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

Abstract

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.

Gravitational lensing by eigenvalue distributions of random matrix models

“…From (23) it is clear that for a lensing model corresponding to a unitary ensemble with potential V(x) of degree 2p > 2 the number of dim images cannot exceed 2p − 1 and that this bound is sharp. On the other hand, from (23) and taking (11) into account, Bezout theorem leads to the upper bound 4p 2 for the number of bright images. Although this bound is sharp for the Gaussian model it would be interesting to improve it for the general case.…”

“…The present paper focuses on lensing models based on eigenvalue distributions of unitary ensembles of random matrices. Nevertheless, lensing models with similar properties can be generated from more general ensembles of random matrices [22][23][24] in which the eigenvalues are constrained to lie on appropriate curves. It remains to know the interpretation of the associated lensing models.…”

Complex saddle points in the Gross-Witten-Wadia matrix model

“…[12] can be completely justified in the large-N limit by using techniques similar to those developed for the determination of the asymptotic support of the zeros of certain non-Hermitian families of orthogonal polynomials [17,18,21].…”

“…(6) determines a non-Hermitian holomorphic matrix model [19,20] which can be analyzed in the same way as the models recently considered in Refs. [17,18,21]. It should be noticed that Eq.…”

“…A detailed discussion of the techniques we have used to carry out this calculations can be found in Refs. [17,18,21], and we have summarized the results of this study schematically in Fig. 1.…”