Hermitian matrix model with plaquette interaction (original) (raw)

Exact solution of the O(n) model on a random lattice

Nuclear Physics B, 1995

We present an exact solution of the O(n) model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the onematrix model is found. In addition we find a large degree of universality with respect to n; namely for n ∈] − 2, 2[ the solution can be presented in a form which is valid not only for any potential, but also for any n (not necessarily rational). The cases n = ±2 are treated separately. We give explicit expressions for the genus zero contribution to the one-and two-loop correlators as well as for the genus one contribution to the oneloop correlator and the free energy. It is shown how one can obtain from these results any multi-loop correlator and the free energy to any genus and the structure of the higher genera contributions is described. Furthermore we describe how the calculation of the higher genera contributions can be pursued in the scaling limit.

Non-hermitian random matrix models

1997

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to nonhermitean random matrix models using the 1/N approximation. A number of one-and two-point functions are evaluated on their holomorphic and nonholomorphic supports to leading order in 1/N . The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic correlations. Generic form for the two-point functions are obtained, generalizing the concept of macroscopic universality to nonhermitean random matrices. We show that the holomorphic and nonholomorphic one-and two-point functions condition the behavior of pertinent partition functions to order O(1/N ). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.

Fine structure in the large n limit of the non-Hermitian Penner matrix model

Annals of Physics, 2015

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 g n n → t, but the product g n n is not necessarily fixed to the value of the 't Hooft coupling t. If t > 1 and the limit l = lim n→∞ | sin(π/g n)| 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 g n n = 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 − Q ln l, which does depend of 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.

On the singular sector of the Hermitian random matrix model in the large N limit

Physics Letters A, 2011

The singular sector of zero genus case for the Hermitian random matrix model in the large N limit is analyzed. It is proved that the singular sector of the hodograph solutions for the underlying dispersionless Toda hierarchy and the singular sector of the 1-layer Benney (classical long wave equation) hierarchy are deeply connected. This property is due to the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations E(a, a), with a = −1/2 for the dToda hierarchy and a = 1/2 for the 1-layer Benney hierarchy.

Deformed Cauchy random matrix ensembles and large N phase transitions

Journal of High Energy Physics

We study a new hermitian one-matrix model containing a logarithmic Penner’s type term and another term, which can be obtained as a limit from logarithmic terms. For small coupling, the potential has an absolute minimum at the origin, but beyond a certain value of the coupling the potential develops a double well. For a higher critical value of the coupling, the system undergoes a large N third-order phase transition.

a Matrix Model Representation of the Integrable Xxz Heisenberg Chain on Random Surfaces

We consider integrable models, i.e. models defined by R-matrices, on random Manhattan lattices (RML). The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. As an example we formulate a random matrix model where the partition function reproduces annealed average of the XXZ Heisenberg chain over all RML. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.

Quantum fluctuations in a dual matrix model

arXiv (Cornell University), 2004

We analyse a specific, duality-based generalization of the hermitean matrix model. The existence of two collective fields enables us to describe specific excitations of the hermitean matrix model. By using these two fields, we construct topologically nontrivial solutions (BPS solitons) of the model. We find the low-energy spectrum of quantum fluctuations around the uniform solution. Furthermore, we construct the wave functional of the ground state and obtain the corresponding Green function.

Pseudo-hermitian random matrix models: General formalism

Nuclear Physics B, 2022

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite dimensional subspace. As such, they can be used, in the usual spirit of random matrix theory, to model chaotic or disordered P T-symmetric quantum systems, or their gain-loss-balanced classical analogs, in the phase of broken P T-symmetry. The eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. In this paper we introduce a family of pseudo-hermitian random matrix models, depending parametrically on their metric. We apply the diagrammatic method to obtain its averaged resolvent and density of eigenvalues as explicit functions of the metric, in the limit of large matrix size N. As a concrete example, which is essentially an ensemble of elements of the non-compact unitary Lie algebra, we choose a particularly simple set of metrics, and compute the resulting resolvent and density of eigenvalues in closed form. The spectrum consists of a finite fraction of complex eigenvalues, which occupy uniformly two two-dimensional blobs, symmetric with respect to the real axis, as well as the complimentary fraction of real eigenvalues, condensed in a finite segment, with a known non-uniform density. The numbers of complex and real eigenvalues depend on the signature of the metric, that is, the numbers of its positive and negative eigenvalues. We have also carried thorough numerical analysis of the model for these particular metrics. Our numerical results converge rapidly towards the asymptotic analytical large-N expressions.

Relaxation of a Simple Quantum Random Matrix Model

arXiv (Cornell University), 2011

We will derive here the relaxation behavior of a simple quantum random matrix model. The aim is to derive the effective equations which rise when a random matrix interaction is taken in the weak coupling limit. The physical situation this model represents is that a quantum particle restricted to move on two sites, where every site has N possible energy states. The hopping from one site to another is then modeled by a random matrix. The techniques used here can be applied to many variations of the model.