Asymptotics of the instantons of Painleve I

TL;DR

This paper analyzes the asymptotic behavior of instanton solutions of Painleve I, extending previous results to higher instantons using Riemann-Hilbert and resurgent analysis, revealing new phenomena and Stokes behaviors.

Contribution

It computes the all-order asymptotics of 1-instanton solutions and formulates formulas for higher instantons, introducing the concept of induced Stokes phenomenon in Painleve I.

Findings

Asymptotics of 1-instanton solutions are obtained to all orders.

Formulas for asymptotics of higher instantons are provided.

Discovery of induced Stokes phenomenon related to Painleve transcendents.

Abstract

The 0-instanton solution of Painlev\'e I is a sequence $(u_{n,0})$ 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 $(u_{n,0})$ for large $n$ were obtained by the third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the $k$-instanton solution of Painlev\'e I is a doubly-indexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of $(u_{n,k})$ 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 tritronqu\'ee Painlev\'e 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.