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Deligne–Mumford compactification M ¯ g , s of moduli spaces of punctured Riemann surfaces.

Fine structure in the large [formula omitted] limit of the non-Hermitian Penner matrix model

In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large n limit in the non-Hermitian Penner matrix model. In these generalizations gnn→t, but the product gnn is not necessarily fixed to the value of the ’t Hooft coupling t. If t>1 and the limit l=limn→∞|sin(π/gn)|1/n exists, then the large n limit is well-defined but depends both on t and on l. This result implies that for t>1 the standard large n limit with gnn=t fixed is not well-defined. The parameter l determines a fine structure of the asymptotic eigenvalue support: for l≠0 the support consists of an interval on the real axis with charge fraction Q=1−1/t and an l-dependent oval around the origin with charge fraction 1/t. For l=1 these two components meet, and for l=0 the oval collapses to the origin. We also calculate the total electrostatic energy E, which turns out to be independent of l, and the free energy F=E−Qlnl, which does depend on the fine structure parameter l. The existence of large n asymptotic expansions of F beyond the planar limit as well as the double-scaling limit are also discussed.

Partition functions and the continuum limit in Penner matrix models

We present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of Penner random matrix models, to obtain their explicit forms in the exactly solvable cases, and to determine the coefficients of their perturbative expansions in the continuum limit. The method relies on identities satisfied by the resolvent of the Jacobi matrix in the three-term recursion relation of the associated families of orthogonal polynomials. These identities lead to a convenient formulation of the string equations. As an application, we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double Penner models can be written as a sum of Gaussian contributions plus linear terms. To illustrate the one-cut case we discuss the linear, double and cubic Penner models, and for the two-cut case we discuss theoretically and numerically the existence of a double-branch structure of the free energy for the Gaussian Penner model.

Parametric dependence of irregular conformal block

Irregular conformal block is an important tool to study a new type of conformal theories, which can be constructed as the colliding limit of the regular conformal block. The irregular conformal block is realized as the $\beta$-deformed Penner matrix model whose partition function is regarded as the inner product of the irregular modules. The parameter dependence of the inner product is obtained explicitly using the loop equation with close attention to singularities in the parameter space. It is noted that the exact singular structure of the parameter space in general can be found using a very simple and powerful method which uses the flow equations of the partition function together with the hierarchical structure of the singularity. This method gives the exact expression to all orders of large $N$ expansion without using the explicit contour integral of the filling fraction.

LargeNPenner matrix model and a novel asymptotic formula for the generalized Laguerre polynomials

The Gaussian Penner matrix model is re-examined in the light of the results which have been found in double-well matrix models. The orthogonal polynomials for the Gaussian Penner model are shown to be the generalized Laguerre polynomials L(α)n(x) with α and x depending on N, the size of the matrix. An asymptotic formula for the orthogonal polynomials is derived following closely the orthogonal polynomial method of Deo (1997 Nucl. Phys. B 504 609). The universality found in the double-well matrix model is extended to include non-polynomial potentials. An asymptotic formula is also found for the Laguerre polynomial using the saddle-point method by rescaling α and x with N. Combining these results a novel asymptotic formula is found for the generalized Laguerre polynomials (different from that given in Szego's book) in a different asymptotic regime. This may have applications in mathematical and physical problems in the future. The density–density correlators are derived and are the same as those found for the double-well matrix models. These correlators in the smoothed large N limit are sensitive to odd and even N where N is the size of the matrix. These results for the two-point density–density correlation function may be useful in finding eigenvalue effects in experiments in mesoscopic systems or small metallic grains. There may be applications to string theory as well as the tunnelling of an eigenvalue from one valley to the other being an important quantity there.

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.

