Uniform asymptotics of Toeplitz determinants with Fisher-Hartwig singularities

We obtain an asymptotic formula for $n\times n$ Toeplitz determinants as $n\to \infty$, for real valued symbols with any fixed number of Fisher-Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.


Introduction
In this paper, we consider the asymptotics as n → ∞ of Toeplitz determinants where the symbol f is of the form satisfying the following conditions: (a) V (z) is real-valued for |z| = 1 and is analytic on an open set containing |z| = 1, (b) z j = e it j , where 0 ≤ t 1 < t 2 < · · · < t m < 2π, (c) α j ≥ 0 and Re β j = 0 for j = 1, 2, . . . , m.
Under these conditions f is a real valued symbol, and we obtain large n asymptotics of D n (f ) (up to a bounded multiplicative term) which are uniform in the parameters z j .
When (α j , β j ) = (0, 0) for all j, one says the Toeplitz determinant possesses a Fisher-Hartwig (FH) singularity at each point z j = e it j , and that the singularity at z j is of root-type if β j = 0 and of jump-type if α j = 0.
The large n asymptotics of Toeplitz determinants were first studied by Szegő in 1915 [52]. They have been intensively studied over the last 70 years, and owe their relevance to applications to physical models. The most prominent such application is the question of spontaneous magnetization of the Ising model on the lattice Z 2 (see e.g. [44] and [17]), but we also mention questions surrounding the momentum of impenetrable bosons in 1 dimension, which we return to in Section 3.
In addition to physical models, a considerable effort has been invested in understanding statistical similarities between the asymptotics of the Riemann zeta function along the critical line Re z = 1/2 and the statistics of the characteristic polynomial of the Circular Unitary Ensemble (CUE) over arcs of the unit circle. Toeplitz determinants appear in this context, and we return to this topic in Section 2.
We now turn to known results for the asymptotics of D n (f ). The simplest case is the special one where ω(z) ≡ 1 (i.e. α j , β j = 0 for all j), in which case as n → ∞, where V k = 2π 0 V e iθ e −ikθ dθ 2π . This is known as the strong Szegő limit theorem (see [31,34,37,53]), and holds for V satisfying condition (a), but also more generally for any V such that ∞ k=−∞ |V k | 2 converges. It was conjectured by Lenard [42] Fisher and Hartwig [21], and proven in subsequent steps by Widom [58] (relying also on work by Lenard [42]) and Basor [4,5], that if f is a symbol of the form as n → ∞, where E is independent of n and given by V + (z) = ∞ j=1 V j z j , and G(z) is the Barnes' G-function (see e.g. [48]). We mention here that although our focus is on real-valued symbols, the analogue of (2), (3) in the case of complex symbols f is interesting and exhibits behaviour with additional subtleties, see [7,16,18], or [17] for a review.
By the proof in [18], it is clear that the asymptotics (2) hold uniformly for e it j and e it k bounded away from each other. It is also clear that the asymptotics (2) are discontinuous if any two points e it j , e it k merge and that the asymptotic formula cannot be correct in this situation. In [36] and [55], for α = 1/2 and β = 0, a part of the transition was considered, corresponding to the box |e it j − 1| < C/n for all j = 1, . . . , m and some fixed, large constant C. More recently, in [11], the authors considered the situation where m = 2 and obtained the full asymptotics of D n (f ), uniformly for 0 ≤ t 1 < t 2 < 2π. It is easily seen that the results of [11] may be presented in the following manner: log D n (f ) = nV 0 + 2 j=1 α 2 j − β 2 j log n+2(α 1 α 2 −β 1 β 2 ) log 1 sin t 1 −t 2 2 + n −1 + F n (t 1 , t 2 )+o (1), (4) uniformly as n → ∞, where F n is an explicit function in t 1 , t 2 which is uniformly bounded as n → ∞. We mention that F n has an interesting and intricate representation involving a solution to the Painlevé V equation when |t 1 − t 2 | = O(1/n) -for the details we refer the reader to [11] (and additionally to [9] for certain simplifications that occur in the specific case where α is integer-valued and β = 0).
We also refer the reader to work on a different but related problem, namely the transition between smooth symbols and those with one singularity, see [10] and [59].
In this paper, we obtain asymptotics for D n (f ) as n → ∞, uniformly in the parameters t 1 , . . . , t m .
Our main result is the following. where the error term is uniform for 0 ≤ t 1 < t 2 < · · · < t m < 2π.
2 The characteristic polynomial of the CUE Let Z 1 , Z 2 , . . . , Z n be random variables, distributed as the eigenvalues of the n × n Circular Unitary Ensemble of random matrices, with the following joint probability density function on the unit circle in the complex plane: Let the characteristic polynomial be denoted by It has long been believed that the statistical properties of the Riemann zeta function on the critical line s = 1/2 + it, t ∈ R, and the statistics of large random matrices are related -it was Dyson who first spotted this possible connection. More recently, possible connections between the behaviour of the characteristic polynomial P n (e iθ ) over the unit circle and the behaviour of the Riemann zeta function along the critical line have been studied intensively (see e.g. [12,13,14,29,28,32,33,38,45] and references therein). In this context the authors of [28,29] were interested in both extreme values and average values of P n (e it ) over the unit circle, namely the random variables In addition, their work sparked interest in connections between the characteristic polynomial of the CUE and Gaussian Multiplicative Chaos, see [57,47] for results on such connections.
In [28] it was conjectured that log Y n − log n + 3 4 log log n converges in distribution to a random variable, and subsequently the asymptotics of Y n have been studied in [1,49,8]. In these works, the terms log n and 3 4 log log n were confirmed. The full conjecture, however, remains open. In [29], Fyodorov and Keating conjectured that for m = 1, 2, . . . , as n → ∞, where E denotes the expectation with respect to (5), and As a corollary to Theorem 1.1, we will prove (6).
To make the connection between Toeplitz determinants and the moments of X n (α), we recall the well-known representation of Toeplitz determinants in terms of multiple integrals from which it follows that where we denote Using (8) and (4), Claeys and Krasovsky were able to prove (6) for m = 2. They were furthermore able to prove that E X n (α = 1/ √ 2) 2 =ĉn log n(1 + o(1)) as n → ∞ for an explicit constantĉ.
Fix δ > 0. We will show that there is an integer N such that for n > N, thus proving the corollary. Given a measurable subset R ⊂ [0, 2π) m , we denote We note that I ǫ (α, R) < I 0 (α, R) for any ǫ > 0 for any R ⊂ [0, 2π) m . For η > 0, divide the integration regime [0, 2π) m into two regions R 1 (η) and R 2 (η), where R 1 (η) is the region where sin > η for all i = j, and R 2 (η) is the complement of R 1 (η). It follows by Theorem 1.1 that as n → ∞, uniformly for 0 < η < η 0 . In particular, since I 0 (α, R 2 (η)) → 0 as η → 0, it follows that there exists η 0 > 0 and N 0 ∈ N such that for n > N 0 and η < η 0 , which gives the desired bound for the integral over R 2 (η). We now evaluate the integral over R 1 (η). By (2), it follows that where the o(1) tends to zero uniformly over R 1 (η) for any fixed η as n → ∞. Thus we may move the error term outside the integral, and since R 2 (η) is the complement of R 1 (η), we have By (11) we obtain where the o(1) tends to zero for any fixed η as n → ∞. Again we use the fact that I 0 (α, R 2 (η)) → 0 as η → 0, from which it follows that we may pick η < η 0 such that the second term on the right hand side of (16) is less than δn mα 2 /4.

