The Classical Compact Groups and Gaussian Multiplicative Chaos

We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a random measure on the unit circle minus small neighborhoods around $\pm 1$. We show that for small enough powers and under suitable normalization, as the matrix size goes to infinity, these random measures converge in distribution to a Gaussian multiplicative chaos measure. Our result is analogous to one on unitary matrices previously established by Christian Webb in [31]. We thus complete the connection between the classical compact groups and Gaussian multiplicative chaos. To prove this we establish appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants with merging singularities. Using a recent formula communicated to us by Claeys et al., we are able to extend our result to the whole of the unit circle.


Introduction
In [38], Hughes, Keating and O'Connell proved that the real and the imaginary part of the logarithm of the characteristic polynomial of a random unitary matrix convergence jointly to a pair of Gaussian fields on the unit circle. Using this result Webb established in [49] a connection between random matrix theory and Gaussian multiplicative chaos (GMC), a theory developed first by Kahane in the context of turbulence in [40] (see [47] for a review). Webb proved that powers of the exponential of the real and imaginary part of the logarithm of the characteristic polynomial of a random unitary matrix converge, when suitably normalized, to Gaussian multiplicative chaos measures on the unit circle. This was achieved using results on Toeplitz determinants with merging Fisher-Hartwig singularities due to Claeys and Krasovsky in [25]. In [49] Webb proved the result only in the so-called L 2 -phase, that is those powers for which the second moment of the total mass of the limiting GMC measure exists. In [45] the result was extended to the whole L 1 -or subcritical phase, i.e. was proven to also hold for the (larger) set of powers for which the limiting GMC measure is non-trivial. Since then the connection between the two fields has been extended to other random matrix ensembles. In [20], Chhaibi and Najnudel proved convergence (in a different sense) of the characteristic polynomials of matrices drawn from the Circular Beta Ensemble to a GMC measure on the unit circle. In [13] Berestycki, Webb and Wong proved that, after suitable normalization, powers of the absolute value of the characteristic polynomial of a matrix from the Gaussian Unitary Ensemble converge to Gaussian multiplicative chaos measures on the real line. They proved this result in the L 2 -phase, however it is likely to also hold in the whole L 1 -phase. The analogous result for powers of the exponential of the imaginary part of the logarithm of the characteristic polynomial of the Gaussian Unitary Ensemble was proven in [22], in the whole L 1 -phase. The connection with GMC is closely related to recent developments concerning the extreme value statistics of the characteristic polynomials of random matrices and the associated theory of moments of moments [1-4, 6, 19, 34-36, 46]. It also has interesting applications to spectral statistics; for example, it implies strong rigidity estimates for the eigenvalues [22].
Our purpose here it to extend Webb's result to the other classical compact groups, i.e. to the orthogonal and symplectic groups. Our starting point is a theorem due to Assiotis and Keating concerning the convergence of the real and imaginary parts of the logarithm of the characteristic polynomials of random orthogonal or symplectic matrices to a pair of Gaussian fields on the unit circle 1 . This is the analogous result to the one for random matrices in [38]. We then complete the connection between the classical compact groups and Gaussian multiplicative chaos, by showing that for the orthogonal and symplectic groups we get statements similar to the one Webb proved for the unitary group. Using the same approach as in [49] we prove our results only in the L 2 -phase, i.e. when the limiting GMC measure's total mass has a finite second moment. We believe that using the techniques in [45] one can extend our results to hold more generally in the whole L 1 -phase, however extra care is needed since the covariance function of the underlying Gaussian field has singularities not just on the diagonal but also on the antidiagonal; in particular at ±1 the field has special behaviour. In order to prove convergence to the GMC measure after restricting all involved measures to ( , π − ) ∪ (π + , 2π − ), i.e. when excluding small neighborhoods around ±1, we computed the uniform asymptotics of Toeplitz and Toeplitz+Hankel determinants with two pairs of merging singularities which are all bounded away from ±1. Our results on these asymptotics are similar to those in [25] and [26], and the proof techniques we employ are strongly influenced by these two papers.
To prove convergence on the full unit circle we need also to know the uniform asymptotics of Toeplitz and Toeplitz+Hankel determinants with 3 or 5 singularities merging at ±1. Claeys, Glesner, Minakov and Yang have recently proved a formula for the uniform asymptotics of Toeplitz+Hankel determinants with arbitrarily many merging singularities, up to a multiplicative constant [23]. Using their formula allows us to extend our analysis to around ±1, and so to cover the full unit circle, however for a slightly smaller set of powers than when neighborhoods around ±1 are excluded.

