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.

Phase transitions in a spin-1 model with plaquette interaction on the square lattice

Physical Review B, 1996

An extension of the Blume-Emery-Griffiths model with a plaquette four-spin interaction term, on the square lattice, is investigated by means of the cluster variation method in the square approximation. The ground state of the model, for negative plaquette interaction, exhibits several new phases, including frustrated ones. At finite temperature we obtain a quite rich phase diagram with two new phases, a ferrimagnetic and a weakly ferromagnetic one, and several multicritical points. ͓S0163-1829͑96͒04222-1͔

Chiral Random Matrix Models: A Novel Intermediate Asymptotic Regime

The Chiral Random Matrix Model or the Gaussian Penner Model (generalized Laguerre ensemble) is re-examined in the light of the results which have been found in double well matrix models [D97,BD99] and subtleties discovered in the single well matrix models [BH99]. The orthogonal polynomial method is used to extend the universality to include non-polynomial potentials. The new asymptotic ansatz is derived (different from Szego's result) using saddle point techniques. The density-density correlators are the same as that found for the double well models ref. [BD99] (there the results have been derived for arbitrary potentials). In the smoothed large N limit they are sensitive to odd and even N where N is the size of the matrix [BD99]. This is a more realistic random matrix model of mesoscopic systems with density of eigenvalues with gaps. The eigenvalues see a brick-wall potential at the origin. This would correspond to sharp edges in a real mesoscopic system or a reflecting boundar...

Conformal field theory techniques in random matrix models

1999

In these notes we explain how the CFT description of random matrix models can be used to perform actual calculations. Our basic example is the hermitian matrix model, reformulated as a conformal invariant theory of free fermions. We give an explicit operator construction of the corresponding collective field theory in terms of a bosonic field on a hyperelliptic Riemann surface, with special operators associated with the branch points. The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are then easily obtained as correlation functions of current, fermionic and twist operators. The result for the spectral kernel is valid both in macroscopic and microscopic scales. At the end we briefly consider generalizations in different directions.

The O(n) model on a random surface: critical points and large-order behaviour

Nuclear Physics B, 1992

In this article we report a preliminary investigation of the large N limit of a generalized one-matrix model which represents an O(n) symmetric model on a random lattice. The model on a regular lattice is known to be critical only for −2 ≤ n ≤ 2. This is the situation we shall discuss also here, using steepest descent. We first determine the critical and multicritical points, recovering in particular results previously obtained by Kostov. We then calculate the scaling behaviour in the critical region when the cosmological constant is close to its critical value. Like for the multi-matrix models, all critical points can be classified in terms of two relatively prime integers p, q. In the parametrization p = (2m + 1)q ± l, m, l integers such that 0 < l < q, the string susceptibility exponent is found to be γ string = −2l/(p + q − l). When l = 1 we find that all results agree with those of the corresponding (p, q) string models, otherwise they are different.

On the remarkable spectrum of a non-Hermitian random matrix model

2003

A non-Hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum. Various attempts have been made to understand the spectrum, but even its dimension is not known. Using the Dyson-Schmidt equation, we show that the spectrum consists of a nondenumerable set of lines in the complex plane. Each line is the support of the spectrum of a periodic Hamiltonian, obtained by the infinite repetition of any finite sequence of the disorder variables. Our approach is based on the "theory of words." We make a complete study of all 4-letter words. The spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe, including this particular paper.