mα 2 > 1
We now study the asymptotics of I ǫ (α) for mα 2 > 1. A lower bound for I ǫ (α) is easily obtained-by integrating over the box |t i − t j | < ǫ for all i = j, it is easily seen that there is a constant c such that for 0 < ǫ < ǫ 0 .
We now take the change of variables s j = t j+1 − t j for j = 1, . . . , m − 1 and find that with integration taken over s 1 , . . . , s m−1 ≥ 0 such that m j=1 s j < 1 − t 1 , from which it follows that where If 2α 2 > 1, then I (2) ǫ (α) is straightforward to evaluate -one simply notes that for any i = j, . . . , k, and thus Separating out the variables, it follows that as ǫ → 0, for 2α 2 > 1. However, if 1 m < α 2 < 1 2 , this approach fails to yield (22), and in fact yields a worse error term 1 . To achieve the optimal error term (22) also in the case 1 m < α 2 < 1 2 , we need to consider ordered integrals.
Since the integral (20) is taken over all possible orderings, it follows that where the integral is taken over . Then S(1), . . . , S(m − 1) are disjoint, and By (21) it follows that 1 As an example where the approach fails to provide optimal error terms, consider m = 3 and α 2 = 2/5. Then we obtain as ǫ → 0. However the optimal bound we are looking to obtain is of order It is easily seen that |S(ℓ)| = m − ℓ, and it follows that By (23), it follows that If m = 2 and mα 2 > 1, then the right hand side is of order ǫ −2α 2 +1 , and we are done. We assume that m > 2, and integrate in s σ(m−1) on the right hand side of (24). The power of s σ(m−1) is −2α 2 , which could very well be equal to −1, so we need to take this into account. Clearly for any fixed x as ǫ → 0. Since as ǫ → 0 where integration on the right hand side is taken over 0 < s σ(m−2) < · · · < s σ(1) < 1. We will next integrate out s σ(m−2) , then s σ(m−3) , etc. To do this, we introduce the following notation for v = 1, 2, . . . , m − 2: with integration taken over 0 < s σ(m−v−1) < · · · < s σ(1) < 1, and where we interpret 0 j=1 j = v j=v+1 j = 0. We observe that the error term on the right hand side of (25) is equal to J ǫ (1). It is easily verified that as ǫ → 0, where we interpret 0 j=1 j = m−2 j=m−1 j = 0. Since mα 2 > 1, it follows that the power of s + ǫ is smaller than −1, namely: for any r = 0, 1, 2, . . . , m − 2. If x < −1, then as ǫ → 0, and thus it follows that as ǫ → 0. Thus by (19) and (24), , as ǫ → 0, which combined with the lower bound (17) and Theorem 1.1 proves Corollary 2.1.

