Large Gap Asymptotics for Airy Kernel Determinants with Discontinuities

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number m of discontinuities. These m-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the m-point determinants can be expressed asymptotically as the product of m 1-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.


Introduction
Airy kernel Fredholm determinants. The Airy point process or Airy ensemble [39,42] is one of the most important universal point processes arising in random matrix ensembles and other repulsive particle systems. It describes among others the eigenvalues near soft edges in a wide class of ensembles of large random matrices [16,21,22,25,40], the largest parts of random partitions or Young diagrams with respect to the Plancherel measure [5,13], and the transition between liquid and frozen regions in random tilings [32]. It is a determinantal point process, which means that correlation functions can be expressed as determinants involving a correlation kernel, which characterizes the process. This correlation kernel is given in terms of the Airy function by (1.1) Let us denote N A for the number of points in the process which are contained in the set A ⊂ R, let A 1 , . . . , A m be disjoint subsets of R, with m ∈ N >0 , and let s 1 , . . . , s m ∈ C. Then, the general theory of determinantal point processes [11,33,42] implies that where the right hand side of this identity denotes the Fredholm determinant of the operator χ ∪ j A j m j=1 (1 − s j )K Ai χ A j , with K Ai the integral operator associated to the Airy kernel and χ A the projection operator from L 2 (R) to L 2 (A). The integral kernel operator K Ai is trace-class when acting on bounded real intervals or on unbounded intervals of the form (x, + ∞). Note that, when s j = 0 for j ∈ K ⊂ {1, . . . , m}, the left-hand-side of (1.2) should be interpreted as In what follows, we take the special choice of subsets 3) The case m = 1 corresponds to the Tracy-Widom distribution [43], which can be expressed in terms of the Hastings-McLeod [29] (if s 1 = 0) or Ablowitz-Segur [1] (if s 1 ∈ (0, 1)) solutions of the Painlevé II equation. It follows directly from (1.2) that F(x; 0) is the probability distribution of the largest particle in the Airy point process. The function F(x; s) for s ∈ (0, 1) is the probability distribution of the largest particle in the thinned Airy point process, which is obtained by removing each particle independently with probability s. Such thinned processes were introduced in random matrix theory by Bohigas and Pato [9,10] and rigorously studied for the sine process in [15] and for the Airy point process in [14]. For m ≥ 1, F( x; s) is the probability to observe a gap on (x m , + ∞) in the piecewise constant thinned Airy point process, where each particle on (x j , x j−1 ) is removed with probability s j (see [18] for a similar situation, with more details provided). It was shown recently that the m-point determinants F( x; s) for m > 1 can be expressed identically in terms of solutions to systems of coupled Painlevé II equations [19,44], which are special cases of integro-differential generalizations of the Painlevé II equations which are connected to the KPZ equation [2,20]. We refer the reader to [19] for an overview of other probabilistic quantities that can be expressed in terms of F( x; s) with m > 1.
Large gap asymptotics. Since F( x; s) is a transcendental function, it is natural to try to approximate it for large values of components of x. Generally speaking, the asymptotics as components of x tend to + ∞ is relatively easy to understand and can be deduced directly from asymptotics for the kernel, but the asymptotics as components of x tend to −∞ are much more challenging. The problem of finding such large gap asymptotics for universal random matrix distributions has a rich history, for an overview see e.g. [35] and [26]. In general, it is particularly challenging to compute the multiplicative constant arising in large gap expansions explicitly. In the case m = 1 with s = 0, it was proved in [4,23] that F(x; 0) = 2 1 24 e ζ (−1) |x| − 1 8 e − |x| 3 12 (1 + o(1)), as x → −∞, (1.4) where ζ denotes the derivative of the Riemann zeta function. Tracy and Widom had already obtained this expansion in [43], but without rigorously proving the value 2 1 24 e ζ (−1) of the multiplicative constant. For m = 1 with s > 0, it is notationally convenient to write s = e −2πiβ with β ∈ iR, and it was proved only recently by Bothner and Buckingham [14] that (1)), as x → −∞, (1.5) where G is Barnes' G-function, confirming a conjecture from [8]. The error term in (1.5) is uniform for β in compact subsets of the imaginary line. We generalize these asymptotics to general values of m, for s 2 , . . . , s m ∈ (0, 1], and s 1 ∈ [0, 1], and show that they exhibit an elegant multiplicative structure. To see this, we need to make a change of variables s → β, by defining β j ∈ iR as follows. If s 1 > 0, we define β = (β 1  and if s 1 = 0, where E denotes the expectation associated to the law of the particles λ 1 ≥ λ 2 ≥ · · · conditioned on the event λ 1 ≤ x 1 . Main result for s 1 > 0. We express the asymptotics for the m-point determinant E( x; β) in two different but equivalent ways. First, we write them as the product of the determinants E(x j ; β j ) with only one singularity (for which asymptotics are given in (1.5)), multiplied by an explicit pre-factor which is bounded in the relevant limit. Secondly, we write them in a more explicit manner.
