Asymptotics for averages over classical orthogonal ensembles
aa r X i v : . [ m a t h - ph ] A ug Asymptotics for averages over classical orthogonal ensembles
Tom Claeys, Gabriel Glesner, Alexander Minakov, and Meng Yang ∗ August 19, 2020
Abstract
We study averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haardistributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. Weobtain new asymptotics for symbols with Fisher-Hartwig singularities in cases where some of thesingularities merge together, and for symbols with a gap or an emerging gap. We obtain theseasymptotics by relying on known analogous results in the unitary group and on asymptotics forassociated orthogonal polynomials on the unit circle. As consequences of our results, we deriveasymptotics for gap probabilities in the Circular Orthogonal and Symplectic Ensembles, and anupper bound for the global eigenvalue rigidity in the orthogonal ensembles.
Consider the classical orthogonal group O N of N × N orthogonal matrices equipped with the Haarmeasure, and its components O ± N of N × N orthogonal matrices with determinant equal to ±
1. If N is even, the eigenvalues of a matrix M ∈ O + N = O +2 n come in complex conjugate pairs e ± iθ , . . . , e ± iθ n with θ , . . . , θ n ∈ [0 , π ], while a matrix M ∈ O − N = O − n +2 has complex conjugate pairs of eigenvalues e ± iθ , . . . , e ± iθ n with θ , . . . , θ n ∈ [0 , π ], and fixed eigenvalues − N = 2 n + 1 is odd, amatrix M ∈ O ± N has complex conjugate pairs of eigenvalues e ± iθ , . . . , e ± iθ n with θ , . . . , θ n ∈ [0 , π ]complemented by the fixed eigenvalue ±
1. Due to Weyl’s integration formula, the joint probabilitydistributions of the free eigenangles θ , . . . , θ n ∈ [0 , π ] are given by (see e.g. [28, p71–72], [37, p76]and [33]) O +2 n : 2 n !(2 π ) n Y ≤ j Our approach will rely on a variantof such existing identities, which is particularly convenient for asymptotic analysis, and which allowsus to write averages over orthogonal ensembles of a symbol f in terms of averages over the unitarygroup U N of Haar distributed N × N unitary matrices for the symbol g ( e it ) := f ( e it ) f ( e − it ) (1.3)and related orthogonal polynomials on the unit circle evaluated at ± 1. Before stating these identities,let us recall that the eigenvalues e iϕ , . . . , e iϕ N , with ϕ , . . . , ϕ N ∈ [0 , π ), of a Haar-distributedmatrix U from the unitary group U N of N × N unitary matrices, often referred to as the CircularUnitary Ensemble (CUE), have the joint probability distribution U N : 1(2 π ) N N ! Y ≤ k 0, characterized by the conditions Z π Φ N ( e it ) e − ikt g ( e it ) dt = 0 for any integer 0 ≤ k < N . (1.8)These polynomials can, by their determinantal representations and Heine’s identity, also be writtenas the averages Φ N ( z ) = det (cid:18) g j − k z k (cid:19) N − ,Nj,k =0 det( g j − k ) N − j,k =0 = E U N [( z − . ) g ( . )] E U N [ g ] . (1.9)2n the next result, we express averages over the orthogonal ensembles in terms of averages overthe unitary group and orthogonal polynomials, and this will be the starting point of our asymptoticanalysis later on. Proposition 1.1. Let f be a function on the unit circle which is such that g defined by (1.3) isnon-negative and integrable on [0 , π ] . Let Φ k be the degree k monic orthogonal polynomial on theunit circle with respect to the weight g ( e it ) . Then for all positive integers n , E (0 , +) n [ f ] = (cid:20) E U n [ g ] − Φ n − (1)Φ n − ( − (cid:21) / , E (2 , − ) n [ f ] = (cid:2) Φ n (1)Φ n ( − E U n [ g ] (cid:3) / , E (1 , ± ) n [ f ] = (cid:20) Φ n ( ± n ( ∓ E U n [ g ] (cid:21) / . (1.10) Asymptotics for averages in orthogonal ensembles. There is a vast literature on asymptoticsfor Toeplitz determinants, and large N asymptotics for (1.5)–(1.6) are well understood for largeclasses of complex-valued symbols g . The most classical result in this context is Szeg˝o’s strong limittheorem, which states that [38, 31, 32] with g ( e it ) = e V ( e it ) and V sufficiently smooth on the unitcircle, as N → ∞ ,det ( g j − k ) N − j,k =0 = e NV e P ∞ k =1 kV k V − k (1 + o (1)) with V k = 12 π Z π V ( e it ) e − ikt dt. (1.11)More precisely, this holds for any V such that P ∞ k =1 k | V k | < ∞ . More general results allow for sym-bols which vanish on an arc of the unit circle [40] or for the presence of Fisher-Hartwig singularities,which are combinations of root-type singularities with jump discontinuities. Such symbols have along history [6, 7, 11, 15, 23, 25, 36, 40], and asymptotics for the associated Toeplitz determinants arenow completely understood in the large N limit, as long as the symbol does not depend on N [21].In cases where the symbol depends