Large gap asymptotics on annuli in the random normal matrix model

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + {\mathcal {F}}_{n} + \mathcal {O}\Big ( n^{-\frac{1}{12}}\Big )\Bigg ), \end{aligned}$$\end{document}exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),where n is the number of points of the process. We determine the constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1},\ldots ,C_{6}$$\end{document}C1,…,C6 explicitly, as well as the oscillatory term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}_{n}$$\end{document}Fn which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1},\ldots ,C_{4}$$\end{document}C1,…,C4 were previously known, (ii) when the hole region is an unbounded annulus, only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1},C_{2},C_{3}$$\end{document}C1,C2,C3 were previously known, and (iii) when the hole region is a regular annulus in the bulk, only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}$$\end{document}C1 was previously known. For general values of our parameters, even \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}$$\end{document}C1 is new. A main discovery of this work is that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}_{n}$$\end{document}Fn is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.


Introduction and statement of results
Consider the probability density function where z 1 , . . . , z n ∈ C and Z n is the normalization constant. We are interested in the gap probability where 0 ≤ r 1 < r 2 < . . . < r 2g ≤ +∞. Thus P n is the probability that no points lie on g annuli centered at 0 and whose radii are given by r 1 , . . . , r 2g . One annulus is a disk if r 1 = 0, and one annulus is unbounded if r 2g = +∞. In this paper we obtain the large n asymptotics of P n , up to and including the term of order 1.
The particular case b = 1 and α = 0 of (1.1) is known as the complex Ginibre point process [40] (or simply Ginibre process, for short) and is the most well-studied two-dimensional determinantal point process of the theory of random matrices. It describes the distribution of the eigenvalues of an n × n random matrix whose entries are independent complex centered Gaussian random variables with variance 1 n . For general values of b > 0 and α > −1, (1.1) is the joint eigenvalue density of a normal matrix M taken with respect to the probability measure [58] 1 Here dM denotes the measure induced by the flat Euclidian metric of C n×n on the set of normal n × n matrices (see e.g. [20,32] for details), M * is the conjugate transpose of M , "tr" denotes the trace, and Z n is the normalization constant.
The limiting mean density (with respect to d 2 z) as n → +∞ of the points z 1 , . . . , z n is given by [67,17] b 2 π |z| 2b−2 , (1.4) and is supported on the disk centered at 0 and of radius b − 1 2b . In particular, for b = 1, the limiting density is uniform over the unit disk; this is a well-known result of Ginibre [40]. Since the points accumulate on a compact set as n → +∞, this means that for large n, P n is the probability of a rare event, namely that there are g "large gaps/holes" in the form of annuli.
The probability to observe a hole on a disk centered at 0 and of radius r < 1 in the Ginibre process was first studied by Forrester, who obtained [37, eq (27)] P n = exp C 1 n 2 + C 2 n log n + C 3 n + C 4 √ n + o( √ n) , as n → +∞, (1.5) where The constant C 1 was also given independently by Jancovici, Lebowitz and Manificat in [48, eq (2.7)]. As noticed in [37, eq (13)], C 1 and C 2 also follow from the asymptotic expansion obtained in an equivalent problem considered in the earlier work [44]. The constants C 1 , C 2 , C 3 have also been obtained in the more recent work [4] using a different method; see also [52, eq (49)] for another derivation of C 1 . Although Forrester's result (1.5) is 30 years old, as far as we know it is the most precise result available in the literature prior to this work.
