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

TL;DR

This paper investigates the detailed asymptotic eigenvalue distribution of a non-hermitian Penner matrix model, revealing a fine structure influenced by a parameter and challenging the standard large n limit assumptions.

Contribution

It introduces a generalized large n limit for the non-hermitian Penner matrix model, incorporating a new parameter that affects eigenvalue support and asymptotic behavior.

Findings

Large n limit depends on both t and l parameters.

Eigenvalue support consists of an interval and an l-dependent oval.

Free energy depends on the fine structure parameter l.

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 $l = \lim_{n\rightarrow \infty} |\sin(\pi/g_n)|^{1/n}$ 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 $l\neq 0$ 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.