Remark 2. The above asymptotics have similarities with the asymptotics for Hankel determinants with m Fisher-Hartwig singularities studied in [17]. This is quite natural, since the Fredholm determinants E( x; β) and E 0 ( x; β 0 ) can be obtained as scaling limits of such Hankel determinants. However, the asymptotics from [17] were not proved in such scaling limits and cannot be used directly to prove Theorem 1.1. An alternative approach to prove Theorem 1.1 could consist of extending the results from [17] to the relevant scaling limits. This was in fact the approach used in [23] to prove (1.4) in the case m = 1, but it is not at all obvious how to generalize this method to general m. Instead, we develop a more direct method to prove Theorem 1.1 which uses differential identities for the Fredholm determinants F( x; s) with respect to the parameter s m together with the known asymptotics for m = 1. Our approach also allows us to compute the r -independent prefactor e −4π 2 1≤k< j≤m β j β k (τ k ,τ j ) in a direct way.
Average, variance, and covariance in the Airy point process. Let us give a more probabilistic interpretation to this result. For m = 1, we recall that E(x; β) = Ee −2πiβ N (x,+ ∞) , and we note that, as β → 0, Comparing this to the small β expansion of the right hand side of (1.11), we see that the average and variance of N (x,+∞) behave as x → −∞ like μ(x) and σ 2 (x). More precisely, by expanding the Barnes' G-functions (see [38, formula 5.17.3]), we obtain where γ E is Euler's constant, and asymptotics for higher order moments can be obtained similarly. At least the leading order terms in the above are in fact well-known, see e.g. [6,28,41]. 1 For m = 2, (1.9) implies that If we expand the above for small β (note that our result holds uniformly for β ∈ iR small), we recover the logarithmic covariance structure of the process N (x,+∞) (see e.g. [11,12,34]), namely we then see that the covariance of N (x 1 ,+∞) and N (x 2 ,+∞) converges as r → ∞ to (τ 1 , τ 2 ). Note in particular that (τ 1 , τ 2 ) blows up like a logarithm as τ 1 − τ 2 → 0, and that such log-correlations are common for processes arising in random matrix theory and related fields. We also infer that, given 0 > τ 1 > τ 2 , as r → +∞. We also mention that asymptotics for the first and second exponential moments Ee −2πiβ N (x,+∞) and Ee −2πiβ N (x 1 ,+∞) −2πiβ N (x 2 ,+∞) of counting functions are generally important in the theory of multiplicative chaos, see e.g. [3,7,37], which allows to give a precise meaning to limits of random measures like e −2πiβ N (x,+∞) +∞) , and which provides efficient tools for obtaining global rigidity estimates and statistics of extreme values of the counting function.
Main result for s 1 = 0. The asymptotics for the determinants F( x; s) if one or more of the parameters s j vanish are more complicated. If s j = 0 for some j > 1, we expect asymptotics involving elliptic θ -functions in analogy to [14], but we do not investigate this situation here. The case where the parameter s 1 associated to the rightmost inverval (x 1 , +∞) vanishes is somewhat simpler, and we obtain asymptotics for E 0 ( x; β 0 ) = F( x; s)/F(x 1 ; 0) in this case. We first express the asymptotics for E 0 ( x; β 0 ) in terms of a Fredholm determinant of the form E( y; β 0 ) with m − 1 jump discontinuities, for which asymptotics are given in Theorem 1.1. Secondly, we give an explicit asymptotic expansion for E 0 ( x; β 0 ). Theorem 1.2. Let m ∈ N >0 , let x = (x 1 , . . . , x m ) be of the form x = r τ with τ = (τ 1 , . . . , τ m ) and 0 > τ 1 > τ 2 > · · · > τ m , and define y = (y 2 , . . . , y m ) by y j = x j −x 1 . For any β 2 , . . . , β m ∈ iR, we have as r → +∞, The error term is uniformly small for β 2 , . . . , β m in compact subsets of iR, and for τ 1 , . . . , τ m such that τ 1 < −δ and min 1≤k≤m−1 {τ k − τ k+1 } > δ for some δ > 0. Equivalently, as r → +∞, with Remark 3. We can again give a probabilistic interpretation to this result. In a similar way as explained in the case s 1 > 0, we can expand the above result for m = 2 as β 2 → 0 to conclude that the mean and variance of the random counting function N (x 2 ,x 1 ) , conditioned on the event λ 1 ≤ x 1 , behave, in the asymptotic scaling of Theorem 1.2, like μ 0 (x) and σ 2 0 (x). Doing the same for m = 3 implies that the covariance of N (x 2 ,x 1 ) and N (x 3 ,x 1 ) converges to 0 (τ 2 , τ 3 ).

