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Title 
Asymptotics of the instantons of Painlevé I 

Authors  
Year  2010  
Description  26 p.  
Abstract  The 0instanton 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 2dimensional quantum gravity. The asymptotics of the 0instanton $(u_{n,0})$ for large $n$ were obtained by the third author using the RiemannHilbert approach. For $k=0,1,2,...$, the $k$instanton solution of Painlev\'e I is a doublyindexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic nonlinear recursion relation. The goal of the paper is threefold: (a) to compute the asymptotics of the 1instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the RiemannHilbert 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 2instanton and beyond exhibits new phenomena not seen in 0 and 1instantons, and their enumerative context is at present unknown.  
Stable URL  http://archiveouverte.unige.ch/unige:11927  
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arXiv: 1002.3634v1 

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