mα 2 = 1
We start by finding a lower bound for I ǫ (α) when mα 2 = 1. Let (10) satisfies for j = 1, 2, . . . , 1/ǫ, assuming for ease of notation that 1/ǫ is an integer. Combined with the fact that B ǫ (j) are disjoint for j = 1, 2, . . . and the fact that mα 2 = 1, it follows that for j sufficiently large, say j > j 0 , it follows that for some constantĉ 0 . Thus we have a lower bound for I ǫ (α) and we look to obtain a corresponding upper bound.

Statistics of impenetrable bosons in 1 dimension
Consider It was proven by Girardeau [30] that it has the following properties: • ψ is the ground-state solution to the general time-independent Schrödinger equation in onedimension with n particles.
• ψ is symmetric with respect to interchange of x i and x j for i = j (Bose-Einstein statistics).
• ψ is translationally invariant with period L.
• ψ vanishes when x i = x j for i = j (mutual impenetrabililty of particles).
In fact Girardeau only proved the above for odd n, but as noted by Lieb and Liniger [43] (footnote 6), it is equally valid for even n. When the system is in ground state, the wave function ψ gives rise to a probability distribution for both the position and momentum. The position of the particles on [0, L) has joint probability density function ψ(x 1 , . . . , x n ) 2 . Following the footsteps of Girardeau, we take as our starting point that the wave function for the momentum is given by the Fourier transform of the wave function of the position: Thus the probability of the j'th particle having momentum 2πM j /L for each j = 1, . . . , n is given by It is easily verified that φ(M 1 , . . . , M n ) is independent of L, and that Thus |φ| 2 may simply be viewed as a probability distribution on Z n , which is the viewpoint we will take in Corollary 3.1 below, where we fix L = 2π without loss of generality.
Since the particles are indistinguishable from one another, it is preferable to characterize the distribution as a point process, which we do as follows. Let The above is only valid for distinct particles, for moments of N M we have where N M = N M (n). Then the expected number of particles with 0 momentum is given by π 1,n (0).
In 1963, Schultz [50] proved that π 1,n (0) = O(n −π/4 ) as n → ∞, which shows that there is no Bose-Einstein condensation according to the Penrose-Onsager criterion (the criterion states that if the proportion of the particles expected to have 0 momentum tends to 0 as n → ∞, then there is no Bose-Einstein condensation). The upper bound obtained by Schultz was not optimal. In 1964 Lenard [41] was able to improve on this, and obtained that E(N 0 (n)) = O(n 1/2 ) as n → ∞. Lenard's approach was to make a connection to Toeplitz determinants with Fisher-Hartwig singularities by observing that if we denote the k particle reduced density matrix by then ρ k,n is a Toeplitz determinant with 2k FH singularities. This observation relies on the multiple integral formula (7). By (29) and (31) it is easily verified that . , x k , y 1 , . . . , y k ). (32) Thus, to obtain the asymptotics of π k,n one must obtain those of ρ k,n . The asymptotics of ρ k,n (x 1 , . . . , x k , y 1 , . . . , y k ) was studied in the limit n → ∞ with L = n and for fixed x j , y j independent of n in [55,36]. This is equivalent to studying Toeplitz determinants with 2k FH singularities with α j = 1/2 for j = 1, . . . , 2k, in the double scaling limit where the singularities are all at a distance of length O(1/n) from each other. This gave rise to some of the first connections to Painlevé V in the study of Toeplitz determinants. To obtain more detailed asymptotics for π k,n however, uniform asymptotics of ρ k,n are required. As mentioned in the introduction, Claeys and Krasovsky [11] obtained uniform asymptotics for ρ 1,n , and they relied on (4) to prove that as n → ∞ (see formula (1.53) of [11]).
We are interested in not just the expectation of N 0 (n)/ √ n, but also the variance and higher moments. By combining (2)-(3) with Theorem 1.1, we obtain the following. Then, as n → ∞, Proof. To study higher moments of N 0 = N 0 (n), we have by (30) that for any fixed k.
Thus the corollary follows from (33).