Statement of Main Result and Strategy of Proof
Denote by O(n) the group of orthogonal n × n matrices, and by Sp(2n) the group of 2n × 2n symplectic matrices, i.e. unitary 2n × 2n matrices that additionally satisfy where J := 0 I n −I n 0 . (2. 2) The characteristic polynomial of U n in O(n) or Sp(2n) (then we have instead I 2n and the product is up to 2n) is taken as a function on the unit circle, where all its zeroes lie.
Using this result, Assiotis and Keating have proved the following theorem 2 : Theorem 3. (Assiotis, Keating) Let p n be the characteristic polynomial of a random U n ∈ O(n), w.r.t. Haar measure. Then for any > 0 the pair of fields ( ln p n , ln p n ) converges in distribution in η j j cos(jθ), η j j sin(jθ).

(2.12)
Similarly, for U n ∈ Sp(2n) and any > 0, the pair of fields ( ln p n , ln p n ) converges in distribution in H − 0 × H − 0 to the pair of Gaussian fields X + x,X +x .
Proof strategy: Let I denote either I or [0, 2π). We first remark that by Theorem 4.2. in [41], weak convergence of µ n,α,β to µ α,β in the space of Radon measures on I equipped with the topology of weak convergence is equivalent to as n → ∞, for any bounded continuous non-negative function g on I.
To prove (2.19) we use Theorem 4.28 in [42]: For k, n ∈ N let ξ, ξ n , η k and η k n be random variables with values in a metric space (S, ρ) such that η k n d − → η k as n → ∞ for any fixed k, and also η k d − → ξ as k → ∞. Then ξ n d − → ξ holds under the further condition lim Our setting corresponds to S = R, ρ = | · |, and n,α,β will now be defined by truncating the Fourier series of ln f n,α,β . We have Tr(U j n ) j (2α cos(jθ) − 2iβ sin(jθ)) , where we used that for U ∈ O(n) or Sp(2n) we have Tr(U −j n ) = Tr(U j n ). Definition 8. For k, n ∈ N, α ∈ R, β ∈ iR and θ ∈ I, let and dθ. (2.24) In order to apply Theorem 7 to verify (2.19), which then implies our main results, we thus need to examine the following three limits: for any bounded continuous non-negative function on I so in particular almost surely (and thus also in distribution) (2.29) The second limit (2.26) will be proved in Section 5, using previously established results on the asymptotics of Toeplitz+Hankel determinants.
To show that the third limit (2.27) holds, we will prove the following lemma in Section 4: Lemma 9 (The L 2 -limit). Let α 2 − β 2 < 1/2. Further let 0 ≤ α < 1/2 in the case I = [0, 2π), and α > −1/4 in the case I = I . Then for any bounded continuous non-negative function g on I the following expectation goes to zero, as first n → ∞ and then k → ∞: dθdθ .
The proof in the case I = I , α 2 − β 2 < 1/2, α > −1/4, works in almost exactly the same way. Instead of Theorem 23, which only holds for α ≥ 0, we use new results on the uniform asymptotics of Toeplitz+Hankel determinants of symbols with two pairs of merging singularities bounded away from ±1, which are stated in Theorems 17 and 22 in the next section. To the best of our knowledge these results have not previously been set out and we believe them to be of independent interest.
Under these assumptions V has a Laurent series, convergent in a neighborhood of the unit circle, (3.5) and the function e V (z) allows the standard Wiener-Hopf decomposition: Note that a symbol f as in Definition 10 that is real on the unit circle and fulfills f (e iθ ) = f (e −iθ ) is of the following form: where r ∈ N ∪ {0}, 0 = θ 0 < θ 1 < ... < θ r < θ r+1 = π, α j > −1/2, β j ∈ iR for j = 0, ..., r + 1, and The Toeplitz and Toeplitz+Hankel determinants we consider are defined as follows: The study of the asymptotics of Toeplitz determinants was initiated by Szegö. The simplest case is the strong Szegö limit theorem (see for example [48] for the most general version), which states that for Subsequently the asymptotics of D n (f ) and D T +H,κ n (f ) have been computed under various assumptions on the symbol f [7-12, 14-16, 30, 31, 50]; see [27] for a recent historical account and [26] for the most general results. The most general results for the case that f is real-valued