Fine structure in the large [formula omitted] 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 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.