Method of proof of Theorem 1.1
Denote ψ 0 (z) = χ 0 = 1/ D 1 (f ) and define the polynomials ψ j for j = 1, 2, . . . by where the leading coefficient χ j is given by By the representation (7), it follows that D j (f ) > 0 and we fix χ j > 0. It is easily seen that ψ j are orthonormal on the unit circle: for j, k = 0, 1, 2, . . . . By (34) and the definition Given U > ǫ > 0, we say that the parameters t 1 , . . . , t m satisfy condition and for each 1 The assumption that t m < 2π − π/m one can make without loss of generality when studying Toeplitz determinants, since the Toeplitz determinant is rotationally invariant (i.e. D n (f (e iθ )) = D n (f (e i(θ+x) )) for all x ∈ [0, 2π)).
• The radius of each cluster is less than ǫ/n. Namely, ǫ n < ǫ, where • The distance between any two clusters is greater than U/n. Namely, u n > U, where In Sections 5-6, we prove the following proposition.
(b) There exists U 1 > U 0 > 0, C > 0 and n 0 > 0, such that if the parameters t 1 , . . . , t m satisfy condition (U 0 , U 1 , n) and n > n 0 , then Using Proposition 4.1, we now compute the asymptotics of D n (f ) as n → ∞, for a specific configuration 0 ≤ t 1 < t 2 < · · · < t m < 2π − π/m, but with error terms which are uniform over all configurations. Let n 0 be a fixed positive integer such that the asymptotics of Proposition 4.1 are valid for n ≥ n 0 . Then D n 0 (f ) is a continuous function in terms of t j on the compact set , and is thus uniformly bounded as t j vary. Thus by (35) log D n = −2 Written differently, we have and it follows that uniformly for t j+1 − t j > 0. Since U 0 and U 1 are fixed, it follows that the right hand side is uniformly bounded, and by Proposition 4.1 (a) and the fact that uniformly for t 1 , . . . , t m ∈ [0, 2π − π/m).
Suppose that t 1 , . . . , t m satisfy condition (U 0 , U 1 , N) for N in an interval N 1 , N 1 + 1, . . . , N 2 . By N 1 , and as a consequence (bearing in mind that Thus, bounding the sum by a suitable integral, Similarly, ǫ N /N = ǫ N 2 /N 2 , and thus Since J c t is composed of at most m − 1 disjoint intervals, it follows that J t is composed of at most m disjoint intervals, and it follows by (40) Since U 0 , U 1 , n 0 are just arbitrary constants, the right hand side is bounded uniformly over S t . Thus, by (38), (39), (43), it follows that uniformly over S t . Since n N =1 as n → ∞, with the implicit constant depending only on U 0 which is fixed, it follows that as n → ∞, uniformly over S t . For such t 1 , . . . , t m , we have with uniform error terms, which yields Theorem 1.1 for t 1 , . . . , t m ∈ S t , and the full theorem follows from the aforementioned rotational invariance of the Toeplitz determinant.
Then the configuration t 1 , . . . , t m will satisfy one of the conditions in the sequence (45), because otherwise, for each k = 1, 2, . . . , m, one would have a corresponding j = j(k) = 2, 3, . . . , m such that , which is a contradiction. Thus, if we denote the maximum of the implicit constants C(U j ) and N 0 (U j ) over j = 1, . . . , m − 1 by C max and N 0,max we obtain |log χ n + V 0 /2| < C max /n, for any n ≥ N 0,max , which proves Proposition 4.1 (a).
We will prove Proposition 4.1 (b) and Proposition 4.2 by applying the Deift-Zhou steepest descent analysis [19] to a Riemann-Hilbert (RH) problem associated to the orthogonal polynomials ψ j . Under the Deift-Zhou steepest descent framework, there are several standard ingredients, including the opening of the lens, and the construction of a main parametrix and local parametrices. Among these ingredients, the opening of the lense and the construction of a local parametrix is the most involved.
Each local parametrix contains a cluster Cl j (u, U, n), and we map a model RH problem to a shrinking disc containing Cl j (u, U, n). We construct and analyze the model RH problem in the next section, Section 5, and use these results in Section 6 to prove Propositions 4.1 (b) and 4.2.

