Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta > 0$$\end{document}θ>0 and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$\end{document}P(gapon[0,s])=Cexp-as2ρ+bsρ+clns(1+o(1))ass→+∞,where the constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}ρ, a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann–Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}θ. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}θ is rational, we find that C can be expressed in terms of Barnes’ G-function. We also show that the asymptotic formula can be extended to all orders in s.


Introduction and Main Results
The Muttalib-Borodin ensembles are joint probability density functions of the form where the n points x 1 , . . . , x n belong to the interval [0, +∞), θ > 0 is a parameter of the model, and Z n is a normalization constant. The positive weight function w is defined on [0, +∞) and is assumed to have enough decay at ∞ to make (1.1) a well-defined density function. The probability density function (1.1) exhibits so-called two-body interactions-in addition to the repulsion between the points x 1 , . . . , x n , there is also repulsion between the points x θ 1 , . . . , x θ n . The models defined by (1.1) were introduced by Muttalib in 1995 in the study of disordered conductors in the metallic regime [26]. They have attracted a lot of attention recently in the random matrix community, partly due to the work of Cheliotis [10] who showed that the squared singular values of certain lower triangular random matrices have the same joint density as (1.1) in the case of the Laguerre weight w(x) = x α e −x , α > −1. Other matrix ensembles whose eigenvalues are distributed according to (1.1) for the Laguerre or Jacobi weight were obtained in [18].
As n → +∞ the macroscopic behavior of the points x 1 , . . . , x n is well described by an equilibrium measure μ which depends on the weight w. Such measures have been studied in detail in [11] for general values of θ . In particular, the authors of [11] found sufficient conditions on w for μ to be supported on a single cut. If there is a hard edge (that is, if part of the points accumulate near the origin as n → +∞), the density of μ behaves as a constant times x − 1 1+θ as x → 0 + . On the other hand, near a soft edge, this density vanishes to the order 1/2 for any value of θ ; this is the usual square root behavior that is often encountered in random matrix theory. We also refer to [5,8,23] for related results on the equilibrium measure.
The Muttalib-Borodin point process is determinantal for any θ > 0. This means that the density (1.1), as well as all the associated correlation functions, can be expressed as determinants involving a function K n (general definitions and properties of point processes can be found in [7,21,28]). This function K n is called the kernel and encodes all the probabilistic information about the point process. In the simplest case θ = 1, the point process is a polynomial ensemble. This means that all the correlation functions can be expressed in terms of orthogonal polynomials (associated to w), and that there exists a Christoffel-Darboux formula which can be utilized to derive asymptotic formulas as n → +∞. For θ = 1, the point process is still determinantal; however the aforementioned properties become more complicated for rational values of θ , and are lost if θ is irrational. In fact, for θ = 1, the kernel is instead expressed in terms of biorthogonal polynomials [6], for which there is no simple analog of the Christoffel-Darboux formula (when θ is an integer, the Christoffel-Darboux formula contains θ terms, see [20]).
As n → +∞, the local repulsion of the points leads to microscopic limit laws that depend on the location. The term microscopic refers to the fact that the correlation is measured in the unit of the mean level spacing. For θ = 1, three different canonical limiting kernels arise: the sine kernel arises in the bulk, the Airy kernel near soft edges, and the Bessel kernel near (typical) hard edges. The three limiting kernels are independent of the fine details of the weight; this phenomenon is called universality in random matrix theory [22]. Also, the kernels are all integrable (of size 2) in the sense of Its-Izergin-Korepin-Slavnov [19], and there are 2 × 2 matrix Riemann-Hilbert (RH) problems available for the asymptotics analysis. Much less is known for θ = 1. In the case of the Laguerre weight w(x) = x α e −x , α > −1, Borodin proved in his pioneering work [6] that for any α > −1 and θ > 0, where the limiting kernel K(x, y) depends on α and θ and can be expressed in terms of Wright's generalized Bessel functions (see also (1.3) below). If θ = p/q with p, q relatively prime integers, then the kernel K is integrable, but of size p + q [31], which means that the associated RH problems involve matrices of size ( p + q) × ( p + q). In the Jacobi case (i.e., the weight is supported on [0, 1] and given by w(x) = x α , α > −1), Borodin proved that the same limiting kernel K appears at the hard edge if a slightly different scaling limit is considered (the terms n 1 θ in (1.2) need to be replaced by n 1+ 1 θ ). It seems reasonable to expect some universality of this kernel, in the sense that K should appear in the hard edge scaling limit for a large class of weights of the form w(x) = x α e −V (x) for a smooth potential V . Moreover, from the behavior of μ described above, one expects the sine kernel in the bulk and the Airy kernel at the soft edge for a large class of weights. This has been proved in the special case θ = 1 2 only recently by Kuijlaars and Molag in [24] using a non-standard analysis of a 3 × 3 matrix RH problem. The case of general θ is still open. We also mention that, in case θ or 1/θ is an integer, the kernel K can be expressed in terms of a Meijer function and coincides with the limiting kernel at the hard edge of certain product random matrices [25,Theorem 5.1].
There are several expressions available in the literature for the kernel K(x, y) in (1.2); in [6] it is written as a series, and also in terms of Wright's generalized Bessel functions. For us, the following double contour integral expression (from [12]) will be important: x, y > 0, (1.3) where the function F is given by with denoting the Gamma function (see [27,Chapter 5]). The contours γ andγ are both oriented upward and do not intersect each other; the contour γ intersects R to the right of the poles of F andγ intersects R to the left of the zeros of F, see Fig. 1. The contour γ tends to infinity in sectors lying strictly in the left half-plane, andγ tends to infinity in sectors lying strictly in the right half-plane. If θ = 1, the kernel K reduces to K(x, y)| θ=1 = 4K Be (4x, 4y), x, y > 0, (1.5) where K Be is the well-known Bessel kernel [29] given by with J α the Bessel function of the first kind of order α. By [12, equation (1.18)], the finite n probability to observe a gap on [0, n − 1 θ s] converges as n → +∞ to the probability to observe a gap on [0, s] in the limiting process with kernel K. This is a slightly stronger result than the convergence of the kernel (1.2). Let x := min{x 1 , . . . , x n } denote the smallest point. Then the limiting distribution of x is given by s > 0, (1.6) where the right-hand side is the Fredholm determinant associated to K on the interval [0, s]. For θ = 1, Tracy and Widom have shown in [29] that the log s-derivative of this Fredholm determinant solves a Painlevé V equation. In the case of θ rational, a more involved system of differential equations has been derived recently in [31]. In the case θ = 1, the large gap asymptotics (i.e., the asymptotics of (1.6) as s → +∞) are known from Deift, Krasovsky and Vasilevska [15,Theorem 4] where it was shown that 1 where G is Barnes' G-function (see [27,Chapter 5]). The study of the general case θ > 0 has been initiated by Claeys, Girotti and Stivigny in the recent paper [12]. They obtained the asymptotic formula det 1 − K| [0,s] = C exp − as 2ρ + bs ρ + c ln s + O(s −ρ ) as s → +∞, (1.8) where the real constants ρ, a, and b are explicitly given by (1.9) It is quite remarkable that, even though the kernel K is known to be integrable only for rational θ , they managed to obtain an asymptotic formula valid for any fixed θ > 0 (we comment on their method below).