are stated below in Theorems 11 and 12. However, all these results are only valid if the singularities of f are bounded away from each other as n → ∞. In recent years advances were made on the asymptotics of D n (f ) when the singularities merge as n → ∞. In [25] the asymptotics were computed for two merging singularities, and were related to a solution of the Painlevé V equation. Using the techniques in [25], we establish new results on Toeplitz determinants in Theorems 16 and 22 that give the asymptotics when there are two conjugate pairs of merging singularities which are bounded away from ±1. Again the asymptotics are related to a Painlevé V equation, in fact the same one as in [25]. In [32] the asymptotics of D n (f ) for arbitrarily many singularities merging were computed up to a factor e O(1) which is uniformly bounded and bounded away from 0. The precise factor is believed to be related to higher-dimensional analogues of Painlevé equations. Using the Riemann-Hilbert analysis of [32], the asymptotics of D T +H,κ n (f ) for f having arbitrarily many singularities merging were computed up to an e O(1) factor in [23]. This result is stated in Theorem 23. For two conjugate pairs of merging singularities which are bounded away from ±1 our Theorem 17 expresses that factor in terms of a Painlevé transcendent, which again is the same one as in [25].
Closely related to Toeplitz and Toeplitz+Hankel determinants are the Hankel determinants , (3.11) where I ⊂ R is an interval on which the weight w(x) is supported. Hankel determinants with Fisher-Hartwig singularities have a weight of the form where W (x) is continuous, ω(x) has Fisher-Hartwig singularities and V (x) is a potential (in case I is unbounded W (x) and V (x) need to fulfill certain integrability conditions). There are three canonical cases: Using the Heine identity those cases of Hankel determinants appear as averages over the (scaled and shifted) Gaussian Unitary Ensemble, Wishart Ensemble and Jacobi Ensemble respectively. [39], [37] and [43] are important early works on the large-n asymptotics of Hankel determinants with Fisher-Hartwig singularities that have greatly contributed to the development of the theory. The most general results have been proven in [17] and [18]. [21] concerns the case when singularities are merging as n → ∞. There are various formulas which relate Hankel determinants with I = [−1, 1] and V (x) = 0 to Toeplitz determinants and Toeplitz+Hankel determinants, for example Theorem 2.6 and Lemma 2.7 in [26]. A combination of those two results gives a relation between Toeplitz and Toeplitz+Hankel determinants and is stated in Lemma 14 below. We use this relation to prove our results on uniform asymptotics of Toeplitz+Hankel determinants.
Theorem 2.6 and Lemma 2.7 in [26] relate Toeplitz+Hankel determinants to Toeplitz determinants and monic orthogonal polynomials, which is how Theorem 12 was proven. This relation can be stated as follows: Lemma 14 (Deift, Its, Krasovsky). Let f be as in (3.7). Then for all n ∈ N: where Φ n (z) = z n + ... are the monic orthogonal polynomials w.r.t. the symbols on the RHS, i.e. f (z), We will use Lemma 14 in Section 9 to prove our results on the asymptotics of Toeplitz+Hankel determinants with merging singularities, stated in Theorems 17 and 22 below, using our results on the asymptotics of Toeplitz determinants with merging singularities, stated in Theorems 16 and 22 below. In those theorems we consider the following class of symbols: let ∈ (0, π/2) and define where • α j ∈ (−1/2, ∞) for j = 0, ..., 5, and α 1 = α 5 , α 2 = α 4 , • V (z) is real-valued on the unit circle, and satisfies V (e iθ ) = V (e −iθ ).
To state our results on the uniform asymptotics of D n (f p,t ) and D T +H,κ n (f p,t ) we further need the following theorem (not in its most general form) from [25], which describes the relevant Painlevé transcendents: Theorem 15. (Claeys, Krasovsky) Let α 1 , α 2 , α 1 + α 2 > − 1 2 , β 1 , β 2 ∈ iR and consider the σ-form of the Painleve V equation