When the hole region is an unbounded annulus centered at 0 and of inner radius r < 1, the following third order asymptotics for P n were obtained by Cunden, Mezzadri and Vivo in [23, eq (51)] for the Ginibre process: P n = exp C 1 n 2 + C 2 n log n + C 3 n + o(n) , as n → +∞, (1.7) where C 1 = r 4 4 − r 2 + 3 4 + log r, C 2 = r 2 −1 2 , Hole probabilities of more general domains have been considered in [2] for the Ginibre process. In particular, for a large class of open sets U lying in the unit disk, Adhikari and Reddy in [1] proved that P # z j : z j ∈ U = 0 = exp C 1 n 2 + o(n 2 ) , as n → +∞, where the constant C 1 = C 1 (U ) is given in terms of a certain equilibrium measure related to a problem of potential theory. When U is either a disk, an annulus, an ellipse, a cardioid, an equilateral triangle or a half-disk, C 1 has been computed explicitly. Some of these results have then been generalized for a wide class of point processes by Adhikari in [1]. For the point process (1.1) (with arbitrary b > 0 but α = 0), he obtained as n → +∞ with 0 < r 1 < r 2 < b − 1 2b and r ∈ (0, b − 1 2b ) fixed, see [1, Theorem 1.2 and eqs (3.5)-(3. 6)]. These are the only works which we are aware of and which fall exactly in our setting. There are however several other works that fall just outside. In [69], Shirai considered the infinite Ginibre process, which, as its name suggests, is the limiting point process arising in the bulk of the (finite) Ginibre process. He proved, among other things, that the probability of the hole event # z j : |z j | ≤ r = 0 behaves as exp −r 4 4 + o(r 4 ) as r → +∞ (see also [47,Proposition 7.2.1] for a different proof). This result can be seen as a less precise analogue of (1.5) for the infinite Ginibre process, and was later generalized for more general shapes of holes in [2] and then for more general point processes in [1]. Hole probabilities for product random matrices have been investigated in [5,3]. The existing literature on large gap problems in dimension ≥ 2 goes beyond random matrix theory. The random zeros of the hyperbolic analytic function +∞ k=0 ξ k z k -here the ξ k 's are independent standard complex Gaussians -form a determinantal point process [63], and the associated large gap problem on a centered disk has been solved in [63,Corollary 3 (i)]. Another well studied two-dimensional point process is the random zeros of the standard Gaussian entire function. This function is given by +∞ k=0 ξ k z k √ k! , where the ξ k 's are independent standard complex Gaussians. In [70], the probability for this function to have no zeros in a centered disk of radius r was shown to be, for all sufficiently large r, bounded from below by exp(−Cr 4 ) and bounded from above by exp(−cr 4 ) for some positive constants c and C. This result was later improved by Nishry in [59], who proved that this probability is exp(− e 2 4 r 4 + o(r 4 )) as r → +∞. A similar result as in [70] was obtained in [46] for a different kind of random functions with diffusing coefficients. Also, for a d-dimensional process of noninteracting fermions, it is shown in [43] that the hole probability on a spherical domain of radius r behaves as exp(−cr d+1 + o(r 3 )) as r → +∞, and an explicit expression for c > 0 is also given.
In its full generality, the random normal matrix model is associated with a given confining potential Q : C → R ∪ {+∞} and is defined by a probability measure proportional to e −ntrQ(M ) dM , where dM is as in (1.3). The random normal matrix model has been extensively studied over the years, see e.g. [20,32] for early works, [38,66,7,53] for smooth linear statistics, [65,9,74,30,36,17] for non-smooth linear statistics, and [11,45,54,55,6] for recent investigations on planar orthogonal polynomials. Despite such progress, the problem of determining large gap asymptotics in this model has remained an outstanding problem. In this work we focus on Q(z) = |z| 2b + 2α n log |z|, which is a generalization of the Gaussian potential |z| 2 known as the Mittag-Leffler potential [8].
In other words, we cover the situations where the hole region consists of 1. g annuli inside the disk of radius b − 1 2b ("the bulk"), 2. g − 1 annuli in the bulk and one unbounded annulus (g ≥ 1), 3. g − 1 annuli in the bulk and one disk (g ≥ 1), 4. g − 2 annuli in the bulk, one unbounded annulus, and one disk (g ≥ 2).
For each of these four cases, we prove that P n = exp C 1 n 2 + C 2 n log n + C 3 n + C 4 √ n + C 5 log n + C 6 + F n + O n − 1 12 , (1.9) as n → +∞, and we give explicit expressions for the constants C 1 , . . . , C 6 .
The quantity F n fluctuates around 0 as n increases, is of order 1, and is given in terms of the Jacobi theta function (see e.g. [61,Chapter 20]) Note that θ(z|τ ) = θ(z + 1|τ ) for all z ∈ C and τ ∈ i(0, +∞); in particular R x → θ(x|τ ) is periodic of period 1. To our knowledge, this is the first time the Jacobi theta function appears in a large gap problem of a two-dimensional point process.