Model RH problem
In this section we introduce and analyze a model Riemann-Hilbert problem, yielding results which we rely on to prove Propositions 4.1 (b) and 4.2 in Section 6.
The model RH problem will later be used to construct a local parametrix at each cluster of points, where µ will be the number of points in the cluster. In particular it means that the ordering of the α j , β j here do not necessarily correspond with those in the definition of the Toeplitz determinant, see Section 6 for details on how the model RH problem is utilized.
(d) Φ(ζ) is bounded as ζ → ±iu. As ζ → iw j for j = 1, . . . , k in the sector arg(ζ − iw j ) ∈ (π/2, π), for some function F j which is analytic on a neighbourhood of iw j , and Note that ±iu are not special points, and therefore the values of u are not particularly important.
However, we present the RH problem in this manner for notational convenience and to make it clear that the local behaviour at each singularity can be presented in the same form, also for the top and bottom singularity.
We also note that g was chosen such that the local behaviour of Ψ at the point iw j is consistent with the jumps.

The case of a single singularity µ = 1
When there is only one singularity µ = 1 and w 1 , u = 0, the RH problem for Φ (and equivalent versions of it) has been studied by many authors. It was first solved by Kuijlaars and Vanlessen in [40,56] for β = 0 in terms of Bessel functions, and brought to the setting of determinants by Krasovsky in [39]. For α = 0 it was solved by Its and Krasovsky in [35], and a solution for general α, β was found in terms of confluent hypergeometric functions by Deift, Its, Krasovsky in [16,18] and Moreno in [46].
Claeys, Its and Krasovsky [10] brought the above solution to the form which we will refer to. In [10] the RH problem is denoted by M, which we will denote by M CIK , and by comparison of RH problems it follows that when one takes u = 0. The solution to M CIK may also be found in [11], Section 4, where we find the following formula where Γ is Euler's Γ function.
5.0.2 The case of multiple singularities µ > 1 In the case of 2 singularities µ = 2, an equivalent version of the RH problem for Φ was proven to have a unique solution by Claeys and Krasovsky [11], and to be connected to the Painlevé V equation.
See also [23] for a reference on Riemann-Hilbert problems connected to the Painlevé equations. The proof of a unique solution by [11] generalizes easily to our situation of µ = 1, 2, 3, . . . singularities, and we have included a proof of the following proposition in the Appendix for the reader's convenience.

Continuity of Φ for varying w j 's
The main result of Section 5 is the following.
The first step in the proof of Lemma 5.2 is to transform the RH problem for Φ to a RH problem for Φ which is analytic except on the imaginary axis Re z = 0, and in particular the jump contour is independent of the locations of the singularities w j (though the jumps themselves will vary with the location of the singularities).

Transformation of RH problem
Let where I-VI are regions in the complex plane given in Figure 2.
Let Φ be defined in terms of Φ as follows.
Then Φ solves the following RH problem.

Steepest descent analysis of Φ
We now prove that Φ is continuous with respect to the parameters w 1 , . . . , w µ .
We plan to approximate the RH problem associated with the w j 's by the RH problem associated with the W j 's, and so for increased clarity we label them as different functions.
Let the RH problem associated with w 1 , . . . , w µ be denoted by and the RH problem associated with W 1 , . . . , W τ by We note that N has the same jumps as Ψ except on neighbourhoods containing W 1 , . . . , W τ , and that Ψ(ζ)N(ζ) −1 → I as ζ → ∞ by condition (c) for the RH problem for Φ and the definition of Φ.
We will additionally need to show that there exists a local parametrix Q(ζ) on fixed neighbour- as ǫ → 0, uniformly for ζ ∈ ∂U W j . Although we only need existence of such a local parametrix, we prove the existence by construction, and we do this in the next subsection, Section 5.3.1. By standard theory of small norm problems, see e.g. [15], it will follow that N approximates Ψ well outside of the neighbourhoods ∪ τ j=1 U W j , which we will subsequently use in Section 5.4 to prove Lemma 5.2.