21)
where the parameters θ 1 , θ 2 , θ 3 , θ 4 are given by Then there exists a solution σ(s) to (3.21) which is real and free of poles for s ∈ −iR + , and which has the following asymptotic behavior along the negative imaginary axis: for some δ > 0.
Our result on the uniform asymptotics of D n (f p,t ) is then the following, which we prove using the Riemann-Hilbert techniques in [25]: Theorem 16. Let f p,t be as in (3.20) with α 1 + α 2 > −1/2, and let σ satisfy the conditions of Theorem 15. Then we have the following large n asymptotics, uniformly for p ∈ ( , π − ) and 0 < t < t 0 , for a sufficiently small t 0 ∈ (0, ): 24) where ln z k zj e iπ = i(θ k − θ j − π) and G denotes the Barnes G-function.
Our result on the uniform asymptotics of D T +H n (f p,t ) is as follows, using the same notation as in Theorem 12: Theorem 17. Let f p,t be as in (3.20) with α 1 + α 2 > −1/2, and let σ satisfy the conditions of Theorem 15. For D T +H,κ n (f p,t ) we get, as n → ∞, uniformly in p ∈ ( , π − ) and 0 < t < t 0 , for a sufficiently small t 0 ∈ (0, ): (3.25) Remark 18. One can probably get similar results if more generally one chooses complex α j , β j with (α j ) > −1/2, but to prove Theorem 5 this is not necessary.
Remark 19. The requirements p ∈ ( , π − ), t 0 ∈ (0, ) are necessary for us to be able to apply the proof techniques in [25]. The results there only hold for two merging singularities, while if p → 0, π we have 5 singularities merging at ±1, and if t → we can have p ± t → 0, π which means 3 singularities are merging at ±1.

Remark 20.
Comparing the uniform asymptotics of D n (f p,t ) in Theorem 16 with the non-uniform asymptotics one gets from Theorem 11, one can see that the different expansions are related in the following way: This is exactly the same relationship as the one between the non-uniform and uniform expansions of D n (f t ) in [25] (see their (1.8), (1.24) and (1.26)).

Remark 21.
The relationship between the uniform asymptotics of D T +H,κ n (f p,t ) in Theorem 17 and the non-uniform asymptotics one gets from Theorem 12 is given by (3.26), with both sides divided by 2, and n replaced by 2n. This is because the uniform asymptotics of D T +H,κ n (f p,t ) are related to the uniform asymptotics of D 2n (f p,t ) 1/2 (with added singularities at ±1) and Φ n (±1) 1/2 , Φ n (0), by Lemma 14. As will be argued in Section 9 the asymptotics of Φ 2n (±1), computed as in [26] are unaffected by the merging of singularities away from ±1, and Φ 2n (0) = o(1) both when singularities merge or not. Thus only the asymptotics of D 2n (f p,t ) 1/2 are different in the merging and non-merging regime, and their relationship is given by (3.26) with both sides divided by 2, and n replaced by 2n.
Our last results corresponds to Theorem 1.11 in [25]. It extends Theorem 11 for the symbol f p,t , and Theorem 12 in the case r = 2: Let ω(x) be a positive, smooth function for x sufficiently large, s.t.

(3.29)
Here e O(1) denotes a function which is uniformly bounded and bounded away from 0 as n → ∞.

The L 2 -Limit
In this section we prove Lemma 9, for which we need to compute asymptotics of all the expectation terms in the integrals, which hold uniformly in θ and θ even as θ → θ . We use the following theorem to express all those expectations as Toeplitz+Hankel determinants, where we let SO(n) : are well-defined. Then with D T +H,κ n defined as in (3.8) we have We define, for φ ∈ [0, 2π) and θ, θ ∈ [0, 2π): where the branch of the logarithm is the principal one (so in particular ln(1 − e i(φ−θ) ) ∈ (−π/2, π/2]).
In the following sections we use the asymptotics obtained in this section to compute the asymptotics of the quotients of expectations that appear in Lemma 9.
In Section 5 we will also need that uniformly in θ ∈ [0, 2π).

Proof of Lemma 9
Now we have all the ingredients necessary to prove Lemma 9. We will only prove it for I = [0, 2π), the proof for I = I = ( , π − ) ∪ (π + , 2π + ) is completely analogous and relies on the fact that Lemma 25 also holds for I = I , α 2 − β 2 < 1/2 and α > −1/4, as explained in Remark 26.  We have (4.54) Since and we see that  Now we use that g is non-negative to apply Fatou's lemma to get the other inequality, which finishes the proof.