Remark 1.3 (The constant d).
The constant d = d(θ, α) (constant in the sense that it is independent of s) is defined by the limit in (1.12) in a similar way as the Euler gamma constant γ E , which appears in the definition of G, is defined by (see [27,Eq. 5.2.3]) The definition of d can also be compared with the following expression for the derivative of the Riemann ζ -function evaluated at −1 (see [27,Eq. 5.17.7]): In fact, comparing the above expression with the definition (1.12) of d, we see that 2 d(1, −1) = ζ (−1).

Proposition 1.4 (Expression for d(θ, α)
when θ = p/q is rational). Let α > −1 and θ = p/q where p, q ∈ N\{0}. Then d(θ, α) admits the following expression: Quantities such as ζ (−1) or G(1 + α) appear in several asymptotic formulas in random matrix theory. For example, ζ (−1) appears in the large gap asymptotics of the Airy point process [13] and in the asymptotics of the partition function for a large class of random matrix ensembles [9, equations (1.38)-(1.40)], while the Barnes G-function appears in the large gap asymptotics of the Bessel point process (see 1.7) and in the asymptotics of large Toeplitz and Hankel determinants with Fisher-Hartwig singularities [9,14]. However, despite its relatively simple definition, we have not been able to express d in terms of known special functions for irrational values of θ . Remark 1.5 (The symmetry θ → 1 θ ). By [6, page 4], the determinant on the left-hand side of (1.8) is invariant under the following changes of the parameters: It follows that the coefficients ρ, a, b, c, and C must obey the following symmetry relations for any θ > 0 and α > −1: where we have indicated the dependence of the coefficients on θ and α explicitly. The first four of these relations are easily verified directly from the definitions (1.9)-(1.10) by simple computations. The relation C(θ, α) = C( 1 θ , α ) can also be verified directly from the definition (1.11) of C, but the computations are more involved. In fact, a long but straightforward computation which uses (1.16) and the functional relation G(z +1) = (z)G(z) implies that the relation C(θ, α) = C( 1 θ , α ) is equivalent to the symmetry relation for d given in the following proposition.
Our second main result shows that the expansion (1.8) of the Fredholm determinant of K on [0, s] can be extended to all orders in powers of s −ρ as s → +∞. More precisely, we have the following. Theorem 1.7 (Asymptotics to all orders). Let N ≥ 1 be an integer and fix θ > 0 and α > −1. As s → +∞, there exist constants C 1 , . . . , C N ∈ R such that where K is the kernel defined in (1.3) and ρ, a, b, c, C are given by (1.9)-(1.11).

Outline of proofs.
Our proof of Theorem 1.1 is based on some preliminary results from [12]. An important and remarkable ingredient of that paper (inspired by [4]) is the identity where the integrable kernel M s of size 2 × 2 is given for any θ > 0 by with χ γ and χγ denoting the indicator functions of γ andγ , respectively. Using some results from [3,4] and following the procedure developed by Its-Izergin-Korepin-Slavnov (IIKS) [19], the authors of [12] obtained a differential identity for in terms of the solution Y of a 2 × 2 matrix RH problem. Moreover, by performing a (non-standard) Deift/Zhou [17] steepest descent analysis of this RH problem, they computed the large s asymptotics of the expression in (1.24). The asymptotic formula (1.8) and the expressions (1.9) for the coefficients a and b were then obtained from the relation where M is a sufficiently large but fixed constant. In principle, the method of [12] can be employed to obtain any number of terms in the large s expansion of (1.24) (even though the computations become technically more involved as higher order terms are included). In particular, it is possible to compute the constant c by extending the expansion of (1.24) to the next order and then substituting the resulting asymptotics into the integrand of (1.25). However, the fact that the quantity is an unknown constant (independent of s) is an essential obstacle to the computation of C, see also [12,Remark 1.3]. Therefore, in the present work we adopt a different approach which takes advantage of the known result for θ = 1 given in (1.7).
Whereas the approach of [12] is based on a differential identity in s, our approach relies on a differential identity in θ . More precisely, using (1.22)-(1.23) and results from [3, Section 5.1], we apply the IIKS procedure [19] to obtain a differential identity for (1. 26) in terms of the solution Y of the RH problem of [12] mentioned above (henceforth referred to as the RH problem for Y ). By recycling the steepest descent analysis of [12], we obtain asymptotics of Y as s → +∞. The steepest descent analysis in [12] was performed for θ fixed, but we can easily show that the resulting asymptotic formulas are in fact valid uniformly for θ in any compact subset of (0, +∞). An integration of (1.26) from θ = 1 to an arbitrary (but fixed) θ > 0 then gives The main advantage of this approach is that the large s asymptotics of (1.28) are known (including the constant term), see (1.7). Therefore, if we compute the asymptotics of (1.26) to sufficiently high order and substitute the resulting expansion into (1.27) (using the uniformity of this expansion with respect to θ ), we can obtain C by performing the integral with respect to θ .
Using the symmetries in (1.19), which we recall can be verified directly from the explicit expressions for ρ, a, b, c, C in (1.9)-(1.11) (see Remark 1.5), the statement of Theorem 1.1 follows also for θ ∈ [1, ∞). A similar argument applies to Theorem 1.7. The upshot is that it is enough to prove Theorem 1.1 and Theorem 1.7 for θ ∈ (0, 1].

1.2.2.
Comparison with the approach of [12] Even though our approach has the major advantage of opening up a path to the evaluation of the constant C, there are several disadvantages of integrating with respect to θ instead of with respect to s. First, in [12] the authors were able to obtain expressions for the constants a and b at the hard edge not only for Muttalib-Borodin ensembles, but also for certain other limiting point processes arising from products of random matrices. This was feasible because s is a common parameter in all of these models and the associated differential identities could be analyzed in a similar way in all cases. Since the parameter θ is not present in the other models, our method of deforming with respect to θ can only be applied in the case of the Muttalib-Borodin ensembles. Second, integration with respect to θ requires significantly more computational effort than integration with respect to s. This can be seen by taking the logarithm of the asymptotic formula (1.8) and differentiating the resulting equation with respect to s and θ respectively: 3 as s → +∞. Note that the differentiation with respect to θ generates additional terms proportional to ln s. Moreover, the expansion in (1.30) involves the rather complicated first-order derivatives of a, b, c, and C with respect to θ . Third, it turns out that the differential identity with respect to θ is more intricate to analyze: While (1.24) is expressed in terms of the first subleading term in the expansion of Y (z) as z → +∞ (see (2.35)), the analogous representation for (1.26) involves an integral whose integrand also contains the digamma function ψ (see (6.1)). The infinitely many poles of the digamma function ψ (which we recall is defined as the log-derivative of , see e.g. [27,Eq. 5.2.2]) complicate the analysis considerably. For all the above reasons, we will in Sect. 5 provide an independent derivation of the expression (1.10) for c by employing the differential identity in s. This derivation is significantly shorter than the derivation based on the differential identity in θ and it can also be generalized to other point processes. In particular, from the formulas we obtain we can straightforwardly determine the constants c (1) and c (2) of [12, formula 1.20] associated with point processes at the hard edge of certain product random matrices, see Remark 5.2. Furthermore, several important aspects of this alternative derivation of (1.10) will be useful in the proofs of Theorem 1.7 and the expression (1.11) for C.
Finally, we note that the fact that the approach based on the differential identity in θ yields the same expressions (1.9) and (1.10) for the coefficients a, b, c as the approach based on the differential identity in s provides a nontrivial consistency check of our results.
1.3. Organization of the paper. In Sect. 2, we introduce some notation and recall some results from [12] that are needed for our analysis. In Sects. 3 and 4 , we establish the existence of large s asymptotics to all orders of three functions which play a pivotal role in the RH formulation. In Sect. 5, we use these expansions to prove Theorem 1.7 and to provide a first proof of the expression (1.10) for c.
In Sect. 6, we derive a differential identity with respect to the parameter θ . This identity expresses the θ -derivative of ln det( 1 − K| [0,s] ) as the sum of four integrals which we denote by I 1 , I 2 , I 3,K , and I 4,K . The arguments required to obtain the large s asymptotics of these integrals are rather long and are presented in Sects. 7-9.
We complete the proof of Theorem 1.1 in Sect. 10 by substituting the above asymptotics into the differential identity in θ and integrating the resulting equation with respect to θ . In addition to yielding the expression (1.11) for C, this also provides independent derivations of the expressions (1.9) and (1.10) for the coefficients a, b, and c.
The proofs of Propositions 1.4 and 1.6 as well as the proofs of two lemmas (Lemma 7.2 and Lemma 8.4) are presented in the four appendices.

