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

In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large nn limit in the non-Hermitian Penner matrix model. In these generalizations gnn→tgnn→t, but the product gnngnn is not necessarily fixed to the value of the ’t Hooft coupling tt. If t>1t>1 and the limit l=limn→∞|sin(π/gn)|1/nl=limn→∞|sin(π/gn)|1/n exists, then the large nn limit is well-defined but depends both on tt and on ll. This result implies that for t>1t>1 the standard large nn limit with gnn=tgnn=t fixed is not well-defined. The parameter ll determines a fine structure of the asymptotic eigenvalue support: for l≠0l≠0 the support consists of an interval on the real axis with charge fraction Q=1−1/tQ=1−1/t and an ll-dependent oval around the origin with charge fraction 1/t1/t. For l=1l=1 these two components meet, and for l=0l=0 the oval collapses to the origin. We also calculate the total electrostatic energy EE, which turns out to be independent of ll, and the free energy F=E−Qlnl, which does depend on the fine structure parameter ll. The existence of large nn asymptotic expansions of FF beyond the planar limit as well as the double-scaling limit are also discussed.