Fermionic quantum gravity

We study the statistical mechanics of random surfaces generated by NxN one-matrix integrals over anti-commuting variables. These Grassmann-valued matrix models are shown to be equivalent to NxN unitary versions of generalized Penner matrix models. We explicitly solve for the combinatorics of 't Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the statistical theory. An examination of the large N and double scaling limits of the theory shows that the genus expansion is a Borel summable alternating series which otherwise coincides with two-dimensional quantum gravity in the continuum limit. We demonstrate that the partition functions of these matrix models belong to the relativistic Toda chain integrable hierarchy. The corresponding string equations and Virasoro constraints are derived and used to analyse the generalized KdV flow structure of the continuum limit.

TWO-LOGARITHM MATRIX MODEL WITH AN EXTERNAL FIELD

We investigate the two-logarithm matrix model with the potential XΛ+α log (1+X)+β log (1-X) related to an exactly solvable Kazakov–Migdal model. In the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich–Penner matrix model and construct the 1/N-expansion solution of this model.

Kazakov–Migdal model with logarithmic potential and the double Penner matrix model

The Kazakov–Migdal (KM) model is a U(N) lattice gauge theory with a scalar field in the adjoint representation but with no kinetic term for the gauge field. This model is formally soluble in the limit N→∞ though explicit solutions are available for a very limited number of scalar potentials. A ‘‘double Penner’’ model in which the potential has two logarithmic singularities provides an example of an explicitly soluble model. The formal solution to this double Penner KM Model is reviewed first. Special attention is paid to the relationship of this model to an ordinary (one) matrix model whose potential has two logarithmic singularities (the double Penner model). A detailed analysis is presented of the large N behavior of this double Penner model. The various one cut and two cut solutions are described and cases in which ‘‘eigenvalue condensation’’ occurs at the singular points of the potential are discussed. Then the consequences of our study for the KM model described above are discussed. The phase diagram of the model is presented and its critical regions are described.

Matrix model for discretized moduli space

We study the algebraic geometrical background of the Penner-Kontsevich matrix model with the potential Nα tr[− 1 2 ΛXΛX+log(1− X+ X]. We show that this model describes intersection indices of linear bundles on the discretized moduli space just in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on Riemann surfaces of arbitrary genus. The special role of the logarithmic potential originating from the Penner matrix model is demonstrated. The boundary effects, which were not essential in the case of the Kontsevich model, are now relevant, and the intersection indices on the discretized moduli space of genus g are expressed through Kontsevich's indices of genus g and of the lower genera using a stratification procedure.

N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant

We discuss gauge theory with a topological N=2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space $\cal M$ and the partition function equals the Euler number of $\cal M$. We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi equations. We interpret the gauge theory actions from the Atiyah-Jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence of these considerations we propose the Euler number of the moduli space of flat connections as a generalization to arbitrary three-manifolds of the Casson invariant. We also comment on the possibility of constructing a topological version of the Penner matrix model.

Random supermatrices and critical behavior

The zero-dimensional M × M random matrix model with N = 2 supersymmetry is presented using a manifest hermitian supermatrix formalism. The general expression for the partition function of such models is derived and analyzed. It is shown that when the superpotential satisfies a particular nonlinear ordinary differential equation, for which the solution is obtained, these models reduce to precisely the Penner matrix model, which is known to be related to c = 1 bosonic string theory. The model is completely solved in the exact double-scaling limit in the case of the cubic superpotential, the critical behavior of which is shown to lie in the same universality class as that of the Penner matrix model. It is argued that, unlike the bosonic case, the solutions of the string equations of the random supermatrix model lie in different universality classes for the cubic and quartic k = 2 potentials.

THE PENNER MATRIX MODEL AND c = 1 STRINGS

The steepest descent solution of the Penner matrix model has a one-cut eigenvalue support. Criticality results when the two branch points of this support coalesce. The support is then a closed contour in the complex eigenvalue plane. Simple generalizations of the Penner model have multi-cut solutions. For these models, the eigenvalue support at criticality is also a closed contour, but consisting of several cuts. We solve the simplest such model, which we call the KT model, in the double-scaling limit. Its free energy is a Legendre transform of the free energy of the c = 1 string compactified to the critical radius of the Kosterlitz–Thouless phase transition.