On the Large N Limit of the Itzykson-Zuber Integral

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 $G_{\pm }(x)\equiv \frac{1}{2}{U}^{\prime }(x)\pm i\pi \rho (x)$. As an illustration we present a number of new particular solutions of the $c=1$ matrix model on a discrete real line.

19 pages

References in corpus (1)

• Induced QCD at Large N

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