Local parametrix
We construct a local parametrix at the point W j which will contain the points w M j , for j = 1, . . . , τ , and are inspired here by a similar construction in [11] in the special case of two singularities.
Throughout the section 5.3.1, j will be fixed, and to reduce the number of superscripts, we denote throughout the section for ν = 1, 2, . . . , M j . Let U W j be a fixed open disc centered at iW j with a fixed radius R > 0.
We first take a transformation Ψ → Ψ j , where Ψ j is analytic for all and similarly a transformation N → N j such that N j is analytic for all On U W j , we define for j = 2, 3, . . . , τ , and On (iy ν+1 , iy ν ), Ψ j has the jumps for ν = 1, . . . , M j−1 where orientation of the contour is taken upwards, where J 0 was defined in condition (b) of the RH problem for Φ, and on arg(ζ − y M j ) = 3π/2, We search for a local parametrix Q j such that Q j has the same jumps as Ψ j on U W j and such that Q j N −1 j = I + O(ǫ) as ǫ → 0, uniformly on the boundary ∂U W j for j = 1, 2, . . . , τ (it follows that Q j differs from Q by right multiplication of J where g was defined in (46), E j is an analytic function given below in (63), arg(ζ−iy ν ) ∈ (−π/2, 3π/2), and We first consider the jumps of Q j . If , and it is easily verified by comparison with (61) that Q −1 j,− Q j,+ = Ψ −1 j,− Ψ j,+ for ζ with arg ζ − iy M j = 3π/2. If in addition h(λ) extends to an analytic function on a an open set containing (ia, ib), then for all ζ ∈ (ia, ib), with upward orientation, so for ζ ∈ (iy ν+1 , iy ν ), ν = 1, 2, . . . , M j − 1, and it follows that By comparison with (60) and the definition of c ν , it follows that Q −1 j,− Q j,+ = Ψ −1 j,− Ψ j,+ on (iy ν+1 , iy ν ), ν = 1, 2, . . . , M j −1. Furthermore, since Q j is bounded on U j and by the fact that (using the definition of Ψ and condition (d) of the RH problem for Φ) is bounded on U j , and the fact that E j is analytic, it follows that Ψ j (ζ)Q j (ζ) −1 is analytic on U j .
We define E j by We recall that A M j = A (j) . Thus, by the definition of N j and condition (d) for the RH problem for for arg(ζ) ∈ (π/2, π), and one verifies that the singularity of N j cancels with that of (ζ − iW j ) −A M j σ 3 . It is easily seen that E j has no jumps on U W j , and thus it is analytic. For ζ ∈ ∂U W j , define We first note that E j are analytic functions on U W j , and since they are independent of ǫ, the are as ǫ → 0, uniformly for ζ ∈ ∂U W j , it follows by (62) and the boundedness of E j on ∂U W j , that as ǫ → 0, uniformly for ζ ∈ ∂U W j .
We define E j by and in a similar manner to the case 2A M j / ∈ N, it follows that E j is analytic on U W j , and that as ǫ → 0, uniformly for ζ ∈ ∂U W j .

Small norm matrix
for Re ζ > 0.
Let R be given by By (64) and (66), it follows that R satsisfies the following RH problem.
By standard small norm analysis, as ǫ → 0 uniformly for the parameters for fixed W 1 , . . . , W τ , with the implicit constant depending only on u, and the parameters α j , β j .
We denote {w 1 , . . . , w µ } = {W 1 , . . . , W τ }, where the points W τ < · · · < W 1 are distinct. Let as k i → ∞. By condition (c) for the RH problem for Φ ζ; (W j , A j , B j ) τ j=1 , and the definition of Φ, it follows that where r(ζ) is given by where the branch cuts of r are a subset of [−iu/2, iu/2] and r(ζ) → I as ζ → ∞. By condition (c) of the RH problem for Φ ζ k i ; (W j , A j , B j ) τ j=1 , and the fact that r(ζ) = O(1/ζ) uniformly in k i as ζ → ∞, it follows that as k i → ∞. Thus the left hand side of (68) is bounded as k i → ∞, which is a contradiction, concluding the proof of Lemma 5.2 (a).