Remark 4.
Another probabilistic interpretation can be given through the thinned Airy point process, which is obtained by removing each particle in the Airy point process independently with probability s = e −2πiβ , s ∈ (0, 1). We denote μ (s) 1 for the maximal particle in this thinned process. It is natural to ask what information a thinned configuration gives about the parent configuration. For instance, suppose that we know that μ (s) 1 is smaller than a certain value x 2 , then what is the probability that the largest overall particle λ 1 = μ (0) 1 is smaller than x 1 ? For x 1 > x 2 , we have that the joint probability of the events μ (s) 1 < x 2 and λ 1 < x 1 is given by (see [19,Section 2]) P μ (s) If we set 0 > x 1 = r τ 1 > x 2 = r τ 2 and let r → +∞, Theorem 1.2 implies that This describes the tail behavior of the joint distribution of the largest particle distribution of the Airy point process and the associated largest thinned particle.
Outline. In Sect. 2, we will derive a suitable differential identity, which expresses the logarithmic partial derivative of F( x; s) with respect to s m in terms of a Riemann-Hilbert (RH) problem. In Sect. 3, we will perform an asymptotic analysis of the RH problem to obtain asymptotics for the differential identity as r → +∞ in the case where s 1 = 0. This will allow us to integrate the differential identity asymptotically and to prove Theorem 1.2 in Sect. 4. In Sect. 5 and in Sect. 6, we do a similar analysis, but now in the case s 1 > 0 to prove Theorem 1.1.

Differential Identity for F
Deformation theory of Fredholm determinants. In this section, we will obtain an identity for the logarithmic derivative of F( x; s) with respect to s m , which will be the starting point of our proofs of Theorems 1.1 and 1.2. To do this, we follow a general procedure known as the Its-Izergin-Korepin-Slavnov method [30], which applies to integral operators of integrable type, which means that the kernel of the operator can be written in x−y where f (x) and g(y) are column vectors which are such that f T (x)g(x) = 0. The operator K x, s defined by is of this type, since we can take .
Using general theory of integral kernel operators, if s m = 0, we have where R x, s is the resolvent operator defined by and where R x, s is the associated kernel. Using the Its-Izergin-Korepin-Slavnov method, it was shown in [19, proof of Proposition 1] that the resolvent kernel R x, s (ξ ; ξ) can be expressed in terms of a RH problem. For ξ ∈ (x m , x m−1 ), we have RH problem for .