on N , various interesting transitions in the large N asymptoticscan take place, such as the emergence of a Fisher-Hartwig singularity [42, 19], the emergence of anarc of vanishing [18, 17], or the merging of Fisher-Hartwig singularities [20, 24].Large N asymptotics for the analogues in the orthogonal ensembles O ± N , namely (1.2), are alsoknown for fixed symbols (i.e. independent of N ) with Fisher-Hartwig singularities, see [21, Theorem1.25] for the most complete result in this respect and [4, 8, 9, 10] for earlier developments. However,the picture for averages in O ± N is incomplete because, as far as we know, asymptotics are not known forsymbols vanishing on an arc, and no results are available about transition asymptotics in situationswhere either several singularities approach each other in the large N limit (except for the resultsfrom [27] obtained simultaneously with ours, see Remark 2.4 below), or parameters tune in such away that a gap in the support emerges as N → ∞ . The objective in this paper is to complete thistask. In order to avoid technical and notational complications, we restrict ourselves to non-negativereal-valued symbols g , although some of the results could be generalized to complex-valued symbols. Outline for the rest of the paper. After stating our main results in Section 2, we will proveProposition 1.1 in Section 3. In Section 4, we will analyze orthogonal polynomials on the unit circlefor symbols with Fisher-Hartwig singularities, which possibly merge in the large degree limit, andthis will allow us to prove Theorem 2.1 and Theorem 2.2 below. In Section 5, we will analyze thecase of symbols with a gap or an emerging gap, and this will lead us to the proof of Theorem 2.5.In Section 6, we will study gap probabilities and global rigidity of eigenvalues in O ( j, ± ) n and proveTheorem 2.12. 3 Statement of results Let V be an analytic function in a neighborhood of the unit circle which is real-valued on the unitcircle and such that V ( e it ) = V ( e − it ), and let 0 < t < . . . < t m < π , with m ∈ N . For any j = 0 , , . . . , m, m + 1, we have parameters α j ≥ j = 1 , . . . , m we have parameters β j ∈ i R . We will consider symbols f such that g given by (1.3) is of the form g ( e it ) = e V ( e it ) | e it − | α | e it + 1 | α m +1 × m Y j =1 (cid:18) e it e i ( π + t j ) (cid:19) β j (cid:18) e − it e i ( π + t j ) (cid:19) β j (cid:12)(cid:12) e it − e it j (cid:12)(cid:12) α j (cid:12)(cid:12) e it − e − it j (cid:12)(cid:12) α j , (2.1)where z β = | z | β e iβ arg z with − π < arg z ≤ π . This is one of the standard forms of a positive symbolwith Fisher-Hartwig singularities, symmetric with respect to the real line and having singularitiesat the points e ± it j , j = 1 , . . . , m , and at the points ± 1. These singularities are combinations ofjump and root singularities whose nature depends on the parameters α j , β j . For instance, if we set m = 1, α = α = α = 0, V ≡ 0, then g is piece-wise constant: g ( e it ) = e − it β for | t | > t and g ( e it ) = e − it β e iπβ for | t | < t . Note that the symmetry with respect to the real line excludes thepossibility of having jump singularities (with non-zero parameters β , β m +1 ) at ± V, m, t j , α j , β j are independent of N , large N asymptotics for E U N [ g ] = det ( g j − k ) n − j,k =0 were ob-tained in [21] (in more general situations where the symbol is complex and not necessarily symmetricwith respect to the real line, where V is not necessarily analytic, and where α j > − / E U n [ g ] = E e nV (2 n ) α + α m +1 +2 P mj =1 ( α j − β j ) (1 + o (1)) , (2.2)as n → ∞ , with E given by E = e P + ∞ k =1 kV k e i P mj =1 α j P mk =1 t k β k e − πi P ≤ j Let m ∈ N , < t < . . . < t m < π , α j ≥ for j = 0 , . . . , m + 1 , β j ∈ i R for j = 1 , . . . , m , and let V be analytic in a neighborhood of the unit circle, real-valued on the unit circleand such that V ( e it ) = V ( e − it ) , with Laurent series V ( z ) = P ∞ k = −∞ V k z k and V k = V − k ∈ R . Let f be such that g is of the form (2.1) . There exists M > such that as n → ∞ , uniformly in the region Mn < t < . . . < t m < π − Mn , we have E (0 , +) n [ f ] = C n (cid:0) E U n [ g ] (cid:1) / (cid:18) O (cid:18) n min { t , π − t m } (cid:19)(cid:19) , E (2 , − ) n [ f ] = C − n (cid:0) E U n [ g ] (cid:1) / (cid:18) O (cid:18) n min { t , π − t m } (cid:19)(cid:19) , E (1 , ± ) n [ f ] = e C ± n (cid:0) E U n [ g ] (cid:1) / (cid:18) O (cid:18) n min { t , π − t m } (cid:19)(cid:19) , (2.5) where C n = 2 α + α m +1 n α αm +12 √ π Γ (cid:18) 12 + α (cid:19) Γ (cid:18) 12 + α m +1 (cid:19) e ( V (1)+ V ( − − V ) m Y j =1 h (2 sin t j ) α j e − iβ j t j e iπ β j i , e C n = n α − αm +12 Γ (cid:0) + α m +1 (cid:1) Γ (cid:0) + α (cid:1) e ( V ( − − V (1)) m Y j =1 "(cid:18) tan t j (cid:19) − α j e − iπ β j . (2.6)In the case where m , the positions of the singularities t j , and the values of the parameters α j , β j are independent of n , we can write the above results in a more explicit form by substituting (2.2)–(2.3). This yields E (0 , +) n [ f ] = C n Ee nV (2 n ) ( α + α m +1 ) / P mj =1 ( α j − β j ) (1 + o (1)) , E (2 , − ) n [ f ] = C − n Ee nV (2 n ) ( α + α m +1 ) / P mj =1 ( α j − β j ) (1 + o (1)) , E (1 , ± ) n [ f ] = e C ± n Ee nV (2 n ) ( α + α m +1 ) / P mj =1 ( α j − β j ) (1 + o (1)) . (2.7)Here, we recover [21, Theorem 1.25] in the case of a positive symbol f (to see this, one needs to usethe doubling formula for Barnes’ G -function, see [21, formula (2.39)]).Let us now consider in more detail the situation where the positions of the Fisher-Hartwig singu-larities are allowed to vary with n . This includes in particular situations where singularities mergein the large n limit or converge to ± 1. For notational convenience, we now set α = α m +1 = 0 in(1.3), but one should note that we can do this without loss of generality because we will now allow t = 0 and t m = π . Although we expect (2.7) to hold whenever the distance between singularitiesdecays slower than 1 /n , the main obstacle to prove this, is that strong asymptotics (including the5alue of the multiplicative constant) for E U n [ g ] have not been established, except for m = 1 [20],when they are related to the Painlev´e V equation. Weak asymptotics, without explicit value forthe multiplicative constant, have been obtained in general [24]. The result of [24] translated to oursetting is E U n [ g ] = F e nV (2 n ) P mj =1 (2 α j − β j ) m Y j =1 (cid:18) sin t j + 1 n (cid:19) − α j − β j e O (1) (2.8)as n → ∞ , uniformly for 0 < t < . . . < t m < π , with F = Y ≤ j Let m ∈ N , ≤ t < . . . < t m ≤ π , α = α m +1 = 0 , α j ≥ , β j ∈ i R for j = 1 , . . . , m ,and let V be analytic in a neighborhood of the unit circle, real-valued on the unit circle and such that V ( e it ) = V ( e − it ) , with Laurent series V ( z ) = P ∞ k = −∞ V k z k and V k = V − k ∈ R . Let f be such that g is of the form (2.1) . Then we have uniformly over the entire region < t < . . . < t m < π , as n → ∞ , E (0 , +) n [ f ] = F e nV m Y j =1 n α j − β j (cid:18) sin t j + 1 n (cid:19) α j − α j − β j × e O (1) , E (2 , − ) n [ f ] = F e nV m Y j =1 n α j − β j (cid:18) sin t j + 1 n (cid:19) − α j − α j − β j × e O (1) , E (1 , ± ) n [ f ] = F e nV m Y j =1 n α j − β j (cid:18) sin t j n (cid:19) ∓ α j − α j − β j (cid:18) cos t j n (cid:19) ± α j − α j − β j × e O (1) , (2.10) with F given by (2.9) . Here e O (1) denotes a function which is uniformly bounded and bounded awayfrom as n → ∞ . These results are also uniform for α j and β j in compact subsets of [0 , + ∞ ) and i R respectively. Remark 2.3. The factors sin t j + n have to be interpreted as follows: whenever t j does not convergetoo rapidly to or π as n → ∞ , the sine is the dominant term; if t j → or t j → π as n → ∞ withspeed of convergence faster than n , the term n will be dominant. Similarly for the factors sin t j + n as t j → and cos t j + n as t j → π . Remark 2.4. As mentioned before, one of the problems in determining the explicit value of the e O (1) factor lies in the asymptotics for E U n [ g ] , which are known only up to a multiplicative constantas n → ∞ . In the case m = 1 where we have only two singularities, this multiplicative constant canbe evaluated explicitly in terms of quantities related to a solution of the fifth Painlev´e equation [20].Simultaneously with this work, Forkel and Keating [27] evaluated the e O (1) factor in (2.10) explicitlyin terms of the same Painlev´e V solution when m = 2 , as long as the singularities e ± it , e ± it do notapproach ± . When there are more than two singularities approaching each other, one might expecta multiplicative constant connected to a generalization of the fifth Painlev´e equation, but the problemof evaluating the constant remains open. .2 Symbols with a gap or an emerging gap Next, we take s ≥ t ∈ (0 , π ). We consider symbols f such that g , defined by (1.3), is of theform g ( e it ) = e V ( e it ) × ( ≤ | t | ≤ t ,s for t < | t | ≤ π, (2.11)and suppose that V is, as before, real on the unit circle, and analytic in a neighborhood of the unitcircle. Note that in view of (1.3), we have V ( e it ) = V ( e − it ). For s > m = 1, α = 0). However, the limit s → β → − i ∞ , and the results stated before do notremain valid in this limit. To state our results, we need the Fourier coefficients e V k of the function e V ( e it ) := V ( e