Proof of the Second Limit
In this section we prove (2.26) for I = [0, 2π), i.e. that for any fixed k ∈ N and bounded continuous function g : which is continuous since the integrand is continuous in z 1 , ..., z n and θ, and bounded in θ for any fixed z 1 , ..., z n . Then we have, with ± corresponding to symplectic/orthogonal:

RHP for Orthogonal Polynomials
By the integral representation for a Toeplitz-determinant and since f p,t > 0 except at z 0 , ..., z 5 , it holds that D n (f p,t ) ∈ (0, ∞) for all n ∈ N. Thus we can define the polynomials where the leading coefficient χ n is given by The above polynomials satisfy the orthogonality relations for k = 0, 1, ..., n, which implies that they are orthonormal w.r.t. the weight f p,t .
Let C denote the unit circle, oriented counterclockwise. It can easily be verified that the matrixvalued function Y (z) = Y (z; n, p, t) given by is the unique solution of the following Riemann-Hilbert problem: The continuous boundary values of Y from inside the unit circle, denoted Y + , and from outside, denoted Y − , exist on C \ {z 0 , ..., z 5 }, and are related by the jump condition (d) As z → z k , z ∈ C \ C, k = 0, ..., 5, we have and From the RHP and Liouville's theorem it follows that det Y (z) = 1 for all z ∈ C \ C. Using this, one can see quickly that the solution is unique. We have Y (z; n, p, t) 21 (0) = χ 2 n−1 and Y (z; n, p, t) 11 (z) = χ −1 n φ n (z) = Φ n (z), thus if we know the asymptotics of Y , we know the asymptotics of Φ n , φ n and χ n .

Differential Identity
The Fourier coefficients are differentiable in t, thus ln D n (f p,t ) is differentiable in t for all p ∈ ( , π − ) and n ∈ N. We calculate: Similarly we obtain Therefore we get where q k = 1 for k = 1, 4 and q k = −1 for k = 2, 5. In the last line we used that k=1,2,4,5 q k α k = 0.
with c j such thatỸ is bounded in that neighborhood. Then we have Proposition 27. Let n ∈ N and α k = 0 for k = 1, 2, 4, 5. Then the following differential identity holds: (z) with z → z k non-tangentially to the unit circle.
Proof: The proof for α k = 0, k = 1, 2, 4, 5 works exactly like the proof of Proposition 2.1 in [25]. We have to modify their (2.16), which we replace with our (6.10). The singularities at ±1 are independent of p and t and thus always stay within f p,t . [25] one can also get a differential identity for ln D n (f p,t ) in the case where α k = 0 for some k ∈ {1, 2, 4, 5}, by letting those α k 's go to zero in (6.10), which is continuous in α k on both sides.

Normalization of the RHP
Then by the RH conditions for Y , we obtain the following RH condition for T : The continuous boundary values of T from the inside, T + , and from outside, T − , of the unit circle exist on C \ {z 0 , ..., z 5 }, and are related by the jump condition and

Opening of the Lens
Define the Szegö function which is analytic inside and outside of C and satisfies We have (see (4.9)-(4.10) in [26]): and and thus The branch of (z − z k ) α k ±β k is fixed by the condition that arg(z − z k ) = 2π on the line going from z k to the right parallel to the real axis, and the branch cut is the line θ = θ k going from z = z k = e iθ to infinity. For any k, the branch cut of the root z α k −β k is the line θ = θ k from z = 0 to infinity, and θ k < arg z < θ k + 2π. By (7.6) we have that f p,t (e iθ ) = D in,p,t (e iθ )D out,p,t (e iθ ) −1 , (7.10) and this function extends analytically to a neighborhood S of the unit circle with the 6 branch cuts z k R + ∩ S, k = 0, ...5, which we orient away from zero. Then we obtain for the jumps of f p,t : f p,t+ (z) =f p,t− (z)e 2πi(αj −βj ) , on z j (0, 1) ∩ S, f p,t+ (z) =f p,t− (z)e −2πi(αj +βj ) , on z j (1, ∞) ∩ S. We factorize the jump matrix of T as follows: We then fix a lens-shaped region as in Figure 1 and define , in the parts of the lenses outside the unit circle, , in the parts of the lenses inside the unit circle. (7.13) The following RH conditions for S can be verified directly: (7.14) (c) S(z) = I + O(1/z), as z → ∞.
(d) As z → z k from outside the lenses, k = 0, ..., 5, we have and The behaviour of S(z) as z → z k from the other regions is obtained from these expressions by application of the appropriate jump conditions. are disjoint for any p ∈ ( , π − ). Let t 0 ∈ (0, ) such that e i(p±t) ∈ U + and e i(2π−(p±t)) ∈ U − for one and hence for all p ∈ ( , π − ). Then one observes that on the inner and out jump contours and outside of U 1 ∪ U −1 ∪ U + ∪ U − the jump matrix J S (z) converges to the identity matrix as n → ∞, uniformly in z, t < t 0 and p ∈ ( , π − ).