(a)
: and oriented as in Fig. 1. (b) (ζ ) has continuous boundary values as ζ ∈ \{y 1 , . . . , y m } is approached from the left (+ side) or from the right (− side) and they are related by where we write y m = 0 and y 0 = +∞. (c) As ζ → ∞, there exist matrices 1 , 2 depending on x, y, s but not on ζ such that has the asymptotic behavior and where principal branches of ζ 3/2 and ζ 1/2 are taken.
We can conclude from this result that (2.5) From here on, we could try to obtain asymptotics for with y replaced by r y as r → +∞. However, we can simplify the right-hand side of the above identity and evaluate the integral explicitly. To do this, we follow ideas similar to those of [14, Section 3].
Lax pair identities. We know from [19, Section 3] that satisfies a Lax pair. More precisely, if we define then we have the differential equation where A is traceless and takes the form for some matrices A j independent of ζ , and where σ + = 0 1 0 0 . Therefore, we have and we can use the relation −i∂ x 1,21 + 2 1,21 = 2 1,11 (see [19, (3.20)]) to see that A takes the form where the matrices A j (x) are independent of ζ and have zero trace. It follows that From the RH problem for , F satisfies the following RH problem (recall that y m = 0): The jumps are given by Thus, by Cauchy's formula, we have Expanding the right-hand-side of (2.12) as ζ → ∞, and comparing it with (2.11), we obtain the identities (2.14) Following again [19], see in particular formula (3.15) in that paper, we can express in a neighborhood of y j as for 0 < arg(ζ − y j ) < 2π 3 and with G j analytic at y j . This implies that where we denoted s m+1 = 1, and also that The above sum can be simplified using the fact that det G j ≡ 1, and we finally get where s m+1 = 1. The only quantities appearing at the right hand side are 1 , 2,21 and G j . In the next sections, we will derive asymptotics for these quantities as x = r τ with r → +∞.

Asymptotic Analysis of RH Problem for with s 1 = 0
We now scale our parameters by setting x = r τ , y = r η, with η j = τ j − τ m . We assume that 0 > τ 1 > · · · > τ m . The goal of this section is to obtain asymptotics for as r → +∞. This will also lead us to large r asymptotics for the differential identity (2.20). In this section, we deal with the case s 1 = 0. The general strategy in this section has many similarities with the analysis in [17], needed in the study of Hankel determinants with several Fisher-Hartwig singularities.

Re-scaling of the RH problem.
Define the function T (λ) = T (λ; η, τ m , s) as follows, The asymptotics (2.4) of then imply after a straightforward calculation that T behaves as as λ → ∞, where the principal branches of the roots are chosen. The entries of T 1 and T 2 are related to those of 1 and 2 in (2.4): we have where A = (η 2 1 + 2τ m η 1 ). The singularities in the λ-plane are now located at the (non-positive) points λ j = η j − η 1 = τ j − τ 1 , j = 1, . . . , m.

Normalization with g-function and opening of lenses.
In order to normalize the RH problem at ∞, in view of (3.2), we define the g-function by once more with principal branches of the roots. Also, around each interval (λ j , λ j−1 ), j = 2, . . . , m, we will split the jump contour in three parts. This procedure is generally called the opening of the lenses. Let us consider lens-shaped contours γ j,+ and γ j,− , lying in the upper and lower half plane respectively, as shown in Fig. 2. Let us also denote j,+ (resp. j,− ) for the region inside the lenses around (λ j , λ j−1 ) in the upper half plane (resp. in the lower half plane). Then we define S by (3.5) In order to derive RH conditions for S, we need to use the RH problem for , the definitions (3.1) of T and (3.5) of S, and the fact that g + (λ)+g − (λ) = 0 for λ ∈ (−∞, 0). This allows us to conclude that S satisfies the following RH problem.
and S oriented as in Fig. 2. (b) The jumps for S are given by Let us now take a closer look at the jump matrices on the lenses γ j,± . By (3.4), we have It follows that the jumps for S are exponentially close to I as r → +∞ on the lenses, and on λ m + e ± 2πi 3 (0, +∞). This convergence is uniform outside neighborhoods of λ 1 , . . . , λ m , but is not uniform as r → +∞ and simultaneously λ → λ j , j ∈ {1, . . . , m}.

