Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge A. Bogatskiy,∗ T. Claeys,y A. Itsz March 29, 2016 Abstract We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discon- tinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory. 1 Introduction Consider the Hankel determinant, 1 !n−1 1 n 1 Y Y H (λ ; β) = det x j+kw(x)dx = (x − x )2 w(x )dx ; (1.1) n 0 ˆ n! ˙ i j k k −∞ j;k=0 −∞ i< j k=1 with respect to a discontinuous Gaussian weight of the form 8 πiβ # 2 >e ; x < λ0 1 1 w x e−x × < 2 − 2 R ( ) = > −πiβ ; Re β ; ; λ0 : (1.2) :>e ; x > λ0 2 2 arXiv:1507.01710v5 [math-ph] 26 Mar 2016 The weight is periodic in β and we can restrict to the case −1=2 < Re β 6 1=2 without loss of generality. If β is purely imaginary, the weight is positive. We also consider the monic orthogonal polynomials pn of degree n with respect to the weight w(x) on the real line, defined by the orthogonality conditions 1 pn(x)pm(x)w(x)dx = hnδnm; hn = hn(λ0; β): (1.3) ˆ−∞ ∗Department of Higher Mathematics and Mathematical Physics, Department of Physics, Saint-Petersburg State University. Saint-Petersburg, Russia. yInstitut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-La-Neuve, Belgium zDepartment of Mathematical Sciences, Indiana University – Purdue University Indianapolis. Indianapolis, IN 46202-3216, USA. 1 1 Introduction Qn−1 Those are connected to the Hankel determinant Hn by the well-known identity Hn(λ0; β) = k=0 hk(λ0; β). We denote by Rn = Rn(λ0; β) and Qn = Qn(λ0; β) the recurrence coefficients in the three-term recurrence relation xpn(x) = pn+1(x) + Qn pn(x) + Rn pn−1(x): (1.4) The question which we are concerned with in this paper is the large n behavior of the Hankel determinants Hn, the polynomials pn(x), andp their recurrence coefficients Rn and Qn, in the regime where the point of discontinuity λ0 behaves like 2n. They can asymptotically be expressed in terms of the Ablowitz-Segur solutions to the Painlevé II equation. As important by-products of the asymptotics for the Hankel determinants, we also conjecture so-called large gap asymptotics for an Airy kernel Fredholm determinant and total integral identities for the Ablowitz-Segur solutions of the Painlevé II equation. Relying on a result of [33], we in addition prove a new result about the poles for those Painlevé transcendents. p If we let λ0 = λ 2n, the large n asymptotics of the orthogonal polynomials (1.3), the recurrence coefficients (1.4), and the Hankel determinant (1.1) depend dramaticallyp p on whether jλj < 1 or jλj > 1, i.e. whether the jump location λ0 is inside or outside of the support [− 2n; 2n] of the equilibrium measure with Gaussian external field. In the case jλj > 1, all the objects of interest behave effectively as they do for the pure Gaussian weight (i.e., the case where we formally set λ0 = +1); the discontinuity yields an exponentially small correction only [25]. In the case jλj < 1, the situation is different; the discontinuity of the weight becomes strongly visible in the large n behavior of the orthogonal polynomials, the recurrence coefficients, and the Hankel determinant [21]. For the Hankel determinant, it was proved in [21, equation (1.5)] that 2 −3β2=2 −β2 Hn(λ0; β) = Hn (λ0; 0) G(1 + β)G(1 − β)(1 − λ ) (8n) × !! p log n 1 × exp 2inβ arcsin λ + λ 1 − λ2 1 + O ; jRe βj < ; (1.5) n1−4j Re βj 4 as n ! 