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Asymptotics of the instantons of Painleve I

Stavros Garoufalidis, Alexander Its, Andrei Kapaev, Marcos Marino

TL;DR

This work advances the understanding of Painlevé I non-perturbative structure by deriving all-orders asymptotics for the 0- and 1-instantons via Riemann-Hilbert analysis and formulating resurgent trans-series predictions for fixed-k instantons. It reveals a refined resurgence framework linking $u_{n,k}$ to neighboring instanton data through explicit algebraic recursions, Stokes constants, and logarithmic corrections, including an induced Stokes phenomenon arising from tritronquée solutions. The results are substantiated with numerical evidence, illustrating the accuracy of the asymptotic expansions and the presence of new phenomena for $krakge2$, with implications for non-perturbative effects in matrix models and non-critical string theory.

Abstract

The 0-instanton solution of Painlevé 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 Painlevé 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 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.

Asymptotics of the instantons of Painleve I

TL;DR

This work advances the understanding of Painlevé I non-perturbative structure by deriving all-orders asymptotics for the 0- and 1-instantons via Riemann-Hilbert analysis and formulating resurgent trans-series predictions for fixed-k instantons. It reveals a refined resurgence framework linking to neighboring instanton data through explicit algebraic recursions, Stokes constants, and logarithmic corrections, including an induced Stokes phenomenon arising from tritronquée solutions. The results are substantiated with numerical evidence, illustrating the accuracy of the asymptotic expansions and the presence of new phenomena for , with implications for non-perturbative effects in matrix models and non-critical string theory.

Abstract

The 0-instanton solution of Painlevé I is a sequence 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 for large were obtained by the third author using the Riemann-Hilbert approach. For , the -instanton solution of Painlevé I is a doubly-indexed sequence 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 to all orders in by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of for fixed and to all orders in 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.

Paper Structure

This paper contains 21 sections, 1 theorem, 203 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Painlevé I equation and its $0$-instanton solution
  3. The Painlevé I equation and its $k$-instanton solution
  4. The predictions of resurgence for the $k$-instanton asymptotics
  5. The asymptotics of the 1-instanton solution via the Riemann-Hilbert method
  6. Acknowledgements
  7. Asymptotics of the $0$-instanton of Painlevé I
  8. A review of the tritronquée solutions of Painlevé I
  9. Gluing the five tritronquée solutions together
  10. An integral formula for the $0$-instanton coefficients and their asymptotic expansion
  11. Asymptotics of the $1$-instanton of Painlevé I
  12. A 5-tuple of differential equations
  13. Gluing the solutions together
  14. An integral formula for the $1$-instanton coefficients and their asymptotic expansion
  15. Tritronquée solutions and the induced Stokes' phenomenon
  16. ...and 6 more sections

Key Result

Theorem 1.1

For $n$ large and $\gamma > 3$, we have:

Figures (7)

  • Figure 1: The sectors of the $z$-complex plane where the tritronquée solutions (\ref{['u0_gen']}) $u_{-2}$, $u_{-1}$, $u_0$, $u_1$, $u_2$ are represented by the formal series $u_f$.In the dotted sectors, the asymptotics at infinity of the tritronquée solutions is elliptic.
  • Figure 2: The sectorially meromorphic function $\hat{u}^{(N)}(t)$\ref{['hat_uN_def']}.
  • Figure 3: Deformation of the contour of integration for computation of the $0$-instanton $N$-large asymptotics
  • Figure 4: A plot of the sequence (\ref{['nll']}) and its Richardson transformations. In this and subsequent plots, the horizontal axis represents the integer $n$, and the vertical axis the values of the sequence.
  • Figure 5: A plot of the sequence (\ref{['ll']}) and its Richardson transformations, for $k=2$ (left) and $k=5$ (right).
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2