Preliminary Results from [12]
All the results presented in this section are taken from [12]. We use the same notation as in [12] except that we use G to denote Barnes' G-function and G to denote the function which is denoted by G in [12]. We start by recalling the RH problem for Y , which is central for this paper.

RH problem for Y.
(a) Y : C\(γ ∪γ ) → C 2×2 is analytic, where γ andγ are the oriented contours shown in Fig. 1. (b) The limits of Y (z) as z approaches γ ∪γ from the left (+) and from the right (-) exist, are continuous on γ ∪γ , and are denoted by Y + and Y − , respectively. Furthermore, they are related by where F is given by (1.4). (c) As z → ∞, Y admits the expansion where the 2 × 2 matrix Y 1 depends on s, α, and θ but not on z.
The solution of the RH problem for Y exists and is unique for any choice of the parameters s > 0, θ > 0, and α > −1, see [12, below (1.21)].
We choose the branch for ln F such that where z → ln (z) is the log-gamma function, which has a branch cut along (−∞, 0]. Therefore, z → ln F(z) has a branch cut along (−∞, − α 2 ] ∪ [1 + α 2 , +∞). Following [12], we introduce a new complex variable ζ by As s ρ ζ → ∞, we have the asymptotics where the logarithms on the right-hand side are defined using the principal branch. The real constants c 1 , . . . , c 8 are computed in [12, equation (3.12)] and are given by 4 We also define G(ζ ) by where The function G above is denoted by G in [12, equation (3.13)], while in this paper G denotes Barnes' G-function. Note that G also depends on s, θ and α, but we omit this dependence in the notation. Following [12, Section 3.3], we define b 1 , b 2 ∈ C by Thus φ ≥ 0 for 0 < θ ≤ 1 while φ ≤ 0 for θ ≥ 1. As explained in Sect. 1.2.1, it is enough to prove Theorems 1.1 and 1.7 for θ ∈ (0, 1] thanks to the symmetry (1.18). Therefore we will henceforth restrict ourselves to the case 0 < θ ≤ 1, for which we have φ ∈ [0, π/2). In the steepest descent analysis of the RH problem for Y , the so-called g-function plays an important role. Using this function, certain jumps of the RH problem can be made exponentially small as s → +∞. The g-function has a jump along the contour 5 , which consists of the two line segments oriented to the right, see Fig. 2, and is defined as follows. Define the function r (ζ ) by where the branch is such that r is analytic in C\ 5 and r (ζ ) ∼ ζ as ζ → ∞. The second derivative of the g-function is given by (2.10) Hence and, as ζ → ∞, The g-function is then obtained by where the integration paths lie in the complement of 5 . The g-function is analytic on C\ 5 and has the following jump across 5 :

Steepest descent analysis.
Let σ 1 and σ 3 denote the first and third Pauli matrices given by The steepest descent analysis of the RH problem for Y involves a sequence of trans- The 2 × 2 matrix-valued function U is analytic on C\(γ U ∪γ U ), where see also [12, Figure 2]. Let { i } 4 1 denote the contours defined by with 0 < < π/10 and oriented from left to right, see Fig. 3.
The second transformation U → T consists of deforming the contour of the RH problem by considering an analytic continuation of U such that T is analytic in C\ ∪ 5 i=1 i ; we refer to [12, Section 3.2] for details.
The third transformation T → S uses the g-function and is defined by The remainder of the steepest descent analysis of [12] consists of finding good approximations of S in different regions of the complex plane. Define the function γ (ζ ) by where the branch is such that γ (ζ ) is an analytic function of ζ ∈ C\ 5 and γ (ζ ) ∼ 1 as ζ → ∞. Define also the function p : C\ 5 → C by where the branch for ln G is such that with ln F defined as in (2.1). Outside small neighborhoods of b 1 and b 2 , S is well approximated by the global parametrix P ∞ defined by The function p satisfies p(ζ ) = p(−ζ ) and where the constants p 0 ∈ R and p 1 ∈ iR are given by dξ, dξ.
2.1.1. The local parametrix P Near the points b 1 and b 2 , S is no longer well approximated by P ∞ , and we need to construct local approximations to S (also called local parametrices and denoted by P). Following [12], these local parametrices are built out of Airy functions and are defined in small open disks D δ (b 1 ) and D δ (b 2 ) centered at b 1 and b 2 , respectively: for some sufficiently small radius δ > 0 which is independent of s. Furthermore, P satisfies the following matching condition with P ∞ on the boundary ∂D δ (b 1 )∪∂D δ (b 2 ): . The local parametrix P obeys the symmetry and let the 2 × 2-matrix valued functions {A j (ζ )} 3 1 be given by These functions satisfy

The solution R
In view of (2.3) and (2.17), the function G(ζ ) is not bounded as s ρ ζ → ∞. Therefore, in the definition of the last transformation S → R, we need to multiply by a conjugation matrix e p 0 σ 3 in order for R to be uniformly bounded on C. 5 More precisely, we define R by (2.32) Then R(ζ ) is analytic for ζ ∈ C\ R where R consists of the parts of ∪ 5 i=1 i lying outside the disks D δ (b j ), j = 1, 2, as well as the two clockwise circles ∂D δ (b j ), j = 1, 2, see Fig. 4. We will show in Sect. 4 that R satisfies a small norm RH problem and that where the matrix R 1 possesses the asymptotics for a certain matrix R (1) 1 independent of s and ζ . 2. Differential identity in s. It was proved in [12] that, for all s > 0, (2.35) Furthermore, it was shown in [12, Section 4.3] that where K is defined via the expansion (see [12, Eq. (4.15)]) Integration of (2.36) yields the expressions in (1.9) for the first two coefficients a and b. Moreover, comparing (2.36) with (1.29) , we infer that the coefficient c can be expressed as Thus, to compute c it is enough to compute K and the (2, 2) entry of R 1 .