Global Parametrix
Define the function One can easily verify that N satisfies the following RH conditions:

Local Parametrix near ±1
The local parametrix near ±1 are constructed in exactly the same way as in [26]. We are looking for a solution of the following RHP: (d) P ±1 satisfies the matching condition P ±1 (z)N −1 (z) = I + o(1) as n → ∞, uniformly in z ∈ ∂U ±1 , p ∈ ( , π − ) and 0 < t < t 0 . P ±1 is given by (4.15), (4.23), (4.24), (4.47)-(4.50) in [26] and one can see from their construction that when all the other singularities are bounded away from ±1, then the matching condition is uniform in the location of the other singularities, i.e. holds uniformly in p ∈ ( , π − ) and 0 < t < t 0 .
For 0 < t ≤ 1/n and 1/n < t ≤ ω(n)/n we will construct local parametrices in U ± which satisfy the same jump and growth conditions as S inside U ± , and which match with the global parametrix N on the boundaries ∂U ± for large n. To be precise, we will construct P ± satisfying the following conditions: RH problem for P ± (z) (a) P ± : U ± \ Σ S → C 2×2 is analytic.

RH Problem for
with the orientation chosen as in Figure 2 ("-" is always on the RHS of the contour).
(c) We have in all regions: (7.30) with the branches corresponding to the arguments between 0 and 2π, and where s ∈ −iR + .
• where ln takes values in (−σ, σ) for some σ > 0, • where E ± is an analytic matrix-valued function in U ± , • and where W is given by for |z| > 1, The singularities z = e i(p±t) for + and z = e i(2π−(p∓t)) for − correspond to the values ζ = ±i. The jumps of W ± follow from (7.11): where Σ is the contour of the RHP for Φ ± , as shown in Figure 2. Inside U ± the combinded jumps of W (z) and Φ are the same as the jumps of S: , z ∈ Σ k,out ∩ U ± , k = 0, 2, 3, 5.