The presence of oscillations in these asymptotics can be explained by the following heuristics. It is easy to see (using Bayes' formula) that P n is also equal to the partition function (= normalization constant) of the point process (1.1) conditioned on the event that #{z j : |z j | ∈ [r 1 , r 2 ] ∪ [r 3 , r 4 ] ∪ ... ∪ [r 2g−1 , r 2g ]} = 0. As is typically the case in the asymptotic analysis of partition functions, an important role is played by the n-tuples (z 1 , . . . , z n ) which maximize the density of this conditional process. One is then left to understand the configurations of such n-tuples when n is large. To be more concrete, suppose for example that g = 1 and 0 < r 1 < r 2 < b − 1 2b . Since the support of (1.4) is the centered disk of radius b − 1 2b , it is natural to expect that the points in the conditional process will accumulate as n → +∞ on two separated components (namely the centered disk of radius r 1 , and an annulus whose small radius is r 2 ). The n-tuples (z 1 , . . . , z n ) maximizing the conditional density may differ from each other by the number of z j 's lying on a given component. This, in turn, produces some oscillations in the behavior of P n . More generally, if the points in the conditional process accumulate on several components ("the multi-component regime"), then one expects some oscillations in the asymptotics of P n . (There exist several interesting studies on conditional processes in dimension two, see e.g. [42,60,68].) In the setting of this paper, there are three cases for which there is no oscillation (i.e. F n = 0): when the hole region consists of only one disk (the case g = 1 of Theorem 1.4), only one unbounded annulus (the case g = 1 of Theorem 1.7), or one disk and one unbounded annulus (the case g = 2 of Theorem 1.9). This is consistent with our above discussion since in each of these three cases the points of the conditional process will accumulate on a single connected component.
It has already been observed that the Jacobi theta function (and more generally, the Riemann theta function) typically describes the oscillations of various statistics of one-dimensional point processes in "the multi-cut regime". Indeed, Widom in [76] discovered that the large gap asymptotics of the one-dimensional sine process, when the gaps consist of several intervals, contain oscillations of order 1 given in terms of the solution to a Jacobi inversion problem. These oscillations were then substantially simplified by Deift, Its and Zhou in [27], who expressed them in terms of the Riemann theta function. Since then, there has been other works of this vein, see [16] for β-ensembles, [22] for partition functions of random matrix models, [35] for the sine process, [12,13,51] for the Airy process, and [14] for the Bessel process. In all these works, the Riemann theta function describes the fluctuations in the asymptotics, thereby suggesting that this function is a universal object related to the multi-cut regime of one-dimensional point processes. Our results show that, perhaps surprisingly, the universality of the Jacobi theta function goes beyond dimension 1.
Another function that plays a predominant role in the description of the large n asymptotics of P n is the complementary error function (defined in (1.6)). This function already emerged in the constant C 4 | (b=1,α=0) of Forrester, see (1.5 and erfc(−y) = 2 − erfc(y), it is easy to check that and that as y → +∞, which implies that the integrals in (1.11), (1.12) and (1.13) are finite, as it must.
We expect that the estimate O n − 1 12 for the error term in (1.9) is not optimal and could be reduced to O n − 1 2 . However, proving this is a very technical task, and we will not pursue that here. We now state our main results, and discuss our method of proof afterwards.
be fixed parameters. As n → +∞, we have The constants C 6 appearing in Theorems 1.7 and 1.9 below are notably different than in the previous two theorems, because they involve a new quantity G(b, α) which is defined by where Γ(z) = ∞ 0 t z−1 e −t dt is the Gamma function. Interestingly, this same object G(b, α) also appears in the large gap asymptotics at the hard edge of the Muttalib-Borodin ensemble, see [18, [18]). It was also shown in [18] that if b is a rational, then G(b, α) can be expressed in terms of the Riemann ζ-function and Barnes' G function, two well-known special functions (see e.g. [61,Chapters 5 and 25] for the definitions of these functions). More precisely, we have the following. Proposition 1.6. (Taken from [18,Proposition 1.4] n2 for some positive integers n 1 , n 2 , then G(b, α) is explicitly given by Figure 3: This situation is covered by Theorem 1.7 with g = 3.