Asymptotics of G(ζ ) and p(ζ )
In this section, we establish asymptotic formulas for the functions G(ζ ) and p(ζ ) defined in (2.5) and (2.16) as s → +∞ with ζ such that s ρ ζ → ∞. More precisely, we will prove that ln G(ζ ) and p(ζ ) admit expansions to all orders in inverse powers of s ρ ζ and we will compute the coefficients of the expansion for p(ζ ) explicitly up to and including the term of order s −ρ ζ −1 (this term plays a role in the derivation of the expressions for both c and C). The results are summarized in the followings two propositions whose proofs are presented in Sects. 3.1 and 3.2 , respectively. We let {c j = c j (θ, α)} 8 1 and {b j = b j (θ, α)} 2 1 be the constants defined in (2.4) and (2.7), respectively. Proposition 3.1 (Asymptotics of ln G(ζ )). Let N ≥ 1 be an integer. Let α > −1 and as s ρ ζ → ∞ uniformly for θ in compact subsets of (0, 1] and ζ ∈ C such that | arg(ζ ) − π 2 | > and | arg(ζ ) + π 2 | > for any fixed > 0. The first coefficient is given by uniformly with respect to ζ ∈ C\ 5 such that s ρ ζ → ∞ and θ in compact subsets of (0, 1], where the functions R(ζ ) and A 1 (ζ ) are given by and is well-defined also for ζ ∈ iR\{0} even though several of the coefficients have jumps across the imaginary axis. Indeed, it can be seen from (3.3) (and more easily from the integral representation (3.38) of R) that R has the following jump across the imaginary axis: where (i∞, 0) and (−i∞, 0) are oriented towards the origin. It follows that the function 2 has no jump across the imaginary axis and hence extends to an analytic function on C\ 5 .
Remark 3.4. The expansion of ln G(ζ ) as s ρ ζ → ∞ up to and including the term of order s −ρ ζ −1 is easily obtained from (2.3) and (2.17), see [12,Eq. (3.15)]. The extension of this expansion to all orders is not straightforward and is the content of Proposition 3.1.
3.1. Proof of Proposition 3.1. We will employ the following exact representation for ln (z) (see [30, Eq. (6.34) with r = N and Eq. (6.38)]): which is valid for | arg z| < π, with N an arbitrary (but fixed) positive integer and where {t} denotes the fractional part of t, i.e., {t} = t − t where t is the largest integer smaller than or equal to t. Here B n is the nth Bernoulli number and B N (x) the N th Bernoulli polynomial given by (see e.g. [1, p. 804]) The first terms on the right-hand side of (3.5) are the same as in Stirling's approximation formula; however (3.5) is an exact identity which is valid for all z ∈ C such that | arg z| < π. It is straightforward to verify that (see [30, last equation on page 78] for details) for any fixed > 0. Using the short-hand notation we have Therefore, for all Hence, by (2.17) and (2.6) we have, for any fixed N ≥ 1, with the real constants a 1 , a 2 ,ã 1 ,ã 2 defined by respectively. The asymptotics off andf as s ρ ξ → ∞ are easily obtained from (3.12): as s ρ ξ → ∞ uniformly for θ in compact subsets of (0, 1], where the constants {a j ,ã j } 4 3 ⊂ R andf 1 ,f 1 ∈ iR are defined by , and {f n ,f n } N n=2 ⊂ C are constants whose exact expressions are unimportant for us. However, we note thatf n (θ, α) andf n (θ, α) are continuous functions of α and θ . From (3.7) and (3.8), we infer that for any > 0 uniformly for θ in compact subsets of (0, 1]. Substituting (3.14)-(3.18) into (3.11), we obtain (3.1) where the coefficients G n are given by G n =f n +f n ; in particular, G 1 = −ic 8 . This completes the proof of Proposition 3.1.

Proof of Proposition Recall that p(ζ ) is defined by
Since 5 passes through the origin, the large s asymptotics for p cannot be straightforwardly obtained from the asymptotics (3.1) of ln G(ζ ). We instead use formula (3.11) to be able to deform the contour 5 . Substituting (3.11) into the definition (3.19) of p(ζ ) yields The remainder of the proof is divided into three lemmas. The first lemma shows that the two integrals in (3.20) involving D N are small whenever s ρ and s ρ ζ are large. Lemma 3.6. For any integer N ≥ 1, it holds that uniformly for ζ ∈ C\ 5 such that s ρ ζ → ∞ and θ in compact subsets of (0, 1].

Fig. 5. The contoursσ andσ for ζ /
Proof. Given ζ ∈ C\ 5 , the integrand in (3.21a) is an analytic function of see (3.13). Using that r + (ξ ) + r − (ξ ) = 0 for ξ ∈ 5 , we can deform the contour 5 into another contourσ which crosses the imaginary axis below the origin such that , a representative choice of σ is shown in Fig. 5, and we obtain where, for a simple closed curve γ ⊂ C, we write int(γ ) and ext(γ ) for the open subsets of C interior and exterior to γ , respectively.
, then we use the jump relation of r (ζ ) to open up the parts ofσ close to the points b 1 and b 2 to two circles in such a way that Fig. 6, and instead of (3.23) we obtain can be treated similarly. Since (3.22) holds, we can apply (3.17), which implies as s → +∞ uniformly for ζ ∈ C\ 5 and θ in compact subsets of (0, 1]. The term for a certain > 0, and since s ρ ζ → ∞ by assumption, we can apply (3.17) to obtain This proves (3.21a). A similar argument based on deforming 5 into a contourσ which crosses the imaginary axis above the origin (see Fig. 5 in the case when ζ / It remains to compute the asymptotics of the two integrals in (3.20) involvingf andf . Since 5 passes through the origin, we cannot immediately use the asymptotic formulas (3.14) forf andf . However, sincef andf are analytic in the regions (3.13), we can deform the contours in the same way as in the proof of Lemma 3.6 and then use (3.14).
only requires minor adaptations of the above arguments, which are similar to those done in the proof of Lemma 3.6, and we omit them here.
It remains to compute the coefficients in the expansion (3.24) of Lemma 3.8 more explicitly.
Lemma 3.9. Let R be defined by (3.3) and let G n be the nth coefficient in the expansion of G given in Proposition 3.1. Then the following identities hold: where the functions A n (ζ ) are defined by In particular, A 1 (ζ ) is given explicitly by (3.4), the functions A n (ζ ) are holomorphic on C\ 5 , satisfy A n = O(ζ n ) as ζ → ∞ uniformly for θ in compact subsets of (0, 1], and depend continuously on α and θ . Proof. Using that r + (ζ ) + r − (ζ ) = 0 for ζ ∈ 5 , we obtain where L is a clockwise loop which encircles 5 but which does not encircle ζ . Deforming L to infinity, picking up a residue at ξ = ζ , and using that which proves (3.33).
In order to prove (3.34), we first establish the identities The function ln(iξ) is not analytic on (0, i∞). Therefore, to prove (3.37a), we first open up the contour 5 and deform it into a loop L 1 which encircles ζ but which avoids the positive imaginary axis as shown in Fig. 7. This gives Deforming the circular part of L 1 to infinity and using that ln(iξ) jumps by 2πi across the positive imaginary axis, the identity (3.37a) follows. The identity (3.37b) follows in a similar way by deforming the contour to a loop which encircles ζ but which does not encircle the negative imaginary axis. Using that which shows that .
Deforming L to infinity (picking up a residue contribution from ζ but no contribution from infinity), we can write the first term in (3.39) as On the other hand, using the jump relation of r on 5 to open up the contour 5,+ and picking up a residue contribution from ξ = 0, we can write the second term in (3.39) as Deforming L to infinity (picking up a residue contribution from ζ but no contribution from infinity), the first term on the right-hand side of (3.41) can be written as Substituting (3.40)-(3.42) into (3.39) and recalling that G n =f n +f n , it follows that the left-hand side of (3.35) equals is analytic for ζ ∈ C\ 5 and of order O(ζ n ) as ζ → ∞. Furthermore, sincef n and G n depend continuously on α and θ , so does A n (ζ ).
The asymptotic formula (3.2) for p(ζ ) follows by substituting the identities of Lemma 3.9 into the expansion (3.24) of Lemma 3.8. This completes the proof of Proposition 3.2.