Global parametrix.
We will now construct approximations to S for large r , which will turn out later to be valid in different regions of the complex plane. We need to distinguish between neighborhoods of each of the singularities λ 1 , . . . , λ m and the remaining part of the complex plane. We call the approximation to S away from the singularities the global parametrix. To construct it, we ignore the jump matrices near λ 1 , . . . , λ m and the exponentially small entries in the jumps as r → +∞ on the lenses γ j,± . In other words, we aim to find a solution to the following RH problem.

RH problem for P
The jumps for P (∞) are given by The solution to this RH problem is not unique unless we specify its local behavior as λ → 0 and as λ → λ j . We will construct a solution P (∞) which is bounded as λ → λ j for j = 2, . . . , m, and which is O(λ − 1 4 ) as λ → 0. We take it of the form with D a function depending on the λ j 's and s, and where we define d 1 below. In order to satisfy the above RH conditions, we need to take For later use, let us now take a closer look at the asymptotics of P (∞) as λ → ∞ and as λ → λ j . For any k ∈ N N >0 , as λ → ∞ we have, 13) and this also defines the value of d 1 in (3.10). A long but direct computation shows that (3.14) To study the local behavior of P (∞) near λ j , it is convenient to use a different representation of D, namely where . (3.16) From this representation, it is straightforward to derive the following expansions. As λ → λ j , j ∈ {2, . . . , m}, λ > 0, we have For j ∈ {2, . . . , m}, as λ → λ k , k ∈ {2, . . . , m}, k = j, j − 1, λ > 0, we have (3.17) Note that T j,k = T k, j for j = k and T j,k > 0 for all j, k. From the above expansions, we obtain, as λ → λ j , λ > 0, j ∈ {2, . . . , m}, that (3.18) where β 1 , . . . , β m are as in (1.6). The first two terms in the expansion of D(λ) as λ → λ 1 = 0 are given by where The above expressions simplify if we write them in terms of β 2 , . . . , β m defined by (1.6).
We also have the identity which will turn out useful later on.

Local parametrices.
As a local approximation to S in the vicinity of λ j , j = 1, . . . , m, we construct a function P (λ j ) in a fixed but sufficiently small (such that the disks do not intersect or touch each other) disk D λ j around λ j . This function should satisfy the same jump relations as S inside the disk, and it should match with the global parametrix at the boundary of the disk. More precisely, we require the matching condition uniformly for λ ∈ ∂D λ j . The construction near λ 1 is different from the ones near λ 2 , . . . , λ m .

