On the Large N Limit of the Itzykson-Zuber Integral

TL;DR

This paper analyzes the large N limit of the Itzykson-Zuber integral, linking it to solutions of the complex Burgers equation and applying it to induced QCD and matrix models, revealing new eigenvalue density solutions.

Contribution

It introduces a novel connection between the large N limit of the Itzykson-Zuber integral and the complex Burgers equation, providing new insights into eigenvalue densities in induced QCD.

Findings

Leading term given by action functional of Burgers equation

Eigenvalue density satisfies a specific functional equation

New solutions for the $c=1$ matrix model on discrete line

Abstract

We study the large N limit of the Itzykson -- Zuber integral and show that the leading term is given by the exponent of an action functional for the complex inviscid Burgers (Hopf) equation evaluated on its particular classical solution; the eigenvalue densities that enter in the IZ integral being the imaginary parts of the boundary values of this solution. We show how this result can be applied to ``induced QCD" with an arbitrary potential $U(x)$. We find that for a nonsingular $U(x)$ in one dimension the eigenvalue density $\rho(x)$ at the saddle point is the solution of the functional equation $G_{+}(G_{-}(x))=G_{-}(G_{+}(x))=x$, where $G_{\pm}(x) \equiv {1\over{2}}U^{\prime}(x)\pm i\pi \rho(x)$. As an illustration we present a number of new particular solutions of the $c=1$ matrix model on a discrete real line.