Asymptotics of the Instantons of Painlevé I

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.

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

“…Resurgence is a general mathematical formalism that unifies perturbative and non-perturbative expansions into a single unified framework in the form of a trans-series, such that the entire trans-series is internally self-consistent with respect to Borel summation and analytic continuation of the couplings, and naturally incorporates Stokes phenomena [135,136,138,139,145]. These ideas have also had a profound impact on the study of matrix models and string theory [140][141][142][143][144].…”

Resurgence matches quantization

“…For the case of the topological-string free energy, the general construction of its resurgent transseries-out of a nonperturbative extension of the holomorphic anomaly equations-was set-up in [4]; with the explicit example of the local P 2 toric Calabi-Yau threefold being fully worked out in [5]. These results were obtained based upon earlier stringy constructions [6][7][8][9][10][11][12][13][14][15], and have since led to a few further developments, e.g., [16,17]. In principle, this construction allows us to obtain fully nonperturbative results for the string-theoretic free energy, at any value of the string coupling constant.…”

Painlevé I double scaling limit in the cubic random matrix model

“…24) see for instance [6], and the orthogonal polynomials satisfy a three-term recurrence relation zp n (z) = p n+1 (z) + β N,n p k (z) + γ 2 N,n p n−1 (z), (1. 25) where β N,n = β N,n (u), and Following the general theory, the existence of such a sequence of orthogonal polynomials is a consequence of the Hankel determinant D n = det[µ j+k ] 0≤j,k≤n being nonzero, where the moments are µ j = Γ z j w(z)dz, see for instance the monographs of Szegő, [32, Chapter II] or Chihara, [13, §1.3]. In this case, however, existence is not guaranteed a priori, since the weight function is not positive on Γ.…”

“…The parameter α that appears in the Riemann-Hilbert problem parametrizes a family of tronquée solutions of Painlevé I, with asymptotic behavior 18) where y 1 (λ) behaves as (2.17). For higher order corrections in the latter formula, k-instanton terms, see the paper [25] and references therein. A similar result holds in the sector arg λ ∈ [π, 7π/5], with respect to y 0 (λ) instead of y 1 (λ).…”