3.4.1.
Local parametrices around λ j , j = 2, . . . , m. For j ∈ {2, . . . , m}, P (λ j ) can be constructed in terms of Whittaker's confluent hypergeometric functions. This type of construction is well understood and relies on the solution HG (z) to a model RH problem, which we recall in "Appendix A.3" for the convenience of the reader. For more details about it, we refer to [17,27,31]. Let us first consider the function defined in terms of the g-function (3.4). This is a conformal map from D λ j to a neighborhood of 0, which maps R ∩ D λ j to a part of the imaginary axis. As λ → λ j , the expansion of f λ j is given by We need moreover that all parts of the jump contour S ∩ D λ j are mapped on the jump contour for HG , see Fig. 6. We can achieve this by choosing 2 , 3 , 5 , 6 in such a way that f λ j maps the parts of the lenses γ j,+ , γ j,− , γ j+1,+ , γ j+1,− inside D λ j to parts of the respective jump contours 2 , 6 , 3 , 5 for HG in the z-plane. We can construct a suitable local parametrix P (λ j ) in the form If E λ j is analytic in D λ j , then it follows from the RH conditions for HG and the construction of f λ j that P (λ j ) satisfies exactly the same jump conditions as S on S ∩D λ j . In order to satisfy the matching condition (3.23), we are forced to define E λ j by (3.27) Using the asymptotics of HG at infinity given in (A.13), we can strengthen the matching condition (3.23) to as r → +∞, uniformly for λ ∈ ∂D λ j , where HG,1 is a matrix specified in (A.14). Also, a direct computation shows that 3.4.2. Local parametrix around λ 1 = 0. For the local parametrix P (0) near 0, we need to use a different model RH problem whose solution Be (z) can be expressed in terms of Bessel functions. We recall this construction in "Appendix A.2", and refer to [36] for more details. Similarly as for the local parametrices from the previous section, we first need to construct a suitable conformal map which maps the jump contour S ∩ D 0 in the λ-plane to a part of the jump contour Be for Be in the z-plane. This map is given by and it is straightforward to check that it indeed maps D 0 conformally to a neighborhood of 0. Its expansion as λ → 0 is given by We can choose the lenses γ 2,± in such a way that f 0 maps them to the jump contours e ± 2πi 3 R + for Be . If we take P (0) of the form with E 0 analytic in D 0 , then it is straightforward to verify that P (0) satisfies the same jump relations as S in D 0 . In addition to that, if we let then matching condition (3.23) also holds. It can be refined using the asymptotics for Be given in (A.7): we have as r → +∞ uniformly for z ∈ ∂D 0 . Also, after a direct computation in which we use (3.19) and (3.32) yields (3.36) 3.5. Small norm problem. Now that the parametrices P (λ j ) and P (∞) have been constructed, it remains to show that they indeed approximate S as r → +∞. To that end, we define (3.37) Since the local parametrices were constructed in such a way that they satisfy the same jump conditions as S, it follows that R has no jumps and is hence analytic inside each of the disks D λ 1 , . . . , D λ m . Also, we already knew that the jump matrices for S are exponentially close to I as r → +∞ outside the local disks on the lips of the lenses, which implies that the jump matrices for R are exponentially small there. On the boundaries of the disks, the jump matrices are close to I with an error of order O(r −3/2 ), by the matching conditions (3.35) and (3.28). The error is moreover uniform in τ as long as the τ j 's remain bounded away from each other and from 0, and uniform for β j , j = 2, . . . , m, in a compact subset of iR. By standard theory for RH problems [21], it follows that R exists for sufficiently large r and that it has the asymptotics is the jump contour for the RH problem for R, and with the same uniformity in τ and β 2 , . . . , β m as explained above. The remaining part of this section is dedicated to computing R (1) (λ) explicitly for λ ∈ C\ m j=1 D λ j and for λ = 0. Let us take the clockwise orientation on the boundaries of the disks, and let us write J R (λ) = R −1 − (λ)R + (λ) for the jump matrix of R as λ ∈ R . Since R satisfies the equation and since J R has the expansion as r → +∞ uniformly for λ ∈ m j=1 ∂D λ j , while it is exponentially small elsewhere on R , we obtain that R (1) can be written as If λ ∈ C\ m j=1 D λ j , by a direct residue calculation, we have We will also need asymptotics for R(0). By a residue calculation, we obtain The above residue at 0 is more involved to compute, but after a careful calculation we obtain (3.45) In addition to asymptotics for R, we will also need asymptotics for ∂ s m R. For this, we note that ∂ s m R(λ) tends to 0 at infinity, that it is analytic in C\ R , and that it satisfies the jump relation This implies the integral equation where the extra logarithm in the error term is due to the fact that ∂ s m |λ j | β j = O(log r ). Standard techniques [24] then allow one to deduce from the integral equation that as r → +∞.

Integration of the Differential Identity
The differential identity (2.20) can be written as where we set s m+1 = 1 as before.

Asymptotics for A τ , s (r ).
For |λ| large, more precisely outside the disks D λ j , j = 1, . . . , m and outside the lens-shaped regions, we have by (3.37). As λ → ∞, we can write for some matrices R 1 , R 2 which may depend on r and the other parameters of the RH problem, but not on λ. Thus, by (3.7) and (3.9), we have Using (3.38) and the above expressions, we obtain as r → +∞, where R (1) 1 and R (1) 2 are defined through the expansion After a long computation with several cancellations using (3.3), we obtain that A τ , s (r ) has large r asymptotics given by Using (1.6), (3.14) and (3.41)-(3.45), we can rewrite this more explicitly as where we recall the definition (3.13) of d 1 = d 1 ( s) and d 2 = d 2 ( s).