i arcsin(sin t sin t ) ) . (2.12)In the cases where either s = 0, or s depends on N and s → N → ∞ , such that s ≤ (cid:0) tan t (cid:1) N , asymptotics for E U N [ g ] were obtained in [17, Theorem 1.1]: E U N [ g ] = N − / (cid:18) sin t (cid:19) N e N e V + ∞ P k =1 k e V k e V − k (cid:18) cos t (cid:19) − / e log 2+3 ζ ′ ( − (1 + o (1)) , as N → ∞ . Setting N = 2 n , we have E U n [ g ] = (2 n ) − / (cid:18) sin t (cid:19) n e n e V + ∞ P k =1 k e V k e V − k (cid:18) cos t (cid:19) − / e log 2+3 ζ ′ ( − (1 + o (1)) , (2.13)as n → ∞ with either s = 0 or s → s ≤ (cid:0) tan t (cid:1) n . This result ismoreover valid uniformly as t → π , as long as n ( π − t ) → ∞ .We also need the function δ ( z ) = exp h ( z )2 πi Z γ V ( ζ )d ζ ( ζ − z ) h ( ζ ) , where h ( ζ ) = (cid:0) ( ζ − e it )( ζ − e − it ) (cid:1) / , (2.14)where γ denotes the counterclockwise oriented circular arc going from e − it to e it and passingthrough 1, and where h is determined by the conditions that it has a branch cut along the comple-mentary circular arc going from e it to e − it and passing through − 1, and that it is asymptotic to ζ for large ζ. We will need in particular the values δ − ( − 1) := lim z → ( − − δ ( z ) = exp − cos t πi Z γ V ( ζ )d ζ ( ζ + 1) h ( ζ ) , δ ( ∞ ) = exp − πi Z γ V ( ζ )d ζh ( ζ ) , which are both positive. We will prove the following in Section 5. Theorem 2.5. Let t ∈ (0 , π ) , let V be real-valued on the unit circle, analytic in a neighborhoodof the unit circle and such that V ( e it ) = V ( e − it ) . Let e V k be the Fourier coefficients of e V definedin (2.12) . Let f be such that g is of the form (2.11) . Then, as n → ∞ , uniformly with respect to ≤ s ≤ (cid:0) tan t (cid:1) n , we have E (0 , +) n [ f ] = C − n − (cid:0) E U n [ g ] (cid:1) / (1 + o (1)) , E (2 , − ) n [ f ] = C n (cid:0) E U n [ g ] (cid:1) / (1 + o (1)) , E (1 , ± ) n [ f ] = e C ± n (cid:0) E U n [ g ] (cid:1) / (1 + o (1)) , (2.15)7 here C n − = 2 n − (cid:18) sin t (cid:19) n (cid:18) cos t (cid:19) n − e − V (1) δ − ( − / δ ( ∞ ) ,C n = 2 n + (cid:18) sin t (cid:19) n (cid:18) cos t (cid:19) n +1 e − V (1) δ − ( − / δ ( ∞ ) , e C n = 2 (cid:18) sin t cos t (cid:19) n + e − V (1) δ − ( − / . (2.16) These asymptotics are also valid as t → π in such a way that n ( π − t ) → ∞ . The o (1) terms canbe written as O (( n ( π − t )) − + ( n ( π − t )) − / s (tan t ) − n ) . Using the known asymptotics for E U2 n [ g ] given by (2.13), we can write the above results in a moreexplicit form: E (0 , +) n [ f ] = 2 e V (1) δ ( ∞ ) e ζ ′ ( − e P ∞ k =1 k e V k e V − k p δ − ( − 1) (sin t ) (cid:0) cos t (cid:1) / · n − (cid:0) sin t (cid:1) n − n + e n e V (cid:0) t (cid:1) n − (1 + o (1)) , E (2 , − ) n [ f ] = 2 e ζ ′ ( − cos t p δ − ( − e P ∞ k =1 k e V k e V − k e V (1) δ ( ∞ ) (cid:0) cos t (cid:1) / · n − (cid:18) sin t (cid:19) n + n (cid:18) t (cid:19) n e n e V (1 + o (1)) , E (1 , ± ) n [ f ] = 2 ± e ζ ′ ( − e ∓ V (1) e P ∞ k =1 k e V k e V − k δ − ( − ∓ / (cid:0) cos t (cid:1) / · n − (cid:18) sin t (cid:19) n ± n (cid:18) t (cid:19) ∓ n e n e V (1 + o (1)) , where ζ is Riemann’s zeta function. The above results can be used to compute asymptotics for gap probabilities and generating functionsin O ( j, ± ) n and also in the Circular Orthogonal Ensemble (COE) and in the Circular SymplecticEnsemble (CSE). These have the joint probability distributionsC β E N : 1 Z [ β ] N Y ≤ k Let t ∈ (0 , π ) . As n → ∞ , with fixed t or with t → in such a way that nt → + ∞ , E ( j, ± ) n ( t ; 0) = 2 − e ζ ′ ( − (cid:0) t (cid:1) ˜ n (cid:0) cos t (cid:1) ˜ n ! ǫ ± j (cid:0) cos t (cid:1) n (cid:0) ˜ n sin t (cid:1) (1 + o (1)) , (2.22) where ˜ n = n + j − and ǫ ± j = 1 if +1 is a fixed eigenvalue of O ± n + j and ǫ ± j = − otherwise. In otherwords, ǫ +0 = − , ǫ +1 = 1 , ǫ − = − , ǫ − = 1 . oreover, as N → ∞ and t ∈ (0 , π ) is either fixed or tends to in such a way that N t → ∞ , wehave E [1] N ( t ; 0) = 2 e ζ ′ ( − (cid:18) cos t t (cid:19) N (cid:0) cos t (cid:1) N ( N sin t ) (1 + o (1)) ,E [4] N ( t ; 0) = 2 − e ζ ′ ( − (cid:18) t cos t (cid:19) N (cid:0) cos t (cid:1) N (cid:0) N sin t (cid:1) (1 + o (1)) . Remark 2.7. We can compare these results with the corresponding result in the CUE, which reads E [2] N ( t ; 0) = 2 e ζ ′ ( − (cid:0) cos t (cid:1) N ( N sin t ) (1 + o (1)) . To compute asymptotics for the right hand sides of (2.21) in the case where s > Corollary 2.8. As ˜ n = n + j − → ∞ , with t ∈ (0 , π ) fixed or such that nt → ∞ , and with ǫ ± j asabove, E ( j, ± ) n ( t ; s ) = s − ǫ ± j (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) s πi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (4˜ n sin t ) log2 s π s ˜ nt π (1 + o (1)) . Moreover, as N → + ∞ with t ∈ (0 , π ) fixed or such that N t → ∞ , we have E [1] N ( t ; s ) = 2 s s (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) sπi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (2 N sin t ) log2 sπ s Nt π (1 + o (1)) ,E [4] N ( t ; s ) = 1 + s s (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) s πi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (4 N sin t ) log2 s π s Nt π (1 + o (1)) . Remark 2.9. The above results should be compared to the CUE analogue E [2] N ( t ; s ) = (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) s πi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (2 N sin t ) log2 s π s Nt π (1 + o (1)) . Finally we can use Theorem 2.2 to obtain weak uniform asymptotics for the generating functionswhen s > 0, see Section 6.2 for the proof of this result. Corollary 2.10. Uniformly for t ∈ [0 , π ] , s in compact sets of (0 , + ∞ ) , as n → ∞ : E ( j, ± ) n ( t ; s ) = ( n sin t + 1) log2 s π s nt π e O (1) , hence for β = 1 , , as N → ∞ , E [ β ] N ( t ; s ) = ( N sin t + 1) log2 sβπ s Nt π e O (1) . Remark 2.11. The above result also holds for β = 2 , see [20] for an expression of the multiplicativeconstant. < θ ≤ . . . ≤ θ n < π in the orthogonal ensembles O +2 n , O − n +2 , O ± n +1 . Given the joint probabilitydistribution of the eigenvalues (1.1) which implies that the eigenvalues repell each other, we canexpect that in a typical situation, the eigenangles are distributed in a rather regular way, in otherwords we can expect that θ j will typically lie not too far from the deterministic value jπn . We canalso expect that the counting function N (0 ,t ) , counting the number of eigenangles in (0 , t ) for t ≤ π ,will behave to leading order typically like ntπ . We prove the following in Section 6.3. Theorem 2.12. In the ensembles O +2 n , O − n +2 , O ± n +1 , we have for any ǫ > n → + ∞ P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) θ k − πkn (cid:12)(cid:12)(cid:12)(cid:12) < (1 + ǫ ) log nn (cid:19) = 1 , and lim n → + ∞ P sup t ∈ (0 ,π ) (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) < (cid:18) π + ǫ (cid:19) log n ! = 1 . Remark 2.13. These results should be compared to concentration inequalities in [37, Section 5.4],which yield probabilistic bounds for (cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12) rather than for its supremum, and to global rigidityresults in the C β E [2, 16, 35] (see in particular Corollary 1.3 of [35]) and the sine β process [30].The method that we use to prove this result is based on a bound for the first exponential momentof the eigenvalue counting function, and this does not allow to get a complementary lower boundfor the maximum and supremum. The question of sharpness of the upper bound is closely related tothe theory of Gaussian multiplicative chaos, see e.g. [2, 12, 39] in general and [27] in this specificsituation. Apart from the positive symbols with Fisher-Hartwig singularities and the symbols with a gap oremerging gap, there are other types of symbols for which Toeplitz determinant asymptotics areknown, and for which one could use Proposition 1.1 in order to generalize them to the orthogonalensembles. One could for instance consider complex-valued symbols or non-analytic symbols withFisher-Hartwig singularities and apply the results from [21]. Another example consists of a situationwhere a symbol is smooth but depends on n and develops a Fisher-Hartwig singularity in the limit n → ∞ , as considered in [19]. In this case, like in the case m = 1 of Theorem 2.2, it is also possibleto evaluate the multiplicative constant in the asymptotic expansion in terms of solutions to thePainlev´e V equation. Yet another example consists of symbols with a gap, but with an additionalFisher-Hartwig singularity inside the gap, as considered in [43]. This situation is related to a systemof coupled Painlev´e V equations.In principle, the results from Theorem 2.2 can also be applied to derive asymptotics for momentsof moments of characteristic polynomials in the orthogonal ensembles, which can be written asmultiple integrals of the multiplicative averages we are considering in Theorem 2.2, in the specialcase where all β j ’s vanish and where all α j ’s are equal. The moments of moments are of interestbecause they reveal some of the statistics of the extrema of characteristic polynomials. In the case ofthe unitary group, their asymptotics were conjectured in [29] and later proved in [20] in the case oftwo singularities, and in [24] in general. Both for unitary and orthogonal ensembles, these momentsof moments have been evaluated exactly in terms of symmetric functions in [5, 3] for α j integer. Itwould be interesting