Asymptotics of R(ζ )
In this section, we establish the existence of an expansion to all orders of R(ζ ) as s → +∞ and derive an explicit expression for the first coefficient R (1) (ζ ) of this expansion. By expanding R (1) (ζ ) as ζ → ∞, we can compute the matrix R (1) 1 defined by (2.34). Even though only the (2, 2) entry of R (1) 1 is needed to compute c, we compute the full matrix R (1) (ζ ), because it will be needed later for the evaluation of C. The results are summarized in the following proposition. There exist holomorphic functions R (n) : uniformly for ζ ∈ C\ R and θ in compact subsets of (0, 1]. As ζ → ∞, R (n) (ζ ) = O(ζ −1 ) for each n = 1, . . . , N . The expansion (4.1) can be differentiated with respect to ζ in the sense that uniformly for ζ ∈ C\ R and θ in compact subsets of (0, 1]. For any N ≥ 1, ) and θ in compact subsets of (0, 1]. Moreover, the first coefficient R (1) (ζ ) is given explicitly by where the constant matrices A and B are defined by (5 + 12c 6 + 12c 5 (1 + 2c 6 )) 48(c 1 + c 2 ) , In particular, the matrix R (1) 1 in (2.34) is given by R (1) 1 = A −Ā and has (2, 2) element The remainder of this section is devoted to the proof of Proposition 4.1. We start by obtaining an asymptotic expansion of the jump matrix J R for the RH problem satisfied by R.

Asymptotics of J R .
We recall from (2.32) that R is given by where S, P, and P ∞ have been defined in Sect. 2. For ζ ∈ R , R satisfies the jump and J S denotes the jump matrix for S (see [12,Eq. (3.21)]): (4.8) The symmetries together with the fact that p 0 , ∈ R imply that the jump matrix J R obeys the symmetry From (4.9) and the symmetry of the behavior of R near the points of self-intersection of R and infinity, as well as the uniqueness of the solution of the RH problem for R, we conclude that R obeys the symmetry Note that |b 2 | = (1 + θ)θ  where the error term is uniform for ζ ∈ ∂D δ (b 1 ) ∪ ∂D δ (b 2 ) and for θ in compact subsets of (0, 1], and are holomorphic functions which satisfy the symmetry i.e., . Proof. Substituting the expressions (2.18), (2.27) and (2.30) for P ∞ , P, and E into the expression (4.7) for J R on ∂D δ (b 1 ), we find (4.14) We can extend the asymptotic formula (2.25) for A k (ζ ) to all orders as follows. The Airy function admits the following well-known uniform asymptotic expansions to all orders (see [27, Eqs. 9.7.5 and 9.7.6]): as ζ → ∞, | arg ζ | < π − δ for any δ > 0, where the coefficients {u l , v l } ∞ l=0 are given by and u 0 = v 0 = 1. Substituting the asymptotic expansions (4.15) into (2.22)-(2.24), it follows that, for k = 1, 2, 3, uniformly in the sector S k defined in (2.26), where the branches of complex powers are as in (2.25). Next note that by combining the expansions (3.1) and (3.2), we find as s → +∞ uniformly for θ in compact subsets of (0, 1] and uniformly for ζ ∈ C\ 5 such that s ρ ζ → ∞, | arg(ζ ) − π 2 | > and | arg(ζ ) + π 2 | > for any fixed > 0, where R(ζ ) is defined by (3.3) andÃ n (ζ ) are holomorphic functions of ζ ∈ C\ 5 defined bỹ Utilizing the large s expansions (4.16) and (4.17) in the expression (4.14) for J R (ζ ), we obtain as s → +∞ uniformly for ζ ∈ ∂D δ (b 1 ) and θ in compact subsets of (0, 1]. The error term O((s ρ q(ζ )) −(N +1) ) can be replaced by O(s −(N +1)ρ ), because |q(ζ )| is uniformly bounded away from zero on ∂D δ (b 1 ) by (2.31). It follows that J R admits an expansion of the form (4.10) with coefficients J (n) R (ζ ), n = 1, . . . , N , which can be computed explicitly from (4.18) by straightforward algebra. In particular, this gives the explicit expression (4.12) for the first coefficient J (1) R (ζ ); using the definition (2.18) of Q ∞ (ζ ) and the fact that r (ζ ) = (ζ − b 2 )γ (ζ ) 2 , the relations in (4.13) follow.
We finally show that J (n) R (ζ ), n = 1, . . . , N , are analytic functions of ζ ∈ D δ (b 1 )\{b 1 }. This will complete the proof of the lemma because the expansion (4.10) for ζ ∈ ∂D δ (b 2 ) and the symmetry (4.11) then follow from (4.9). Clearly, the coefficients J (n) R are analytic on D δ (b 1 )\ 5 . In fact, it follows from (4.18) that they have no jump across 5 , because for ζ ∈ 5 we have This shows that the coefficients J (n) R (ζ ) are analytic on D δ (b 1 )\{b 1 } (note however that the J

Existence of an expansion to all orders.
In the following lemma, we show that the L p norm of w R := J R − I on R is small for any 1 ≤ p ≤ ∞ uniformly for θ in compact subsets of (0, 1], whenever s is large enough. Proof. In this proof, c and C denote generic positive constant which may change within a computation. Since ∂D δ (b 1 ) ∪ ∂D δ (b 2 ) is compact, the estimate (4.19a) follows from Lemma 4.2.
For the reader's convenience, we recall some well-known facts from the theory of singular integral operators. For a function u ∈ L 2 ( R ) we define the Cauchy integral Cu by and we denote the non-tangential limits of Cu from the left-and right-hand side of R by C + u and C − u, respectively. The Cauchy operator C w R : This operator is bounded and linear and, assuming that is invertible, the solution of the RH problem for R is given by (see e.g. [16, Section 7]) where (4.28) In particular, if C w R has sufficiently small L 2 -operator norm, I − C w R can be inverted in terms of a Neumann series, that is, Hence it follows from Lemma 4.3 and the estimate that I − C w R is invertible for all sufficiently large s. Here · L 2 ( R )→L 2 ( R ) denotes the operator norm of bounded linear operators L 2 ( R ) → L 2 ( R ). The standard theory for asymptotics of small norm RH problems (see e.g. [16]) together with Lemma 4.3 implies that R satisfies (4.1) and that this expansion can be differentiated with respect to ζ . The basic idea here is to combine (4.27)-(4.29) and the expansion (4.10) of the jump matrix. This immediately gives (4.1) uniformly for ζ bounded away from the contour R . For ζ close to R , one uses analyticity of the jump matrix in a neighborhood of R to deform the contour in such a way that ζ is bounded away from the deformed contour.
Using the jump relation R + = R − J R , the left-hand side of (4.3) can be written for The estimate (4.3) is then a consequence of the estimates (4.19b) and (4.19c) of J R (ζ ) and J R (ζ ), as well as the expansions (4.1) and (4.2) of R(ζ ) and R (ζ ). (1) (ζ ). We next derive the explicit expression (4.4) for the coefficient R (1) (ζ ). We have R = I + C(μ R w R ) and, by Lemma 4.3 and (4.28),