Asymptotics of the orthogonal polynomials
Define Y = Y (z) in terms of the orthogonal polynomials: with the integration taken in counter-clockwise direction on the unit circle C, and where ψ n−1 (z) = ψ n−1 (z). The function Y uniquely solves the following Riemann-Hilbert Problem (a) Y : C \ C → C 2×2 is analytic; That Y defined in (70) solves the RH problem for Y is easily verified, and is a result due to Baik, Deift, Johansson [6], who were inspired by a similar observation by Fokas, Its, Kitaev [22] concerning orthogonal polynomials on the real line. It is immediate that We rely on the Deift-Zhou [19] steepest descent analysis for RH problems to obtain the asymptotics of Y (0) as n → ∞. See e.g. [15] for an introduction to analysis of RH problems. z − e it j e it j e πi α j +β j for z ∈ C \ {z : arg z = arg t j , |z| ≥ 1}, analytic on C \ {z : arg z = arg t j }. In [16], it was noted that for |z| < 1, we have D(z) = D in (z) and for z > 1 we have D(z) = D out (z). Furthermore, for z ∈ C \ ∪ m j=1 e it j , and we extend the definition of f by letting f be defined by (73) on C \ ({0} ∪ {z : arg z = arg t j }). It follows that on {z : arg z = arg t j }, with the orientation taken away from 0 and toward ∞.
6.1 Transformation of the RH problem for Y , and opening of the lens Let U > 0 be such that the asymptotics of Lemma 5.2 (a) hold for ζ > U /3, for any µ = 1, 2, . . . , m. Recall the notation from Section 4. Assume that t 1 , . . . , t m satisfies condition (u, U, n).
In this section we denote Cl j = Cl j (u, U, n). Denote the number of points in each set Cl j by µ j for j = 1, 2, . . . , r, and let We denote the elements of Cl j = {t  i , and order the parameters so that t (j) 1 > · · · > t (j) µ j . In this way we have a natural partition It is clear that the parameters which we label α in this section are not in general in direct correspondence with the parameters of the same notation in Section 5.
We let Λ j = z : |z| = 1, −u ≤ n arg(z) − t j ≤ u , so that {e it : t ∈ Cl j } ⊂ Λ j and so that e it (j) 1 , e it (j) µ j are not the endpoints of the arc Λ j , and define We open a lens around each arc comprising C \ Λ, where C is the unit circle, as in Figure   3. Let   inside the lenses and outside the unit disc,   inside the lenses and inside the unit disc.
By noting that and by using the factorisation it is easily verified that S uniquely solves the following RH problem.
6.1.1 RH problem for S (a) S is analytic on C \ Σ S , where Σ S is the union of the unit circle and the contours of the lenses.
(b) S has the following jumps on Σ S : z −n f (z) −1 1 on the contours of the lenses, |z| > 1, on the contours of the lenses, |z| < 1,

Main parametrix
We define M by Then M is analytic for z ∈ C \ C, by (73) and M(z) = I + O(z −1 ) as z → ∞. Thus MS −1 solves an RH problem with jumps that converge pointwise to I as n → ∞ except on the shrinking contour Λ, and (MS −1 )(z) → I as z → ∞, so we take M to be our main parametrix.

Local parametrix
We define open sets U 1 , . . . , U r containing each cluster Cl 1 , . . . , Cl r respectively by where we recall t j from (76) and u n from (37). Let for z ∈ U j .
Recall the model RH problem Φ from Section 5. On U j , we define recalling that χ w (ζ) was defined in condition (c) of the RH problem for Φ, and the branches of Thus E j is analytic on U j , and uniformly bounded on ∂U j as n → ∞.
By the jumps of f in (74) and condition (b) for the RH problem for Φ, it follows that P j and S have the same jumps on U j . By condition (c) for the RH problem for Φ, and the boundedness of as n → ∞, uniformly for z ∈ U j , and by (48) the error term is also uniform for t 1 , . . . , t m satisfying condition (u, U, n).

