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. The multi-instanton amplitudes are computed at both one and two loops. 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 Painleve I equation.