Explicit expression for R
as s → +∞, where the error terms are uniform with respect to ζ ∈ R and θ in compact subsets of (0, 1]. This implies where the matrices A and B are defined by It follows from equations (4.31)-(4.33) that R (1) (ζ ) satisfies (4.4) with A and B given by (4.34). We next show that the matrices A and B can be written as in (4.5). Expanding (2.10) in powers of √ ζ − b 1 and recalling the definition (2.28) of q, we obtain Expansion of (3.3) gives where r is any fixed large radius; substituting in (3.2), this gives the following extension of (2.37) to all orders as s → +∞: Similarly, by the definition (2.33) of R 1 (s) and the expansion (4.1) of R(ζ ), where the function g obeys the bound |g(ζ, s)| ≤ C s −(N +1)ρ (1+|ζ |) −1 . The coefficients R (n) (ζ ) are analytic at ζ = ∞ by Proposition 4.1. Hence where R

Proof of the expression (1.10) for c. Comparing (2.37) and (5.1), we see that
where r > 0 is any large radius, i.e., K is the term of order 1 in the large ζ expansion of the function A 1 (ζ ) defined in (3.4). A direct computation shows that and therefore Substituting the expressions (4.6) and (5.2) for (R (1) 1 ) 2,2 and K into (2.38) and recalling the definition (2.4) of the constants {c j } 8 1 , we obtain the expression (1.10) for c.
Remark 5.1. The above evaluation of the constant c is based on the differential identity (2.35) in s. In Sect. 10, we will obtain an independent second proof of (1.10) by using a differential identity in θ .
Remark 5.2 (The constant c for two other models). Our approach to obtain the constant c presented in Sects. 3 and 4 is based on the differential identity in s derived in [12]. Hence, it also applies to two other random matrix models studied in [12]. The first model consists of random matrices of the form where * denotes the complex conjugate transpose operator, and each G j is an independent (n + ν j ) × (n + ν j−1 ) complex Ginibre matrix, with integers r ≥ 1, ν 0 = 0, and ν j ≥ 0, j = 1, . . . , r . The second model consists of products of the form where each T j is an (n + ν j ) × (n + ν j−1 ) upper left truncation of an j × j Haar distributed unitary matrix U j . Here U 1 , . . . , U r are assumed to be independent and ν 0 = 0, r ≥ 1, and ν j ≥ 0, j = 1, . . . , r , are integers. Furthermore, it is assumed that j ≥ n +ν j +1 and r j=1 ( j −n −ν j ) ≥ n. In the second model, a subset J ⊂ {2, . . . , r } of cardinality q < r is fixed such that μ j := k j − n > ν j for k j ∈ J and k − n → +∞ for k ∈ {1, . . . r }\J as n and 1 , . . . , r go to infinity. In [12], it is shown that these two models admit large gap asymptotics for the eigenvalues of the form where the first and second model corresponds to j = 1 and j = 2, respectively. Moreover, explicit expressions are derived for the constants ρ ( j) , a ( j) , and b ( j) .
A straightforward modification of our approach yields the existence of constants C as s → +∞ for j = 1, 2, and shows that the constants c (1) and c (2) are given explicitly by Let K (1) be the hard edge limiting kernel for the eigenvalues associated to the first model presented above (this is the same notation as in [12]). For certain particular choices of the parameters ν 1 , . . . , ν r and θ , the kernel K (1) defines the same point process (up to rescaling) as the one associated to K 7 -this is a result of Kuijlaars and Stivigny, see [25,Theorem 5.1]. More precisely, if r ≥ 1 is an integer, α > −1 and Therefore, if the parameters satisfy (5.5), we obtain the following relations: 8 The three relations in (5.6) can be verified from [12], and the relation c (1) = c can be verified directly from (1.10) and (5.4). This provides a non-trivial consistency check of the results from [12] and of our result for c and c (1) .

Differential Identity in θ
In this section, we derive an identity for the derivative of ln det(1 − K| [0,s] ) with respect to θ . As explained in Sect. 1.2, this differential identity is needed for the derivation of the expression (1.11) for C. Our proof of Lemma 6.1 below is inspired by the derivation of the differential identity (2.35) given in [12]. θ , 1st version). For every α > −1, θ > 0 and s > 0, the following identity holds:
A computation gives from which it follows that Since J is triangular and J − I is off-diagonal we infer that using also the jump relations for Y , from which we obtain

4)
A similar computation yields By substituting (6.5) in (6.4), we obtain Using (6.5) and (6.6), we arrive at Substitution of the above identity into (6.3) finishes the proof.
In the following lemma, we rewrite the differential identity (6.1) in a form which is more convenient for the asymptotic analysis. Let us define the sequence {ζ j } ∞ 0 ⊂ iR by (6.7) and the meromorphic function H (ζ ) by Note that H has a simple pole at each of the points ζ j , j = 1, 2, . . . , and no other poles in C; the point ζ 0 is a simple pole of ψ( 1+α 2 −is ρ ζ θ ) but not of H . Given K > |b 1 |, we let σ K denote the closed s-dependent counterclockwise contour displayed in Fig. 8. The contour σ K surrounds 5 once in the positive direction, but does not surround any of the poles ζ j of H . The circular part of σ K has radius K and its horizontal part has a length of order O(1) as s → +∞ and crosses the imaginary axis at the point ζ 0 /2. If K = 2|b 1 |, we write σ for σ K , i.e., σ = σ 2|b 1 | . We also define the contour K as the union of the parts exterior to σ K of the rays { i } 4 1 defined in (2.14), i.e., where , (6.10) 11) , (6.12) Proof. Using the change of variable z = is ρ ζ + 1 2 in (6.1), we obtain an integral over γ U ∪γ U whose integrand is expressed in terms of U via (2.13). By deforming the contour of this integral using the analytic continuations of U + and U − (i.e., using T ), we arrive at Another contour deformation gives 2πi , (6.14) where we have used that ζ → H (ζ ) Tr[T −1 (ζ )T (ζ )σ 3 ] is analytic for all ζ in the interior region of σ K such that ζ / . Therefore, inverting the transformations T → S → R for ζ ∈ σ K , we find The first two terms on the right-hand side of (6.15) are analytic in the region between σ and σ K . Therefore, substituting (6.15) into the first term on the right-hand side of (6.14) and deforming the contour from σ K to σ in the integrals involving the first two terms on the right-hand side of (6.15), we find that this term equals I 1 + I 2 + I 3,K .
Similarly, by inverting the transformations T → S → R for ζ ∈ K , we find that the second term on the right-hand side of (6.14) equals I 4,K . Remark 6.3. In the application of the differential identity (6.9) to the proof of Theorem 1.1, we will choose K = s ρ ; that is, the radius K will be s-dependent and growing to infinity as s → +∞.
The remainder of the paper is devoted to the proof of Theorem 1.1. The proof is divided into two steps. The first step consists of obtaining large s asymptotics of the differential identity (6.9) uniformly for θ in compact subsets of (0, 1]. This is achieved by computing the large s asymptotics of each of the four terms I 1 , I 2 , I 3,K , and I 4,K on the right-hand side of (6.9). These computations are presented in Sects. 7-9. The second step is presented in Sect. 10 and consists of integrating the resulting asymptotic expansion from θ = 1 to an arbitrary θ ∈ (0, 1].

