Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

Theorem 1. Let u(t; κ) be the solution to the Painlevé II equation (1.8) characterized by (1.9). If κ ∈C \ ((−∞, −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 = −i∞) is known as the Hastings-McLeod solution. The function y(t; β) = u(t; κ)2 solves the Painlevé XXXIV equation ytt = 4y2 + 2ty + (yt)2 2y . (1.12) The function y(t, β) and equ

Theorem 2. Let | Re β| < 1/2 and let Hn(λ0, β) be the Hankel determinant (1.1) corresponding to the weight (1.2), with λ0 given by (1.7). If κ2 = 1 −e−2πiβ, we have Hn λ0, β) = eiπβnHn (λ0, 0) exp

Theorem 2 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 det  1 −κ2KAi [t,+∞)  , where KAi [t,+∞) is the integral operator with kernel KAi (x, y) = Ai (x)Ai ′(y) −Ai (y)Ai ′(x) x −y (1.14) acting on [t, +∞). Indeed, it was shown in [31] that

Theorem 4. Let u(t; κ) be the solution to the Painlevé II equation (1.8) characterized by (1.9) and let κ2 = 1 −e−2iπβ = 1 + e2πγ, β = 1/2 + iγ, γ ∈R. Then y (t; β) = u (t; κ)2 is a solution to the Painlevé XXXIV equation (1.12) and has the following asymptotics as t →−∞, away from the zeros of trigonometric functions appearing in the denominators: y

Theorem 5. Let Rn and Qn be the recurrence coefficients defined in (1.4), associated to the orthogonal polynomials with respect to the weight (1.2). Let | Re β| < 1/2 and let λ0 be given by (1.7). Then, as n →∞, the recurrence coefficients have the following expansions, Rn(λ0, β) = n 2 −1 2u(t; κ)2n1/3 + O(1), (1.24) and Qn(λ0, β) = −1√ 2 u(t; κ)2n−1/6 + O  n−1/2 , (1.25)

Theorem 8. Let pn(x) be the degree n monic orthogonal polynomial with respect to the weight (1.2), and let λ0 be given by (1.7). Let |Re β| < 1/2. Then, as n →∞, pn (λ0) = √ 2π κ ne 2 n/2 n1/6etn1/3u (t; κ)  1 + O  n−1/3 ,

Theorem 2 can be proved in two different ways. The first one is very short and relies on the Tracy-Widom formula (1.15) and on known asymptotic results in the Gaussian Unitary Ensemble. This proof will be given in Section 2. The second proof, given in Section 6, is lengthy but has the advantage of being self-contained. It relies on the RH analysis which we need anyways for the asymptotics of the orthogonal polynomials and their recurrence coefficients. As is always the case in the asymptotic analysi

Theorem 2 and Conjecture 3 where s = 1 −κ2. This relation, without the Hankel determinant, was discussed previously in [5, 6], where a transition was observed from the Tracy-Widom distribution (at s = 0) to the Weibull distribution (at s = 1). It is challenging, however, to describe explicitly the transition asymptotic regime from the behavior (1.17) corresponding to β = −i 2π ln s, 0 < s ⩽1 to the Tracy-Widom asymptotic behavior, ln det  1 −KAi [t,+∞)  = 1 12t3 −1 8 ln (−t) + 1 24 ln 2 + ζ′ (

Theorem 2 and Conjecture 3 2.1

Theorem 2 and Conjecture 3 built out of normalized degree k Hermite polynomials Hk, orthonormal with respect to the weight e−x2. Define Gλ0,n(κ) by Gλ0,n(κ) = Mλ0,n(log(1 −κ2)) = 1 Zn ˆ Rn Y 1⩽i<j⩽n (xi −x j)2 n Y j=1 e−x2 j ×

Proposition 13. We have ∂ ∂λ0 log Hn(λ0, β) = 1 π sin πβ  Y−1Y′ 21 (λ0)e−λ2 0. (6.1) Here ′ is the derivative of Y(z) with respect to z.

Proposition 14. Let r be defined by (6.16), Ψ0 as introduced in Section 3.3, and let u be the Painlevé II solution characterized by (1.9). The following identity holds, ∂ ∂τr(τ; β) = −2πi 1 −e−2iπβu(τ; κ)2, (6.18) where κ and β are related by (1.16).