We now state our next theorem.
be fixed parameters. As n → +∞, we have P n = exp C 1 n 2 + C 2 n log n + C 3 n + C 4 √ n + C 5 log n + C 6 + F n + O n − 1
Method of proof. The problem of determining large gap asymptotics is a notoriously difficult problem in random matrix theory with a long history [50,39,41]. There have been several methods that have proven successful to solve large gap problems of one-dimensional point processes, among which: the Deift-Zhou [29] steepest descent method for Riemann-Hilbert problems [49,26,25,10,28,21,18,19,24], operator theoretical methods [75,33,34], the "loop equations" [15,16,56,57], and the Brownian carousel [72,73,64,31]. Our method of proof shows similarities with the method of Forrester in [37]. It relies on the fact that (1.1) is determinantal, rotation-invariant, and combines the uniform asymptotics of the incomplete gamma function with some precise Riemann sum approximations. Our method is less robust with respect to the shape of the hole region than the one of Adhikari and Reddy [1,2], but allows to give precise asymptotics. We also recently used this method of Riemann sum approximations in [17] to obtain precise asymptotics for the moment generating function of the disk counting statistics of (1.1). However, the problem considered here is more challenging and of a completely different nature than the one considered in [17]; most of the difficulties that we have to overcome here are not present in [17]. These differences will be discussed in more detail in Section 3.

Preliminaries
By definition of Z n and P n (see (1.1) and (1.2)), we have where the weight w is defined by We will use the following well-known formula to rewrite Z n and P n in terms of one-fold integrals.
The proof of Lemma 2.1 is standard and we omit it, see e.g. [74], [17, Lemma 1.9] and the references therein. The argument relies on the fact that the point process on z 1 , . . . , z n ∈ C with density proportional to 1≤j<k≤n |z k − z j | 2 n j=1 w(z j ) is determinantal and rotation-invariant.
This exact formula is the starting point of the proofs of our four theorems. To analyze the large n behavior of log P n , we will use the asymptotics of γ(a, z) in various regimes of the parameters a and z. These asymptotics are available in the literature and are summarized in the following lemmas. Section 11.2.4]). The following hold: where erfc is given by and the principal branch is used for the roots. In particular, .
By combining Lemma 2.3 with the large z asymptotics of erfc(z) given in (1.14), we get the following.
To analyze asymptotically as n → +∞ the sum on the right-hand side, we will split it into several smaller sums, which need to be handled in different ways. For j = 1, . . . , n and = 1, . . . , 2g, we define Let M be a large integer independent of n, and let > 0 be a small constant independent of n. Define where x denotes the smallest integer ≥ x, and x denotes the largest integer ≤ x. We take sufficiently small such that A natural quantity that will appear in our analysis is It is easy to check that for each k ∈ {1, . . . , g}, t 2k lies in the interval (br 2b 2k−1 , br 2b 2k ). For reasons that will be apparent below, we also choose > 0 sufficiently small such that br 2b Using (2.4) and (3.4), we split the j-sum in (3.1) into 4g + 2 sums with , We first show that the sums S 0 and S 1 , S 5 , S 9 , . . . , S 4g+1 are exponentially small as n → +∞.
The next lemma makes apparent the terms that are not exponentially small.
, and where t k is defined in (3.5).
The reason why we have split the sum in (3.15) into two parts (denoted S (1) 2k−1 and S (2) 2k−1 ) around the value j = j k, is the following. As can be seen from the proof of Lemma 3.3, we have as n → +∞ uniformly for j ∈ {j k−1,+ + 1, . . . , j k,− − 1}. The two above right-hand sides are exponentially small. To analyze their sum, it is relevant to know whether η 2 holds. It is easy to check that the function j → η 2 j,k − η 2 j,k−1 , when viewed as an analytic function of j ∈ [j k−1,+ + 1, j k,− − 1], has a simple zero at j = j k, . In fact, we have for all sufficiently large n by (3.3), (3.5) and (3.6), which implies that the number of terms in each of the sums S (1) 2k−1 and S (2) 2k−1 is of order n. When j is close to j k, , the two terms (3.16) and (3.17) are of the same order, and this will produce the oscillations in the asymptotics of log P n . We will evaluate S (1) 2k−1 and S (2) 2k−1 separately using some precise Riemann sum approximations. We first state a general lemma.
