Asymptotics of the instantons of Painleve I

@article{Garoufalidis2010AsymptoticsOT,
  title={Asymptotics of the instantons of Painleve I},
  author={Stavros Garoufalidis and Alexander Its and Andrei A. Kapaev and Marcos Mari{\~n}o},
  journal={arXiv: Classical Analysis and ODEs},
  year={2010}
}
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… 

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