Critical edge behavior and the Bessel to Airy transition in the

PROPOSITION 1. The equation (1.12) is the compatibility condition Ψζs = Ψsζ of the following Lax pair Ψζ(ζ, s) =  A0(s) + A1(s) ζ + A2(s) ζ2  Ψ(ζ, s), (1.13) Ψs(ζ, s) = B1(s)

PROPOSITION 2. By a change of unknown function v(s) = sr′(s), (1.19) the third-order equation (1.12) for r(s) is reduced to a particular PIII equation for v(s), namely, v′′ = v′2 v −v′ s + v2 s2 + α s −1 v; (1.20) cf., e.g., [6, (3.12)] for the PIII equation. We note that the coefficients l and α, respectively in (1.12) and (1.20), are determined by initial values of r(s); see Section 2.2 below. In the present paper, the constant l in (1.12) and (1.18) is equal to 0.

PROPOSITION 3. There exists a solution r(s) of (1.12), analytic for s ∈(0, +∞), with the following boundary behaviors r(0) = 1 8 1 −4α2
, and r(s) = 3 2s 2 3 −αs 1 3 + O(1) as s →+∞. (1.21) The existence of such an r(s) follows directly from the vanishing lemma of the corresponding RH problem in Section 2.3 below.

THEOREM 1. Let Kn(x, y; t) be the kernel given in (1.6), then it has the Ψ-kernel asymptotic approximation 1 4nKn  u 4n, v 4n; t  = KΨ(u, v, 2nt) + O  1 n2  (1.25) as n →∞, uniformly for u, v in compact subsets of (0, ∞) and uniformly for t in (0, d], where d is a positive constant. And the Ψ-kernel is given by

COROLLARY 1. Let Kn(x, y) be the kernel given in (1.6). If the parameter t →0 and n →∞in the way such that lim n→∞2nt = τ, τ ∈(0, ∞), we have the double scaling limit for Kn(x, y) given in terms of the Ψ-kernel defined in (1.26) lim n→∞ 1 4nKn  u 4n, v 4n; t  = KΨ(u, v, τ)

THEOREM 2. We obtain the Bessel type limit for small parameter. (a) The Ψ-kernel is approximated by the Bessel kernel as s →0+ KΨ(u, v, s) = Jα(u, v) + O(s), (1.28) where the error term is uniform for u and v in compact subsets of (0, ∞). The Bessel kernel Jα is defined in (1.10). (b) If the parameter t →0+ and n →∞such that lim n→∞2nt = 0, we have the Bessel kernel limit for Kn: lim n→∞ 1 4nKn  u

THEOREM 3. We obtain the Airy type limit for large parameter. (a) The Ψ-kernel is approximated by the Airy kernel as s →+∞ s4/9 c KΨ  s2/3  1 − u cs2/9  , s2/3  1 − v cs2/9 

LEMMA 1. Assume that the homogeneous RH problem for Ψ(1)(ζ, s) adapting the same jump conditions (2.1) and the same boundary condition (2.3) as Ψ(ζ, s), with the behavior (2.2) at infinity being altered to Ψ(1)(ζ, s) = O 1 ζ  ζ−1 4 σ3 I + iσ1 √ 2 e √ζσ3, arg ζ ∈(−π, π), ζ →∞.

LEMMA 2. For s ∈(0, ∞), there exists a unique solution to the RH problem (2.1)-(2.3) for Ψ(ζ, s). 16

COROLLARY 2. The initial values for the nonlinear equations for q(s), r(s) and t(s) in (2.8) and (2.9) can be determined, namely,        q(0) = 1 128(4α2 −1)(4α2 −9) r(0) = 1 8 1 −4α2