(7.34)
By the condition (d) of the RHP for S, the singular behaviour of W near z k , k = 1, 2, 4, 5 and condition (d) of the RHP for Φ ± , the singularities of S(z)P ± (z) −1 at z 1 , z 2 for +, and at z 4 , z 5 for −, are removable.
We see that we have a normalized RHP with small jumps, which by the standard theory on RHP implies that as n → ∞, uniformly for z off the jump contour and uniformly in p ∈ ( , π − ), 0 < t < t 0 .
7.6 ω(n)/n < t < t 0 . Local Parametrices near e ±ip We now transfer the construction from Section 7.5 in [25] to our setting in a completely straightforward manner. Although the parametrices P ± from the previous section are valid for the whole region 0 < t < t 0 we need to construct more explicit parametrices for the case ω(n)/n < t < t 0 to get a simpler large n expansion for Y , which is needed for the analysis in the next section.
In the case ω(n)/n < t < t 0 ζ = 1 t ln z e ±ip is not necessarily large on ∂U ± . But we can construct a large s = −int expansion for Y , as |s| = nt is large.
We modify the S-RHP by now also opening up lenses around the arcs (p − t, p + t) and (2π − p − t, 2π − p + t), i.e. we choose the contour Σ S as in Figure 4. Figure 4: The modified jump contour Σ S of S in the case ω(n)/n < t < t 0 . The difference compared to Figure 1 is that here there are also lenses around the arcs (p − t, p + t) and (2π − p − t, 2π − p + t).
Let U 1 , U 2 be small non-intersecting disks around ±i, those are the same neighborhoods as in Section 5 of [25]. We surround the points z 1 = e i(p−t) , z 2 = e i(p+t) by small neighborhoodsŨ 1 ,Ũ 2 , withŨ 1 being the image of U 2 under the inverse of the map ζ = 1 t ln z e ip , andŨ 2 being the image of U 1 under the same map. Similarly we surround z 4 = e i(2π−(p+t)) by a small neighborhoodŨ 4 , which is the image of U 2 under the inverse of the map ζ = 1 t ln z e −ip , and z 5 = e i(2π−(p−t)) we surround byŨ 5 which is the image of U 1 under the same map. Since the disks U 1 , U 2 are fixed in the ζ-plane, the neighborhoods U 1 ,Ũ 2 ,Ũ 4 ,Ũ 5 contract in the z-plane if t decreases with n.
Choose Σ S such that 1 t ln Σ S z k ⊂ e ± πi 4 R ∪ iR ∪ R inŨ k . Then one can easily verify, as in (7.34), thatP k has the same jumps as S inŨ k , so that S(z)P k (z) −1 is meromorphic inŨ k , with at most an isolated singulary at z k . The singular behaviour of S and W ± near z k , and of M (α k ,β k ) near 0 (given in (D.8) and (D.12)), imply that S(z)P −1 k (z) is bounded at z k , which shows that thatP k is a parametrix for S inŨ k with the matching condition (7.52) with N (z) at ∂Ũ k . 7.6.1 ω(n)/n < t < t 0 . Final Transformation Figure 5: The jump contour ΣR ofR in the case ω(n)/n < t < t 0 .
We transfer Section 7.5.1 in [25] to our case. Figure 5 shows the contour chosen for the RHP ofR, which we define as follows: ThenR is analytic, in particular has no jumps inside any of the local parametricesŨ k , k = 1, 2, 4, 5, U ± , or on the unit circle. On the rest of the lenses we can see that the jump matrix is I + O(e −δnt ) for some δ > 0, uniformly in p ∈ ( , π − ) and ω(n)/n < t < t 0 . Because of the matching condition (d) of P ±1 we have as in the case 0 < t < ω(n)/n that uniformly for z ∈ ∂U ±1 , p ∈ ( , π − ) and 0 < t < t 0 . Using (7.52), we get that uniformly for z ∈ ∂Ũ k , p ∈ ( , π − ) and ω(n)/n < t < t 0 . Finally we have that lim z→∞R (z) = I, which by standard theory for RHPs with small jumps and RHPs on contracting contours implies that uniformly for z off the jump contour ofR, and uniformly in p ∈ ( , π − ) and ω(n)/n < t < t 0 .
8 Asymptotics of D n (f p,t ) This section is a transfer of Section 8 in [25] to our case.
Then the following asymptotic expansion holds: where for the error term n,p,t for some δ > 0, uniformly in p ∈ ( , π − ) and 0 < t < t 0 , and where cos p sin p .

(8.4)
Proof: The proof is analogous to the proof of Proposition 8.1 in [25]. As is done there, we assume below that α k > 0, k = 1, 2, 4, 5, for simplicity of notation. Once (8.2) is proven under this assumption, the case where α k = 0 for some k then follows from the uniformity of the error terms in α k , k = 1, 2, 4, 5.
Extending to the case where α k < 0 for some k is straightforward. We prove the proposition first in the regime 0 < t ≤ ω(n)/n and then in the regime ω(n)/n < t < t 0 .
Using the transformation Y → T → S inside the unit circle, outside the lenses, we can rewrite the differential identity (6.11) in the form with q k = 1 for k = 1, 4 and q k = −1 for k = 2, 5, and where the limit z → z k is taken from the inside of the unit circle and outside the lenses.
Remark 32. Integrating (8.37) from t to t 0 with ω(n)/n < t < t 0 and using Theorem 11 for the expansion of D n (f p,t0 ), we get the same expansion for D n (f p,t ) that Theorem 11 gives. The error term is then O(ω(n) −1 ) and uniform for p ∈ ( , π − ) and ω(n)/n < t < t 0 . Thus we have proven the statement on Toeplitz determinants in Theorem 22.
The asymptotics of Φ n (1) and Φ n (−1) can be calculated exactly as in Chapter 7 of [26]. As is apparent from (7.13) there, these asymptotics are uniform for all other singularities bounded away from ±1. Thus when using Lemma 14 to calculate from the asymptotics of D n (f p,t ) given in Theorem 22, Φ n (±1) and Φ n (0), the asymptotics of the corresponding Toeplitz+Hankel determinants, then uniformity of the error terms in p, t is preserved. This proves the statement on Toeplitz+Hankel determinants in Theorem 22.
Further, since we know the relation (3.26) between the asymptotic expansion of D n (f p,t ) which is uniform for 0 < t < t 0 , and the asymptotic expansion for t > t 0 , Lemma 14 immediately gives the relationship between the expansions of D T +H,κ n (f p,t ) in the two regimes 0 < t < t 0 and t > t 0 . In view of the way the uniform asymptotics of D T +H,κ n (f p,t ) are derived from the uniform asymptotics of D 2n (f p,t ) 1/2 , Φ n (±1) 1/2 and Φ n (0), the relationship between the 0 < t < t 0 asymptotics of D T +H,κ n (f p,t ) and the t > t 0 asymptotics one gets from Theorem 17 is given by (3.26), with both sides divided by 2, and n replaced by 2n. This proves Theorem 17.

