Oscillatory Asymptotics for the Airy Kernel Determinant on Two Intervals

Abstract

We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta $-functions.

Gap probabilities in the bulk of the Airy process

“…A classical and much harder problem is to find large gap asymptotics, which are asymptotics for F (rI) as r → +∞. Large r asymptotics for F (rI (e) 0 ) and F (rI (e) 1 ) have been analysed in [2,16,29] and [6,26], respectively. In this work, we focus on F (rI…”

“…The above RH problem was solved in [6] (the points y 1 , y 2 and 0 in [6] should be identified with x 0 , x 1 and x 2 in the present paper, respectively). Consider the function ϕ : C \ (−∞, x 0 ] → C defined by…”

“…Note that the holomorphic differential ω 1 of (1.4) with g = 1 coincides with ω. By [6], P (∞) is given by…”

“…) and E ′ x2 (x 2 ) contain the θ-function and its derivatives at the points 0, 1 2 , τ 2 and 1+τ 2 . We use θ-function identities to simplify these expressions following the strategy of [6].…”

Airy-kernel determinant on two large intervals

“…log|q(a)q(b)q(c)| + 4ζ ′ (−1). (25) In the recent work [4], Blackstone, Charlier and Lenells have simultaneously and independently analyzed the large-s asymptotics of log P Ai (sJ). They found the expansion −α 2 s 3 − 1 2 log s + log θ 3 (s 3/2 Ω) + χ ′ + O(1/s) with an undetermined constant term χ ′ = χ ′ (a, b, c).…”