Asymptotics for B
( j) τ , s (r ) with j = 1. Now we focus on (ζ ) with ζ near y j . Inverting the transformations (3.37) and (3.5), and using the definition (3.26) of the local parametrix P (λ j ) , we obtain that for z outside the lenses and inside D λ j , j ∈ {2, . . . , m},

Evaluation of B
( j, 3) τ , s (r ). The last term B ( j, 3) τ , s (r ) is the easiest to evaluate asymptotically as r → +∞. By (3.38) and (3.46), we have that Moreover, from (3.29), since β j ∈ iR, we know that E λ j (λ j ) = O (1). Using also the fact that HG (0; β j ) is independent of r , we obtain that 3) τ , s (r ) = O(r −3/2 log r ), r → +∞. (4.7) Evaluation of B τ , s (r ), we need to use the explicit expression for the entries in the first column of HG given in (A.19). Together with (1.6), this implies that Using also the function relations for m ≥ 3; for m = 2 the formula is correct only if we set β 1 = 0, which we do here and in the remaining part of this section, such that the first term vanishes.

Proof of Theorem 1.2.
We now prove Theorem 1.2 by induction on m. For m = 1, the result (1.4) is proved in [14], and we work under the hypothesis that the result holds for values up to m − 1. We can thus evaluate F(r τ ; s 0 ) asymptotically, since this corresponds to an Airy kernel Fredholm determinant with only m − 1 discontinuities. In this way, we obtain after another straightforward calculation the large r asymptotics, uniform in τ and β 2 , . . . , β m , This implies the explicit form (1.14) of the asymptotics for E 0 (r τ ; β 0 ) = F(r τ ; s)/ F(r τ 1 ; 0). The recursive form (1.9) of the asymptotics follows directly by relying on (1.4) and (1.11). Note that we prove (1.11) independently in the next section.

Asymptotic Analysis of RH Problem for with s 1 > 0
We now analyze the RH problem for asymptotically in the case where s 1 > 0. Although the general strategy of the method is the same as in the case s 1 = 0 (see Sect. 3), several modifications are needed, the most important ones being a different g-function and the construction of a different local Airy parametrix instead of the local Bessel parametrix which we needed for s 1 = 0. We again write x = r τ and y = r η, with η j = τ j − τ m .

Re-scaling of the RH problem.
We define T , in a slightly different manner than in (3.1), as follows, Similarly as in the case s 1 = 0, because of the triangular pre-factor above, we then have as λ → ∞, but with modified expressions for the entries of T 1 and T 2 : The singularities of T now lie at the negative points λ j = τ j , j = 1, . . . , m.
where j,± are lens-shaped regions around (λ j , λ j−1 ) as before, but where we note that the index j now starts at j = 1 instead of at j = 2, and where we define λ 0 := 0, see Fig. 3 for an illustration of these regions. Note that λ 0 is not a singular point of the RH problem for T , but since λ 3/2 = 0 on (−∞, 0), it plays a role in the asymptotic analysis for S. S satisfies the following RH problem.

Global parametrix.
The RH problem for the global parametrix is as follows.
(c) As λ → ∞, we have This RH problem is of the same form as the one in the case s 1 = 0, but with an extra jump on the interval (λ 1 , λ 0 ). We can construct P (∞) in a similar way as before, by setting with We emphasize that the sum in the above expression now starts at j = 1. For any positive integer k, as λ → ∞ we have This defines the value of d 1 in (5.11), and with these values of d 1 , d 2 , the expressions (3.14) for P (∞) 1 and P (∞) 2 remain valid. As before, we can also write D as This expression allows us, in a similar way as in Sect. 3, to expand D(λ) as λ → λ j , λ > 0, j ∈ {1, . . . , m}, and to show that with T k, j as in (3.17) and the equations just above (3.17) (which are now defined for k, j ≥ 1). The first two terms in the expansion of D(λ) as λ → λ 0 = 0 are given by where Note again, for later use, that for all ∈ {0, 1, 2, . . .}, we can rewrite d in terms of the β j 's as follows,