Therefore, by isolating the sum as n → +∞. In the same way as (3.21), by replacing f successively by f , f and f , we also obtain as n → +∞. The integral on the right-hand side of (3.21) can be expanded using again Taylor's theorem; this gives bn n an n bn n an n f (x)dx can be expanded in a similar way using Taylor's Theorem. After substituting these expressions in (3.25) and using some elementary primitives, we find the claim.
We introduce here a number of quantities that will appear in the large n asymptotics of S and for k = 1, 2, . . . , 2g, define where t k is given in (3.5) and k−1,+ are given in (3.26)-(3.29). Proof. Recall from (3.11) that λ j,k−1 remains in a compact subset of (0, 1) as n → +∞ uniformly for j ∈ {j k−1,+ + 1, . . . , j k,− − 1}, and that λ j,k remains in a compact subset of (1, +∞) as n → +∞ uniformly for j ∈ {j k−1,+ + 1, . . . , j k,− − 1}. Hence, by Lemma 2.4 (i)-(ii), as n → +∞ we have Since the number of terms in S 2k−1 , namely #{j k−1,+ + 1, . . . , j k, }, is of order n as n → +∞, the above asymptotics can be rewritten as where where E n = O(n −1 ) and E n = O(n −1 ) as n → +∞ uniformly for j ∈ {j k−1,+ + 1, . . . , j k, }. The large n asymptotics of S n , S n and S n can be obtained using Lemma 3.4 with a n = j k−1, and with f replaced by f 1,k−1 , − 1 2 , f 2,k−1 and f 3,k−1 respectively. Thus it only remains to obtain the asymptotics of S n . We can estimate the E n -part of S n using (3.18) as follows: where we recall that M is a large but fixed constant (independent of n). Thus we have S n = S 0 + O( log n n ) as n → +∞, where By changing the index of summation in (3.32), and using again (3.18), we get as n → +∞, (3.33) where the error term has been estimated in a similar way as in (3.31), and To estimate the remaining sum in (3.33), we split it into two parts as follows For the second part, we have provided M is chosen large enough. For the first part, since f 0 is analytic in a neighborhood of 0, as n → +∞ we have Hence, we have just shown that as n → +∞.
where t k is given in (3.5) and k,− are given in (3.26)-(3.29). Substituting the asymptotics of Lemmas 3.6 and 3.7 in (3.13), we directly get the following.
and where t k is given in (3.5) and Proof. It suffices to substitute the quantities f 1,k , f 2,k , f 3,k , A k , B k by their definitions (3.14), (3.26)-(3.29), and to perform some simple primitives.
We now turn our attention to the sums S 2k , k = 1, . . . , 2g. Their analysis is very different from the analysis of S 2k−1 . We first make apparent the terms that are not exponentially small.

Thus, by (3.10) and Lemma 2.4 (i)-(ii), we have
as n → +∞ for some constant c > 0, and the claim follows.
Let M = n 1 12 . We now split the sums on the right-hand sides of (3.35) and (3.36) into three parts S 2k , which are defined as follows where 2k and S 2k depend on this new parameter M , but their sum S 2k does not. Note also that S (2) 2k is independent of the other parameter , while S (1) 2k and S (3) 2k do depend on . The analysis of S (2) 2k is very different from the one needed for S (1) 2k and S (3) 2k . For S (1) 2k and S (3) 2k , we will approximate several sums of the form j f (j/n) for some functions f , while for S (2) 2k , we will approximate several sums of the form j h(M j,k ) for some functions h, where M j,k := √ n(λ j,k − 1). As can be seen from (3.38) and (3.39), the sum S 2k , S 2k is that all the functions f and h will blow up near certain points. To analyze S (2) 2k , it would be simpler to define M as being, for example, of order log n, but in this case the sums j f (j/n) involve some 2k are even empty sums), but in this case the sums j h(M j,k ) involve some M j,k 's that are too close to the poles of h. Thus we are tight up from both sides: M of order log n is not large enough, and M of order √ n is too large. The reason why we choose exactly M = n 1 12 is very technical and will be discussed later. We also mention that sums of the form j h(M j,k ) were already approximated in [17], so we will be able to recycle some results from there. However, even for these sums, our situation presents an important extra difficulty compared with [17], namely that in [17] the functions h are bounded, while in our case they blow up near either +∞ or −∞.
