Asymptotic Properties of a Special Solution to the (3,4) String Equation

We analyze the asymptotic properties a special solution of the (3, 4) string equation, which appears in the study of the multicritical quartic 2-matrix model. In particular, we show that in a certain parameter regime, the corresponding τ -function has an asymptotic expansion which is 'topological' in nature. Consequently, we show that this solution to the string equation with a specific set of Stokes data exists, at least asymptotically. We also demonstrate that, along specific curves in the parameter space, this τ -function degenerates to the τ -function for a tritronquée solution of Painlevé I (which appears in the critical quartic 1-matrix model), indicating that there is a 'renormalization group flow' between these critical points. This confirms a conjecture of Crncovic, Ginsparg, and Moore (1990).