1, uniformly for λ in compact subsets of (−1; 1). Here G is the Barnes’ G-function, and n−1 n=2 −n2=2 Y Hn (λ0; 0) = (2π) 2 k! (1.6) k=1 denotes the Hankel determinant corresponding to the pure Gaussian weight e−x2 . Asymptotics for the recurrence coefficients Qn and Rn in the case −1 < λ < 1 are also given in [21]. In this paper,p we analyze the transition regime where the point λ0 of discontinuity of the weight is (relatively) close to 2n. More precisely we let p t λ = λ 2n; λ = 1 + n−2=3; (1.7) 0 2 where t 2 R. We will see that the asymptotic behavior of Hn, pn, Rn, and Qn depends in a non-trivial way on the parameter t in (1.7). The asymptotic behavior is described in terms of a family of solutions to the Painlevé II equation 3 utt = tu + 2u ; (1.8) with the asymptotic behavior u(t; κ) ∼ κAi(t); t ! +1; (1.9) where Ai denotes the Airy function, and ! 1 p 1 u(t; κ) = 2iβ sin φ(t; β) + O ; t ! −∞; (1.10) (−t)1=4 t2−3j Re βj 2 1 Introduction with π Γ(−β) 2 3 1 φ(t; β) = − − i log + (−t)3=2 − iβ log (−t) − 3iβ log 2; κ2 = 1 − e−2πiβ; j Re βj < : (1.11) 4 Γ(β) 3 2 2 For 0 < κ < 1, these solutions are known as the Ablowitz-Segur solutions [1] of the second Painlevé equation. They are uniquely characterized either by (1.9) or by (1.10). Moreover, it is known that u(t; κ) has no singularities for t on the real line if κ 2 iR or if jκj < 1. For κ 2 R n [−1; 1], or equivalently j Re βj = 1=2, it is known that u(τ; κ) does have real poles [4]. Relying on a result from [33], we will prove the following result, stating that u has no real poles for any κ 2 C n ((−∞; −1] [ [1; +1)), or equivalently for any β with j Re βj < 1=2. Theorem 1. Let u(t; κ) be the solution to the Painlevé II equation (1.8) characterized by (1.9). If κ 2 C n ((−∞; −1] [ [1; +1)) is fixed, then u(t; κ) has no poles at real values of t. In the case κ = 0, we simply have u(t; κ) = 0; the unique Painlevé II solution satisfying (1.9) with κ = ±1 (which means formally that β = −i1) is known as the Hastings-McLeod solution. The function y(t; β) = u(t; κ)2 solves the Painlevé XXXIV equation (y )2 y = 4y2 + 2ty + t : (1.12) tt 2y The function y(t; β) and equation (1.12) are, in fact, the objects which directly appear in our double scaling analysis of Hn, pn, Rn and Qn. Our next result describes the asymptotics or the Hankel determinants Hn(λ0; β). Theorem 2. Let j Re βj < 1=2 and let Hn(λ0; β) be the Hankel determinant (1.1) corresponding to the weight 2 −2πiβ (1.2), with λ0 given by (1.7). If κ = 1 − e , we have 1 ! iπβn 2 Hn λ0; β) = e Hn (λ0; 0) exp − (τ − t)u(τ; κ) dτ (1 + o(1)); n ! 1; (1.13) ˆt uniformly for t 2 [−M; 1) for any M > 0 and for β in compact subsets of j Re βj < 1=2, where Hn (λ0; 0) is given in (1.6). Theorem2 has two consequences which are not directly related to the Hankel determinants or orthogonal polynomials studied in this paper, but which are of independent interest. To describe them, we note first that the exponential in (1.13) can be recognized as the Tracy-Widom formula for the Fredholm determinant − κ2K K det 1 Ai [t;+1) , where Ai [t;+1) is the integral operator with kernel Ai (x)Ai 0(y) − Ai (y)Ai 0(x) K (x; y) = (1.14) Ai x − y acting on [t; +1). Indeed, it was shown in [31] that 1 ! − κ2K − τ − t u τ κ 2 τ : det 1 Ai [t;+1) = exp ( ) ( ; ) d (1.15) ˆt This observation, together with a strengthened version of the Hankel determinant asymptotics (1.5), allows us t ! −∞ − κ2K to formulate the following conjecture about the asymptotics of det 1 Ai [t;+1) . 3 1 Introduction Conjecture 3. Let κ 2 C n ((−∞; −1] [ [1; +1)) and define β by κ2 = 1 − e−2πiβ; j Re βj < 1=4: (1.16) As t ! −∞, we have 2 4 3=2 3 2 2 log det 1 − κ KAi = − iβ (−t) − β log (−t) + log (G (1 + β) G (1 − β)) − 3β log 2 + o(1); (1.17) [t;+1) 3 2 or equivalently in form of a total integral identity 1 ! 4 3 lim − (τ − t)u(τ; κ)2dτ + iβ (−t)3=2 + β2 log (−t) = log (G (1 + β) G (1 − β)) − 3β2 log 2: (1.18) t→−∞ ˆt 3 2 Remark. Similar asymptotics for the Airy kernel determinant in the case κ = 1 were proved in [2, 12]: we then have 3 t 1 −3 log det I − KAi = − log jtj + c0 + O t ; t ! −∞; (1.19) [t;+1) 12 8 0 where c0 = log 2=24 + ζ (−1) and ζ is the Riemann ζ-function.