Small norm matrix
Let R be given by (83) R satsisfies the following RH problem.
RH problem for R (a) R is analytic on C \ Σ R , where Σ R is the union of the edges of the lenses and ∪ r j=1 ∂U j .
Proof. Small-norm analysis of RH-problems with fixed contours is standard material, see e.g. [15], but for RH-problems with shrinking contours the theory is less developed. In the following, we follow [26], where a slightly more detailed description may be found for a similar problem.
It is easily verified that Consider R max = sup z∈C,j,k∈{1,2} |R j,k (z)|, and assume this maximum is acheived at z max ∈ C ∪ {∞} (or that R + or R − acheives this supremum at z max ). We piecewise analytically continue R − and ∆ to strips of width of order 2c u n /n containing Σ R , for some fixed but sufficiently small c > 0. On these strips the bounds on ∆ from (84) still hold.
Furthermore, on these strips R is either equal to R − or R − (I + ∆), either way it follows by (84) that max j,k∈{1,2} for n sufficiently large, for all z in the strips. By deforming the contour of integration Σ R , but keeping it in the strips, we may assume that z max is of distance greater than c u n /n from Σ R . Crucially, (86) still holds on this deformed contour, and combined with (85), it follows that where we now assume that z max is of distance greater than c u n /n from Σ R . Thus By the fact that z max is of at least distance c u n /n from Σ R for some c > 0, by (84), and by the fact that ∂U j is of length of order u n /n, it follows that as n → ∞, uniformly for t 1 , . . . , t m satisfying condition (u, U, n). Let Σ out Edge denote the edges of the lenses in the exterior of the unit disc in the complex plane. Then for u n > U and some U > 0. It follows that Now consider (R − I) max = z∈C,j,k∈{1,2} |R(z) − I| j,k , and assume this supremum is acheived at z max,2 ∈ C ∪ {∞}. By deforming the contour of integration, we may assume that z max,2 is of distance greater than c u n /n from Σ R , for some constant c > 0. Thus, by (89), (86), (85), it follows that The lemma follows upon integration, by similar arguments to (87) and (88).
where the orientation of the integral is clockwise. Since E j (z) = 0 1 1 0 g σ 3 E,j (z), for some analytic function g E,j , it follows that ∆ 1,22 (z) = Φ 1,11 /ζ j (z), and thus we have proven part (b) of the lemma.
Then V is analytic on C \ Γ V , where Γ V = ∪ 4 j=0 Γ j . On Γ V \ ±iu ∪ µ j=1 w j , the jump matrix of V factorizes into where J V,− is upper triangular and piecewise analytic, and J V,+ is lower triangular and piecewise analytic, and For RH problems of the form V , it is well known (see [60] and [23,24,25]) that V has a unique solution if and only if the homogenous RH problem V Hom has a unique solution, namely 0, where V Hom is analytic on C \ Γ V , has jumps J V on Γ V , and satisfies V Hom (ζ) = O(ζ −1 ), as ζ → ∞.
We will find it easier to work with Φ Hom which we define below. It is easily verified that if Φ Hom has the zero solution as its unique solution, then the same holds for V Hom .
We will prove that Φ Hom (ζ) = 0 is the only function satisfying these conditions, and by the discussion above, it follows that the RH problems for Φ and Φ have unique solutions. Let U(ζ) = Φ Hom (ζ)e where * denotes the conjugate transpose. Then W is analytic on C \ (−i∞, i∞), and we take the orientation of (−i∞, i∞) upwards. We note that if x ∈ R, then as ζ → ix from the "+" side, it follows that −ζ → ix from the "-" side. Thus, by conditions (b) and (d) of the RH problem for Φ and the definitions of Φ, U, W , as ζ → iw j from the + side, Thus W + (ζ) is integrable for ζ ∈ (−i∞, i∞), and since W (ζ) = O (|ζ| −2 ) as ζ → ∞, it follows from Cauchy's theorem that i∞ −i∞ W + (ζ)dζ = 0.
By the definitions of g j and U, and condition (b) of the RH problem for Φ, it follows that g j is analytic on C \ {z : Re z ≥ 0, Im z ∈ [w µ , w 1 ]}. Furthermore, if h j (ζ) = g j (−(ζ + u) 3/2 ), then h j is analytic and bounded for Re ζ > 0, and h j (ζ) = O e −|ζ|/2 as ζ → ±i∞. Thus it follows by Carlson's theorem (see e.g. [54]), that h j (ζ) = 0 for Re ζ > 0, and by analytic continuation it follows that g j (ζ) = 0 for ζ in the domain of g j . It follows that Φ Hom = 0.