Remark 34.
(H s , ·, · s ) is a Hilbert space for all s ∈ R. For s ≥ 0 H s is a subspace of H 0 , i.e. the space of square-integrable functions on the unit circle. For s < 0, H s can be interpreted as the dual space of H −s , and as a space of generalized functions.

Proof of Theorem 3:
The proof strategy is as follows: we treat ( ln p n , ln p n ) n∈N as a sequence in H − 0 × H − 0 and show that if any of its subsequences has a limit then that limit has to be (X ± x,X ±x), with + when the underlying matrices are symplectic and − when they are orthogonal. We do this by showing that the finite-dimensional distributions of ( ln p n , ln p n ) n∈N , i.e. the distributions of finite sets of pairs of Fourier coefficients, converge to those of (X ± x,X ±x). We then show that the set { ln p n , ln p n } n∈N is tight in H − 0 × H − 0 . Since H − 0 × H − 0 is complete and seperable, Prokhorov's theorem implies that the closure of { ln p n , ln p n } n∈N is sequentially compact w.r.t. the topology of weak convergence. In particular this means that every subsequence of ( ln p n , ln p n ) n∈N has a weak limit in H − 0 × H − 0 . Since any such limit has to be (X ± x,X ±x) it follows that the whole sequence ( ln p n , ln p n ) n∈N must converge weakly to (X ± x,X ±x).
We recall that for |z| ≤ 1, where for z = 1 both sides equal −∞. By using the identity ln det = Tr ln we see that the Fourier expansions of ln p n , ln p n , ln p n in H − 0 are given as follows: Tr(U k n ) k (e ikθ + e −ikθ ), ln p n (θ) = 1 2i ∞ k=1 Tr(U k n ) k (e ikθ − e −ikθ ).

(B.2)
Define the measures µ where σ (N 1 , ..., N k−1 ) is the σ-algebra generated by N 1 , ..., N k−1 . Thus, since also for any k ∈ N the random variable µ Since the sequence is non-negative it converges a.s. to a random variable which will be denoted by µ α,β (A). One can show that a.s. the map A → µ α,β (A) is a measure and we have that a.s. µ which implies that the event {µ α,β is the 0 measure} does not have probability 1. Since that event is independent of any finite number of the N k , k ∈ N, Kolmogorov's zero-one law implies that this event then has probability 0. Thus µ α,β is a.s. non-trivial for α ≥ 0, α 2 − β 2 < 1/2 and 4α 2 < 1, while when restricted to I the measure µ α,β is almost surely non-trivial for α > −1/4 and α 2 − β 2 < 1/2. In both cases though, one can expect µ α,β to be a.s. non-trivial for a larger set of values α, β.

C Riemann-Hilbert Problem for Ψ
This appendix is a mostly verbatim transfer from the beginning of Section 3 of [25]. We include it here to make our account self-contained. We use Ψ to construct local parametrices for the RHP for the orthogonal polynomials in Section 7.5. We always assume that α 1 , α 2 > − 1 2 and β 1 , β 2 ∈ iR (in [25] also the more general case of α 1 , α 2 , β 1 , β 2 ∈ C was considered).

RH Problem for
with the orientation chosen as in Figure 6 ("-" is always on the RHS of the contour).
with the branches corresponding to the arguments between 0 and 2π, and where s ∈ −iR + .