We now introduce some new quantities that will appear in the large n asymptotics of S (1) 2k , S Clearly, θ Remark 3.12. Note thatm j,n depends on k, although this is not indicated in the notation.
for some ξ j,n (x) ∈ [M j , x]. For the summand of the last term, we have the bound Since the following asymptotics (3.45) hold as n → +∞ uniformly for j ∈ {g k,− + 1, . . . , g k,+ }, by rearranging the terms in (3.44) we obtain |M j |m j,n (h) + (1 + M 2 j )m j,n (h ) + |M j |m j,n (h ) +m j,n (h ) for all large enough n and for a certain C > 0. The sums appearing on the right-hand side of (3.46) can be analyzed similarly. For example, we have for some C > 0 and for all sufficiently large n. After substituting (3.47) in (3.46) we obtain where | E 6 | is also bounded by (3.48) for a certain C > 0 and for all large enough n. We then use to obtain asymptotics for the integrals in (3.49). The errors in these expansions are smaller than (3.48). The claim then follows after a computation.  (3.53) and , if k ∈ {2, 4, 6, . . . , 2g}, In the above equation for h 2,k , the first line reads for k ∈ {1, 3, 5, . . . , 2g − 1} and the second line reads for k ∈ {2, 4, 6, . . . , 2g}. Proof. We only do the proof for k odd (the case of k even is similar). For convenience, in this proof we will use λ j , η j and M j in place of λ j,k , η j,k and M j,k . From (2.6), (3.38) and (3.41), we see that Recall from (3.42) that for all j ∈ {j : λ j ∈ I 2 }, we have Hence, using (3.2) we obtain (3.55) as n → +∞ uniformly for j ∈ {j : λ j ∈ I 2 }. By Taylor's theorem, for each j ∈ {j : λ j ∈ I 2 } we have and (3.55), we infer that there exists a constant C > 0 such that holds for all sufficiently large n and all j ∈ {j : λ j ∈ I 2 }. Similarly, by Taylor's theorem, for each j ∈ {j : λ j ∈ I 2 } we have . Furthermore, R a (η) is analytic with respect to λ (see [71, p. 285]), in particular near λ = 1 (or η = 0), and the expansion (2.8) holds in fact uniformly for | arg z| ≤ 2π − for any > 0 (see e.g. [62, p. 325]). It then follows from Cauchy's formula that (2.8) can be differentiated with respect to η without increasing the error term. Thus, differentiating twice (2.8) we conclude that there exists C > 0 such that  .9), we obtain after a computation that as n → +∞. Each of these three sums can be expanded using Lemma 3.11. The errors in these expansions can be estimated as follows. First, note that the function e − ponentially small as x → +∞, and is bounded by a polynomial of degree 1 as x → −∞. Hence, the functions h 0,k (x), h 1,k (x) and h 2,k (x) also tend to 0 exponentially fast as x → +∞, while as x → −∞ they are bounded by polynomials of degree 2, 3 and 6, respectively. The derivatives of h 0,k (x), h 1,k (x) and h 2,k (x) can be estimated similarly. Using Lemma 3.11, we then find that the fourth term in the large n asymptotics of Similarly, the third term in the asymptotics of 1 , and the second term in the asymptotics of 1 . All these errors are, in particular, O(M 9 n −1 ). Hence, by substituting the asymptotics of these three sums in (3.60), we find the claim. 7,k appearing in (3.50) depend quite complicatedly on M . The goal of the following lemma is to find more explicit asymptotics for S (2) 2k . We can do that at the cost of introducing a new type of error terms. Indeed, the error O(M 9 n −1 ) of (3.50) is an error that only restrict M to be "not too large". In Lemma 3.16 below, there is another kind of error term that restrict M to be "not too small". (3.62) Let k ∈ {2, 4, . . . , 2g}. As n → +∞, we have Proof. We only do the proof for k odd. As already mentioned in the proof of Lemma 3.14, h 0,k (x), h 1,k (x) and h 2,k (x) are exponentially small as x → +∞, and since M = n 1 12 , this implies that there exists c > 0 such that ), as n → +∞, j = 0, 1, 2, = 0, 1, 2.