Asymptotics of I 1
In this section, we prove the following proposition which establishes the large s asymptotics of I 1 .
Proposition 7.1 (Large s asymptotics of I 1 ). Let α > −1. As s → +∞, the function I 1 defined in (6.10) satisfies uniformly for θ in compact subsets of (0, 1], where the coefficients I (1) 1 , I 1 , I 1 , I (3) 1 are given by (7.2a) where ζ ∈ C\(−i∞, ζ 0 ] is some point at which is normalized to vanish; we will choose this normalization point below. Then is analytic in C\(−i∞, ζ 0 ]. In particular, is analytic on σ . Using the explicit expression (2.10) for g , an integration by parts therefore gives We assume that σ is big enough to enclose the straight line segment [b 1 , b 2 ] and move the branch cut for r (ζ ) upwards from 5 to the horizontal line segment [b 1 , b 2 ]; this does not change the value of the integral. We letr denote the analytic continuation of r defined byr where the branch is such thatr is analytic in Thenr (ζ ) is equal to r (ζ ) except for ζ in the region enclosed by 5 where we instead haver (ζ ) = −r (ζ ). Deforming σ upwards through the origin, a residue contribution is generated by the simple pole of iIm b 1 /(ζr (ζ )) at ζ = 0. We find where [b 1 , b 2 ] is oriented from b 1 to b 2 with + and − sides to the left and right as usual, andr (0) = r − (0) = −i|b 2 |. Thus, 2πi , (7.5) where γ b 2 b 1 denotes the part of the circle of radius |b 2 | centered at the origin going from b 2 to b 1 and oriented counterclockwise. Let us choose ζ = 0; then (0) = 0, so the first term on the right-hand side of (7.5) vanishes. The choice ζ = 0 implies that the term 1+α 2 −is ρ ξ θ is not uniformly large for ξ ∈ [0, ζ ] with ζ ∈ γ b 1 b 2 as s → +∞, so the large s behavior of (ζ ) does not follow immediately from (6.8) and (7.3); however, we can determine the large s asymptotics of (ζ ) as follows. Using the change of variables we can write Integrating by parts, we get Using the well-known identity (see e.g. [27,Eq. 5.17.4]) The above expression is convenient since the large z asymptotics of (z) and G(z) are known (see e.g. [27, Eqs. 5.11.1 and 5.17.5]): (7.11) as z → ∞ with | arg z| < π, where ζ is Riemann's zeta function. 9 Expanding (7.9) as s ρ ζ → ∞, we get (7.12) Substituting (7.12) into (7.5), we find that I 1 satisfies (7.1) as s → +∞ with coefficients given by 2πi , , , 2πi . (7.13) It only remains to show that the coefficients in (7.13) can be expressed as in (7.2). This requires the evaluation of several integrals; we have collected the necessary results in the next lemma. α > −1 and θ ∈ (0, 1]. Let r (ζ ) denote the square root defined in (2.9). Then the following identities hold:

Asymptotics of I 2
The large s asymptotics of I 2 is a consequence of the following three propositions whose proofs are given in Sects. 8.1, 8.2, and 8.3 , respectively. Proposition 8.1 (Splitting of I 2 ). The function I 2 defined in (6.11) can be written as where X and Z are defined by The quantity X defined in (8.1) admits the following asymptotic expansion as s → +∞: uniformly for θ in compact subsets of (0, 1], where the coefficients are given by

Proof of Proposition Recall that the integral I 2 is given by
where H is given by (6.8) and σ is a closed curve surrounding 5 once in the positive direction which does not surround any of the poles ζ j of H . A straightforward computation gives where we have used that has trace zero in the last step. Hence The function p(ζ ) is analytic for ζ ∈ C\ 5 and satisfies the following jump condition across 5 : Integrating by parts, deforming the contour, and using the jump condition for p, we find which completes the proof.
Let m = 1+α 2 ; the exact value of m is not essential as long as m > 0. We split X as follows: where 10) and the functionĝ(w) is defined bŷ Note that all the s-dependence of X 2 and X 3 is in the contour. The termĝ(w) has been added and subtracted so that the integrand in the definition of X 2 is O(w −2 ln w) as w → ∞. This can be verified by using the asymptotic expansion (see [27,Eq. 5.11.2]) where B 2k is the 2kth Bernoulli number, which implies that as w → ∞ away from the negative real axis, as well as the expansion The integrals defining X 2 and X 3 converge because the functionĝ(w) is analytic except for a double pole at w = m > 0. We will show that X 1 , X 2 , X 3 satisfy the large s asymptotics 2 + O(s −ρ ln(s ρ )), (8.16) 3 s ρ ln(s ρ ) + X 3 ln(s ρ ) + X where the constants c j are given by (2.4). Substituting these expansions into the definition (8.8) of X 1 , we find (8.15).

Asymptotics of X 2 Since the integrand in the definition of
where the contour crosses the real line at 0 andĝ(w) is given by (8.11). It remains to show that X 2 can be written as in (8.4). Since ψ(z) = −1 z+k + O (z + k) −2 as z → −k for each k = 0, 1, 2, . . . , it follows that ψ 1+α 2 − w θ has a simple pole with residue θ at each of the points 1+α 2 + kθ , k = 0, 1, 2, . . . . For k ≥ 1, the associated residue gives a contribution to X (3) 2 equal to (taking into account that after the deformation, the loop is going in the clockwise orientation around 1+α 2 +kθ ) On the other hand, the residue at m = 1+α 2 is given by where γ E is Euler's gamma constant. By (8.13) and (8.14), we have as w → ∞ away from the negative real axis. Moreover, by (8.12), (8.20) as w → ∞ away from the positive real axis; in fact, combining (8.12) with the reflection formula ψ(1−z) = ψ(z)+π cot(π z), we see that (8.20) holds also as w → ∞ in a sector containing the positive real axis as long as w stays away from the poles { 1+α 2 + jθ } ∞ j=0 . Thus, deforming the contour in (8.19) to infinity in the right half-plane along curves which stay away from the set { 1+α 2 + jθ } ∞ j=0 , the contribution from infinity vanishes and we find where the series is convergent because it originates from a convergent integral. Using the series representation for the Euler gamma constant (see [27,Eq. 5.2.3]) we conclude that X 2 can be written as in (8.4). This proves (8.16).

