Asymptotics of the Instantons of Painlevé I

The 0-instanton solution of Painlevé I is a sequence (un,0) of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models, and 2-dimensional quantum gravity. The asymptotics of the 0-instanton (un,0) for large n were obtained by the third author using the Riemann–Hilbert approach. For k=0,1,2,…, the k-instanton solution of Painlevé I is a doubly indexed sequence (un,k) of complex numbers that satisfies an explicit quadratic nonlinear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence (un,1) to all orders in 1/n by using the Riemann–Hilbert method, (b) to present formulas for the asymptotics of (un,k) for fixed k and to all orders in 1/n using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronquée Painlevé transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.