On the other hand, as x → −∞, we have as n → +∞, and the claim follows. explicitly. Indeed, as can be seen from the statement of Lemma 3.14, h 2,k consists of two parts, and it is easy to check that each of these two parts is of order x 6 as x → −∞. Thus the fact that actually we have h 2,k (x) = O(x 4 ) (see (3.65)) means that there are great cancellations in the asymptotic behavior of these two parts, and this is not something one could have detected in advance without computing explicitly G 1 − 5y 2 3 + 11 3 y 3 + 2y log y + 1 2 + 2 log(2 √ π) y dy.
Proof. For k ∈ {1, 3, . . . , 2g − 1}, using the change of variables y = − xr b k √ 2 and splitting the two integrals around 0 instead of −1, we obtain It is also possible to split the integrals in G (M ) 6,k around 0 by regrouping some integrands as follows We obtain the claim for k odd after substituting these expressions in the definitions ( 2k . The analogue of these sums in [17] were relatively simple to analyze, see [17, Lemmas 2.5 and 2.6]. In this paper, the sums {S    68) and the last expansion holds as n → +∞ uniformly for j ∈ {g k,+ + 1, . . . , j k,+ }. We recall that the functions c 0 and c 1 are defined in (2.9).
Proof. This follows from a direct application of Lemma 2.3.

The asymptotic analysis of {S
2k } k even is challenging partly because, as can be seen from the statement of Lemma 3.21, there are four types of n-dependent parameters which vary at different speeds. Indeed, as n → +∞ and j ∈ {g k,+ + 1, . . . , j k,+ }, the quantities √ a j , η j,k , j/n and α/n are of orders √ n, j/n − br 2b k , 1 and 1 n respectively. In particular, for j close to g k,+ + 1, η j,k is of order M √ n , while for j close to j k,+ , it is of order 1. In the next lemma, we obtain asymptotics for the right-hand sides of (3.66) and (3.69). These asymptotics will then be evaluated more explicitly using Lemma 3.4.
By applying Lemma 3.4 with f replaced by g k,1 , . . . , g k, 6 , h k, 3 and h 4,k , we can obtain the large n asymptotics of the various sums appearing in the above Lemma 3.22. Note that, as already mentioned in Remark 3.23, the functions g 4,k , g 5,k , g 6,k and h k, 4 where we have used that g k, 3 (A) n = O( log n n ) to estimate the error term. After expanding g k, 3 (A), we find (3.88). The expansion (3.89) direct follows from Lemma 3.4 with f replaced by g k, 4 and with A, a 0 , B, b 0 as in (3.94), and from the fact that . Next, applying Lemma 3.4 with f replaced by g k, 5 and with A, a 0 , B, b 0 as in (3.94), we get where we have used to estimate the error term. Since as n → +∞, and (3.90) follows. Similarly, using Lemma 3.4 with f replaced by g k, 6 and with A, a 0 , B, b 0 as in (3.94), we get The expansion g k,6 (x) = − 37b 6 r 6b  4 . This finishes the proof of the formulas corresponding to k odd. The formulas corresponding to k even can be proved in a similar way.
Lemma 3.28. We have where θ is the Jacobi theta function given by (1.10).
Proof. In the same way as in Lemma 3.3, as n → +∞ we find .
The large n asymptotics of S (1) 2k−1 can be obtained in a similar (and simpler, because there are no theta functions) way than in Lemma 3.6 using Lemma 3.4. We omit further details.
By substituting the asymptotics of Lemmas 3.1, 3.2, 3.8, 3.26 and 4.1 in (4.1), and then simplifying, we obtain the statement of Theorem 1.4. 5 Proof of Theorem 1.7: the case r 1 = 0 We use again (2.5), but now we split log P n into 4g − 1 parts as follows log P n = S 3 + S 4 + .