Multi–Instantons and Multi–Cuts

We discuss various aspects of multi–instanton configurations in generic multi–cut matrix models. Explicit formulae are presented in the two–cut case and, in particular, we obtain general formulae for multi–instanton amplitudes in the one–cut matrix model case as a degeneration of the two–cut case. These formulae show that the instanton gas is ultra–dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi–instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi–instanton contributions in two–dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ–branes, which take into full account their back–reaction on the target geometry. Finally, we also derive structural properties of the trans–series solution to the Painlevé I equation.