5.4.1.
where the principal branch of (−λ) 3/2 is chosen. This is a conformal map from D λ j to a neighborhood of 0, satisfies f λ j (R ∩ D λ j ) ⊂ iR, and its expansion as λ → λ j is given by Similarly as in Sect. 3.4.1, we define where HG is the confluent hypergeometric model RH problem presented in "Appendix A.3" with parameter β = β j . The function E λ j is analytic inside D λ j and is given by (5.24) We will need a more detailed matching condition than (5.20), which we can obtain from (A.13): as r → +∞ uniformly for λ ∈ ∂D λ j . Moreover, we note for later use that

5.4.2.
Local parametrices around λ 1 = 0. The local parametrix P (0) can be explicitly expressed in terms of the Airy function. Such a construction is fairly standard, see e.g. [21,22]. We can take P (0) of the form for λ in a sufficiently small disk D 0 around 0, and where Ai is the Airy model RH problem presented in "Appendix A.1". The function E 0 is analytic inside D 0 and is given by A refined version of the matching condition (5.20) can be derived from (A.2): one shows that as r → +∞ uniformly for z ∈ ∂D 0 , where Ai,1 is given below (A.2). An explicit expression for E 0 (0) is given by 5.5. Small norm problem. As in Sect. 3.5, we define R as and we can conclude in the same way as in Sect. 3.5 that (3.38) and (3.46) hold, uniformly for β 1 , β 2 , . . . , β m in compact subsets of iR, and for τ 1 , . . . , τ m such that τ 1 < −δ and min 1≤k≤m−1 {τ k − τ k+1 } > δ for some δ > 0, with where J R is the jump matrix for R and J (1) R is defined by (3.39). A difference with Sect. 3.5 is that J (1) R now has a double pole at λ = 0, by (5.30). At the other singularities λ j , it has a simple pole as before. If λ ∈ C\ m j=0 D λ j , a residue calculation yields where we set s m+1 = 1. We assume in what follows that m ≥ 2.
For the computation of A τ , s (r ), we start from the expansion (4.3), which continues to hold for s 1 > 0, but now with P (∞) 1 and P (∞) 2 as in Sect. 5 (i.e. defined by (3.14) but with d 1 , d 2 given by (5.18)), and with R (1) 1 and R (1) 2 defined through the expansion τ , s (r ), we proceed as before by splitting this term in the same way as in (4.6). We can carry out the same analysis as in Sect. 4 for each of the terms. We note that the terms corresponding to j = 1 can now be computed in the same way as the terms j = 2, . . . , m. This gives, analogously to (4.10), 2β j ∂ s m log j Substituting this identity and the fact that λ j = τ j , we find after a straightforward calculation [using also (1.6)] that, uniformly in τ and β as r → +∞, We are now ready to integrate this in s m . Recall that we need to integrate s m = e −2πiβ m from 1 to s m = e −2πiβ m , which means that we let β m go from 0 to − log s m 2πi , and at the same time β m−1 go fromβ m−1 := − log s m−1 2πi to β m−1 = log s m 2πi − log s m−1 2πi . We then obtain, using (4.19) and ( From this expansion, it is straightforward to derive (1.11). The expansion (1.9) follows from (1.5) after another straightforward calculation. This concludes the proof of Theorem 1.1.

A Model RH Problems
In this section, we recall three well-known RH problems: (1) the Airy model RH problem, whose solution is denoted Ai , (2) the Bessel model RH problem, whose solution is denoted by Be , and (3) the confluent hypergeometric model RH problem, which depends on a parameter β ∈ iR and whose solution is denoted by HG (·) = HG (·; β).
A.1 Airy model RH problem.
(a) Ai : C\ A → C 2×2 is analytic, and A is shown in Fig. 4. 3) The Airy model RH problem was introduced and solved in [24] (see in particular [24, equation (7.30)]). We have (c) As z → ∞, z / ∈ Be , we have Be (z) = (2π z where Be,1 = 1 16 −1 −2i −2i 1 . (d) As z tends to 0, the behavior of Be (z) is (A.8) This RH problem was introduced and solved in [36]. Its unique solution is given by where * denotes entries whose values are unimportant for us.

A.3 Confluent hypergeometric model RH problem.
(a) HG : C\ HG → C 2×2 is analytic, where HG is shown in Fig. 6.  Fig. 6. The jump contour HG for HG . The ray k is oriented from 0 to ∞, and forms an angle with R + which is a multiple of π