Asymptotics of X 3
Deforming the contour is ρ 5 in the definition (8.10) of X 3 in the left half-plane to the contour is ρ γ b 2 b 1 , we get The above representation is convenient for the asymptotic analysis of X 3 , because the argument ( 1+α 2 − w)/θ of ψ is large as s → +∞ uniformly for w ∈ is ρ γ b 2 b 1 . As w → ∞ away from the positive real axis, we have Letting w = is ρ ζ , we obtain 3 ln(s ρ ) + X 3 + O s −ρ ln(s ρ ) (8.23) uniformly for θ in compact subsets of (0, 1], where the coefficients are given by Using that we see that the coefficients X 3 , X 3 , X 3 , X 3 can be written as in (8.5). This proves (8.17) and thus completes the proof of Proposition 8.2.

Proof of Proposition We have
uniformly for ζ ∈ γ b 2 b 1 . Moreover, by Proposition 3.2, uniformly for ζ ∈ γ b 2 b 1 , where the coefficients B(ζ ) and A(ζ ) are defined by with A 1 and R given by (3.4) and (3.3). Substitution into the definition (8.1) of Z shows that Z admits an expansion of the form (8.6) as s → +∞, uniformly for θ in compact subsets of (0, 1], with coefficients given by It only remains to show that the coefficients in (8.26) can be expressed as in the statement of Proposition 8.3. Inspection of (8.26) shows that there are five different integrals that need to evaluated: These integrals are evaluated in the following lemma.

Asymptotics of I 3,K and I 4,K
In this section, we prove two propositions (Proposition 9.1 and Proposition 9.2) which establish the large s asymptotics of I 3,K and I 4,K , respectively, where we henceforth choose K = s ρ . Proposition 9.1 (Large s asymptotics of I 3,K ). Let α > −1 and let K = s ρ . As s → +∞, the function I 3,K defined in (6.12) satisfies uniformly for θ in compact subsets of (0, 1], where the coefficients I 3 and I (3) 3 are given by Proof. By the cyclicity of the trace, we can write the definition (6.12) of I 3,K as where K = s ρ , and σ K and H are defined in Lemma 6.2. All the s-dependence of the trace lies in the factor R −1 (ζ )R (ζ ), since by (2.18), the quantity is independent of s. As s → +∞, we have by Proposition 4.1 that uniformly for ζ ∈ C\ R and that this expansion can be differentiated with respect to ζ . The asymptotics in (9.4) as well as all other asymptotic expansions in the rest of this section are uniform with respect to θ in compact subsets of (0, 1]. From the explicit expression (4.4) for R (1) , we see that R (1) (ζ ) and R (1) (ζ ) are O((1 + |ζ |) −1 ) and O((1 + |ζ |) −2 ), respectively, uniformly for ζ ∈ σ K as s → +∞. Therefore, uniformly for ζ ∈ σ K . Hence, for large s and ζ ∈ σ K , we have where the function W (ζ ) is defined by Substituting (9.5) into (9.2), we see that where I 3 and I 3,K are defined by We first estimate I 3,K . Since we have (see also Fig. 8) where c , C > 0 are two sufficiently large constants. We next consider I 3 . Substituting (4.4) and (9.3) into (9.6), it follows that Replacing r byr in W does not change the value of I 3 . Deforming σ (which surrounds 0) into another contourσ which surrounds the cut [b 1 , b 2 ] ofr once in the positive direction but which does not surround 0, it transpires that where W is defined by the expression obtained by replacing r byr in the right-hand side of (9.10). Assuming thatσ is bounded away from 0, we can replace H (ζ ) by its large s asymptotics (8.18); this gives We split the leading term as follows: where I 3,1 and I 3,2 are given by From (9.10), we have the expansion By deforming the contourσ to infinity, we get , (9.12) while We have can be computed explicitly using (9.10). After simplification this gives Substituting this expression for I 3,2 into (9.11) and recalling (9.7), (9.9), and (9.12), equation (9.1) follows.
Remark 10.2. Lemma 10.1 provides an alternative proof of the expressions (1.9) and (1.10) for a, b, and c based on the differential identity in θ . Note that this method yields an error term in (10.2) of order O s −ρ ln(s ρ ) for θ ≤ 1 (and for θ ≥ 1 we would get O(s − 1 2 ln s)), which is slightly worse than the optimal bound O(s −ρ ) (which was proved via the differential identity in s in [12]).
To complete the proof of Theorem 1.1 it only remains to verify that the sum of the terms of order 1 on the right-hand side of (10.2) equals ln C, where C is given by (1.11). In order to verify this we need to compute the three integrals on the right-hand side of (10.2). These integrals are computed in the following three lemmas.

Proof.
A simple integration by parts shows that Using the identities (see [27,Eq. 5.17.4] for the first identity) Proof. This follows from a long but straightforward computation.
Remark 10.6. Note that the reasoning leading to (10.8) cannot be applied to the sum for general values of θ . In fact, this is the only finite N sum in (10.7) which we are not able to evaluate in terms of known special functions.
Replacing the three integrals on the right-hand side of (10.2) with the expressions derived in Lemmas 10.3-10.5, we obtain the following expression for the term of order 1 in the large s asymptotics of ln det( 1 − K| [0,s] ): which is precisely ln C, where C is defined by (1.11). This finishes the proof in the case when θ ∈ (0, 1]; as explained in Sect. 1 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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A. Proof of Proposition 1.4
In this "Appendix", we establish the formula (1.17) for d(θ, α) for rational values of θ stated in Proposition 1.4. Let θ = p/q where p, q ≥ 1 are two (not necessarily relatively prime) integers. Let N = mq where m ≥ 1 is an integer (later we will take m → +∞). We have

(B.3)
All the special functions on the right-hand side of (B.3) have uniform expansions for large N whenever the argument of 1 + 1 + α θ + N is bounded away from ±π ; in particular, this is the case for (θ, α) ∈ K . Furthermore, by (3.7), there are constants C , C > 0 (that only depend on K ) such that and thus the series ∞ k=1 D 1 (1+α+kθ) converges uniformly for (θ, α) ∈ K . We conclude that the sequence of functions d N converges to d uniformly for (θ, α) in compact subsets of U and thus the proof of Proposition 1.6 is complete.
On the other hand, if j ≤ −1 in (C.1), then ζ j r (ζ ) has no residue at ∞ but has a pole of order | j| at 0. By deforming the contour σ through ∞, we obtain where C denotes a small circle of radius centered at 0 oriented positively. Therefore the right-hand side of (C.1) is equal to the coefficient of ζ −1 in the expansion of ζ j r (ζ ) as ζ → 0. Since as ζ → 0, this proves (7.14d).
To prove the remaining four identities, we note that the same kind of argument that gave (C.1), shows that, for any j ∈ Z, If j ≥ 0, then by deforming σ to C R ∪ (−R, 0) + i0 + ∪ (0, −R) − i0 + where R > 0 is any large radius, and noting that r =r over the range of integration, we get Since the left-hand side is independent of R, by taking the limit R → +∞, we obtain R j e i jϕ ln(Re iϕ ) 1 Taking j = 0 in (C.3), we find dζ .
Using that where we have fixed the branch of Substituting the above expansion into (C.4), we obtain (7.14g).
Suppose now that j = −1. Using that d dζ as → 0 + . Also, by a straightforward computation, ln ζ ζ r (ζ ) which proves the eighth and last identity (7.14h). The proof of Lemma 7.2 is complete.