On the Large N Limit of the Itzykson – Zuber Integral

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 ρ(x) at the saddle point is the solution of the functional equationG+(G−(x)) = G−(G+(x)) = x, whereG±(x) ≡ 2U ′(x)± iπρ(x). As an illustration we present a number of new particular solutions of the c = 1 matrix model on a discrete real line.