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 $g_{n}n\to t$, 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 $l={\lim }_{n\to \infty }|\sin (\pi /g_n){|}^{1/n}$ 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 $l\ne 0$ 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 $\mathcal{E}$, which turns out to be independent of $l$, and the free energy $\mathcal{F}=\mathcal{E}-Q\ln l$, which does depend of the fine structure parameter $l$. The existence of large $n$ asymptotic expansions of $\mathcal{F}$ beyond the planar limit as well as the double-scaling limit are also discussed.

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Cited by in corpus (4)

• Complex saddles in the Gross-Witten-Wadia matrix model
• Phase space and phase transitions in the Penner matrix model with negative coupling constant
• The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models
• Gravitational lensing by eigenvalue distributions of random matrix models