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 Ԑ which turns out to be independent of l, and the free energy Ƒ = Ԑ - Ǫ ln l, which does depend of the fine structure parameter l. The existence of large n asymptotic expansions of Ƒ beyond the planar limit as well as the double-scaling limit are also discussed.