to see if Theorem 2.2 can be used to generalize the asymptotics to any α j ≥ Proof of Proposition 1.1 Given a real-valued integrable function f on the unit circle, define the symbol g ( e it ) = f ( e it ) f ( e − it ),symmetric with respect to complex conjugation of the variable, and define its Fourier coefficientsas in (1.7). It is known that the averages (1.2) can be written as determinants of Toeplitz+Hankelmatrices. More precisely, we have (see e.g. [4, theorem 2.2], [28, p212], or [32]) for all n ∈ N , E (0 , +) n [ f ] = 12 det ( g j − k + g j + k ) n − j,k =0 , E (2 , − ) n [ f ] = det ( g j − k − g j + k +2 ) n − j,k =0 , E (1 , ± ) n [ f ] = det ( g j − k ∓ g j + k +1 ) n − j,k =0 . (3.1)Moreover, there exist identities expressing products of two such Toeplitz+Hankel determinants as aToeplitz determinant (see e.g. [41], [4, Corollary 2.4], or [28, p211]) E U n [ g ] = det ( g j − k ) n − j,k =0 = det ( g j − k − g j + k +1 ) n − j,k =0 det ( g j − k + g j + k +1 ) n − j,k =0 , E U n +1 [ g ] = det ( g j − k ) nj,k =0 = 12 det ( g j − k + g j + k ) nj,k =0 det ( g j − k − g j + k +2 ) n − j,k =0 . (3.2)We would like to invert such factorizations, and write a single Toeplitz+Hankel determinant in termsof a Toeplitz determinant. To that end, we need in addition analogues of the above identities, butwith slightly different products of Toeplitz+Hankel determinants at the right. As above, let Φ N bethe degree N monic orthogonal polynomial associated with the symbol g . Proposition 3.1. Φ n ( ± 1) det ( g j − k ) n − j,k =0 = det ( g j − k − g j + k +2 ) n − j,k =0 det ( g j − k ∓ g j + k +1 ) n − j,k =0 , Φ n +1 ( ± 1) det ( g j − k ) nj,k =0 = ± det ( g j − k ∓ g j + k +1 ) nj,k =0 det ( g j − k − g j + k +2 ) n − j,k =0 . (3.3) Proof. The representation of the monic orthogonal polynomials in terms of the determinant (1.9)yields Φ N ( ± 1) det( g j − k ) N − j,k =0 = det (cid:0) g j − k | ( ± j (cid:1) N,N − j,k =0 . Setting N = 2 n and subtracting the (2 n − j )-th row from the j -th row of the matrix at the righthand side for j = 0 , ..., n − 1, we obtainΦ n ( ± 1) det( g j − k ) n − j,k =0 = det ( g j − k − g n − j − k ) n − , n − j,k =0 (0) n − , j,k =0 ( g n + j − k ) n, n − j,k =0 (cid:0) ( ± n + j (cid:1) n, j,k =0 ! . Then, adding the (2 n − k )-th column to the k -th column for k = n, ..., n − k = n , we get ( g being symmetric, we have g m = g − m )Φ n ( ± 1) det( g j − k ) n − j,k =0 = 12 det ( g j − k − g n − j − k ) n − ,n − j,k =0 (0) n − ,n − j,k =0 (0) n − , j,k =0 ( g n + j − k ) n,n − j,k =0 ( g j − k + g j + k ) n,n − j,k =0 (cid:0) ( ± n + j (cid:1) n, j,k =0 ! . This yieldsΦ n ( ± 1) det( g j − k ) n − j,k =0 = 12 det( g j − k − g j + k +2 ) n − j,k =0 det (cid:16) ( g j − k + g j + k ) n,n − j,k =0 (cid:0) ( ± n + j (cid:1) n, j,k =0 (cid:17) . Adding or subtracting the ( j + 1)-th row from the j -th for j = 0 , . . . , n − 1, and then expanding withrespect to the last column, we end up withdet (cid:16) ( g j − k + g j + k ) n,n − j,k =0 (cid:0) ( ± n + j (cid:1) n, j,k =0 (cid:17) = det ( g j − k + g j + k ∓ ( g j − k +1 + g j + k +1 )) n − j,k =0 . g j − k ∓ g j + k +1 ) n − j,k =0 . To see this, it suffices in the latter matrixto subtract or add the ( k − k -th for k = 1 , . . . , n − 1, and to multiply the firstcolumn by 2. This indeed givesdet( g j − k ∓ g j + k +1 ) n − j,k =0 = 12 det( g j − k ∓ g j + k +1 ∓ ( g j − k +1 ∓ g j + k ) n − j,k =0 , thus proving the first identity. For the second, one proceeds similarly by subtracting or adding the(2 n + 1 − j )-th row from the j -th for j = 0 , ..., n , and then adding or subtracting the (2 n + 1 − k )-thcolumn from the k -th for k = n + 1 , . . . , n , leading toΦ n +1 ( ± 1) det( g j − k ) nj,k =0 = det( g j − k ∓ g j + k +1 ) nj,k =0 det (cid:16) ( g j − k ± g j + k +1 ) n,n − j,k =0 (cid:0) ( ± n +1+ j (cid:1) nj =0 (cid:17) . As before, subtracting or adding the next row to each row except the last one, and then expandingwith respect to the last column, we getdet (cid:16) ( g j − k ± g j + k +1 ) n,n − j,k =0 (cid:0) ( ± n +1+ j (cid:1) nj =0 (cid:17) = ± det( g j − k ± g j + k +1 ∓ ( g j − k +1 ± g j + k +2 )) n − j,k =0 . Also similarly as before, we havedet( g j − k − g j + k +2 ) n − j,k =0 = det( g j − k − g j + k +2 ∓ ( g j − k +1 − g j + k +1 )) n − j,k =0 , and the two above equations allow us to conclude the proof.We can now combine the factorizations (3.2) and (3.3) to obtain (cid:18) 12 det ( g j − k + g j + k ) n − j,k =0 (cid:19) = 1 − Φ n − ( − n − (1) det( g j − k ) n − j,k =0 , (cid:16) det ( g j − k − g j + k +2 ) n − j,k =0 (cid:17) = Φ n (1)Φ n ( − 1) det( g j − k ) n − j,k =0 , (cid:16) det ( g j − k ± g j + k +1 ) n − j,k =0 (cid:17) = (cid:18) Φ n ( − n (1) (cid:19) ± det( g j − k ) n − j,k =0 . (3.4)To prove Theorem 1.1, it then suffices to use (3.1) and to note that since g is non-negative, thezeros of the orthogonal polynomials are symmetric with respect to the real line and lie inside theunit disk, hence the right hand sides of (3.4) are positive. In this section, we let, as in Theorem 2.1 and Theorem 2.2, V be an analytic function in a neighbor-hood of the unit circle, real-valued on the unit circle and such that V ( e it ) = V ( e − it ), with Fouriercoefficients V k = V − k ∈ R , and we let m ∈ N , 0 < t < . . . < t m < π , and for any j = 1 , . . . , m , α j ≥ β j ∈ i R . Then we let g be of the form (2.1). This is a positive symbol with 2 m + 2Fisher-Hartwig singularities e ± it j and ± N ( ± N = 2 n and N = 2 n − .1 Asymptotics for Φ N ( ± The large N asymptotics for Φ N ( ± 1) are not readily available in the literature, but can be computedusing the RH analysis from [24], which was inspired by the analysis of [21]. Both those RH methodsare based on an asymptotic analysis of the function Y ( z ) = Φ N ( z ) πi R C Φ N ( ζ ) g ( ζ )d ζζ N ( ζ − z ) − χ N − z N − Φ N − ( z − ) − χ N − πi R C Φ N − ( ζ − ) g ( ζ )d ζζ ( ζ − z ) , (4.1)where χ − N − = π R π (cid:12)(cid:12) Φ N − ( e it ) (cid:12)(cid:12) g ( e it ) dt and C is the unit circle. This is the standard solution ofthe following RH problem for orthogonal polynomials on the unit circle [26]. RH problem for Y (a) Y is analytic in C \ C , where the unit circle C is oriented counterclockwise.(b) Y has continuous boundary values Y ± as z ∈ C\{± , e ± it , . . . , e ± it m } is approached from inside(+) or outside ( − ) the unit circle, and they are related by Y + ( z ) = Y − ( z ) (cid:18) z − N g ( z )0 1 (cid:19) . (c) Y ( z ) = ( I + O ( z − )) z Nσ as z → ∞ , where σ = (cid:18) − (cid:19) .If one imposes moreover suitable conditions near the points ± , e ± it j , the solution to the above RHproblem is unique, and one can derive asymptotics for it as N → ∞ using the Deift/Zhou steepestdescent method [22].In the following result, we restrict ourselves to symbols g of the form (1.3) with α = α m +1 = 0,i.e. the case where there are no Fisher-Hartwig singularities at the points ± 1, in the region neededfor Theorem 2.1. Proposition 4.1. Let g be of the form (2.1) with α = α m +1 = 0 . Define u + = t and u − = π − t m .We have Φ N (1) = e ( V − V (1)) m Y j =1 (cid:18) t j (cid:19) − α j e it j β j e − iπβ j (cid:18) O (cid:18) N u + (cid:19)(cid:19) Φ N ( − 1) = ( − N e ( V − V ( − m Y j =1 (cid:18) t j (cid:19) − α j e it j β j (cid:18) O (cid:18) N u − (cid:19)(cid:19) (4.2) as N → ∞ , uniformly over the region M/N < t < . . . < t m < π − M/N with M > sufficientlylarge, and uniformly for α j and β j in compact subsets of [0 , + ∞ ) and i R respectively.Proof. The analysis in [24] is based on partitioning the 2 m singularities e ± it j in different clusters.To do this, let us define for any 0 < M < M the clustering condition ( M , M , N ) as follows. Wesay that clustering condition ( M , M , N ) is satisfied if the set A = { t , . . . , t m , − t , . . . , − t m } canbe partitioned into ℓ ≤ m clusters A , . . . , A ℓ such that the following holds:(a) for any two values x, y ∈ A belonging to the same cluster A k , we have | x − y | ≤ M /N or || x − y | − π | ≤ M /N | , which means that singularities corresponding to the same clusterapproach each other fast enough as N → ∞ ,14b) for any two values x, y ∈ A belonging to a different cluster, we have | x − y | > M /N and || x − y | − π | > M /N | , which means that singularities corresponding to different clusters donot approach each other too fast as N → ∞ .Note that any clustering condition is trivially satisfied if m = 0. Observe also that different values of M may lead to a different number of clusters ℓ . Indeed, one cluster corresponding to a bigger valueof M may consist of the union of several clusters corresponding to a smaller value of M . Given M > 0, partition the ± t j ’s in ℓ = ℓ ( M ) clusters A j as above, and define µ ( M , M , N ) := min x ∈ A k ,y ∈ A j ,j = k | x − y | , (4.3)i.e. µ ( M , M , N ) is the minimal distance between arguments belonging to different clusters. Next,define arguments ˆ t , . . . , ˆ t ℓ , also depending on M and N , where ˆ t j is the average of the arguments t k belonging to the cluster A j . Under the clustering condition ( M , M , N ) and if in addition M > M / M ≥ M , we have 3 M N ≤ M N ≤ µ ( M , M , N ) ≤ u ± . (4.4)The RH analysis in [24] (which is inspired by the one from [21]) consists of explicit transformations Y T S R, where Y is given by (4.1), such that in particular we have Y ( z ) = Φ N ( z ). The transformations Y T and T S are similar as in [21] and are fairly standard; the transformation S R consistsof constructing local parametrices P in disks U j of radius µ ( M , M , N ) / e i ˆ t j for j = 1 , . . . , ℓ , and a global parametrix P ∞ elsewhere in the complex plane. By (4.4), everysingularity e ± it k is contained in one of the disjoint disks U j , and the points ± Step 1. Define T ( z ) = ( Y ( z ) , | z | < ,Y ( z ) z − Nσ , | z | > . Step 2. Define S ( z ) = T ( z ) z − N g ( z ) − ! , when z inside the lenses and outside the unit disc ,T ( z ) − z N g ( z ) − ! , when z inside the lenses and inside the unit disc ,T ( z ) , when z outside the lenses , where g is the analytic extension of g defined in (2.1) to the interior parts of the lenses, see Figure1 for the shape of the lenses and [21, Section 4] or [24, Section 6] for an explicit expression of thisanalytic continuation. Step 3. Define R ( z ) = ( S ( z ) P ∞ ( z ) − , z ∈ C \ ( S j U j ) ,S ( z ) P j ( z ) − , z ∈ U j . Here P ∞ is the global parametrix and P j ’s are local parametrices . We will not need their generalexpressions, but we will need the value of the global parametrix evaluated at z = ± ± , which isdefined in (4.6) below. 15 z z z z Figure 1: Opening of lenses in the case of 4 singularities z , z , z , z partitioned into four clusters.For z → + or z → − − , the transformations Y T S R imply (see [24, formulas (70),(75), (78), (83)]) Y ( ± 1) = T ( ± ± )( ± ± ) N = ( ± ± ) N S ( ± ± ) − f ( ± ± ) − S ( ± ± )= ( ± ± ) N ( RP ∞ ) ( ± ± ) − f ( ± − ( RP ∞ ) ( ± ± ) , (4.5)where (see [24, formulas (79) and (72)]) P ∞ ( ± ± ) = e − P − k = −∞ V k · ( ± k σ m Y j =1 (cid:0) ∓ e it j (cid:1) ( β j − α j ) σ (cid:0) ∓ e − it j (cid:1) ( − β j − α j ) σ , (4.6)with the principal branch of the roots.The final conclusion of the RH analysis in [24] is the following: for any M > 0, then forlarge enough M > 0, we have R ( z ) = I + O (cid:16) Nµ ( M ,M ,N ) (cid:17) , uniformly in z as N → ∞ , anduniformly under clustering condition ( M , M , N ). Let us now choose any value of M > 0, andlet M ≥ M be a constant induced by the above statement, i.e. let M = M ( M ) be such that R ( z ) = I + O (cid:16) Nµ ( M ,M ,N ) (cid:17) , uniformly in z as N → ∞ under clustering condition ( M , M , N ).Next, we iterate by defining M = M ( M ) ≥ M as some value such that (noting that µ ( M , N ) ≥ µ ( M , N )) R ( z ) = I + O (cid:18) N µ ( M , M , N ) (cid:19) = I + O (cid:18) N µ ( M , M , N ) (cid:19) , uniformly in z and under clustering condition ( M , M , N ) as N → ∞ . We iterate this procedure,which allows us to conclude that R ( z ) = I + O (cid:16) Nµ ( M ,M ,N ) (cid:17) uniformly as N → ∞ under the m disjoint clustering conditions( M , M , N ) , ( M , M , N ) , . . . , ( M m , M m +1 ) , M < . . . < M m +1 .We now take M > M m +1 / t j ’s such that M/N < t <. . . < t m < π − M/N and for sufficiently large N , at least one of the m above clustering conditionshold. By contraposition, if this were false, there would be for any k = 1 , . . . , m a different value j k ∈ { , . . . , m − } such that M k /N ≤ t j k +1 − t j k < M k +1 /N, since t , π − t m ≥ M m +1 / (2 N ). This yields a contradiction by the pigeonhole principle. We canconclude that we have the uniform bound R ( z ) = I + O (cid:16) Nµ ( M ,M ,N ) (cid:17) as N → ∞ , where we recallthat the constant M > M needs to be.This estimate is weaker than the one needed for (4.2), but we know in addition from [24, formulas(84)–(85), (86), and (89)] that k R ( z ) − I k ≤ π (cid:13)(cid:13)(cid:13)(cid:13)Z Σ R − ( s )(∆( s ) − I ) z − s ds (cid:13)(cid:13)(cid:13)(cid:13) , (4.7)where ∆( s ) is a matrix-valued function (the jump matrix), and Σ is the jump contour consisting ofthe circles ∂ U , . . . , ∂ U ℓ , and 2 ℓ arcs connecting neighbouring circles by one arc inside and one arcoutside the unit circle. On ∂ U j , we have the uniform bound ∆( s ) − I = O ( Nµ ( M ,M ,N ) ) as N → ∞ ,on the arcs inside (+) or outside ( − ) we have ∆( s ) − I = O ( | s | ± N ) as N → ∞ . Substituting this in(4.7) and setting z = ± 1, we obtain after straightforward estimates the uniform bound k R ( ± − I k = O (cid:18) N u ± (cid:19) , N → ∞ . Finally, after all these preparations, the result (4.2) follows upon substituting the asymptotics for R ( ± 1) and (4.6) in (4.5).We will now extend the above result to α , α m +1 > Proposition 4.2. Writing u + = t and u − = π − t m , we have Φ N (1) = e ( V − V (1)) √ πN α α + α m +1 Γ( α + ) m Y j =1 (cid:18) t j (cid:19) − α j e it j β j e − iπβ j (cid:18) O (cid:18) N u + (cid:19)(cid:19) , Φ N ( − 1) = ( − N e ( V − V ( − √ πN α m +1 α m +1 + α Γ( α m +1 + ) m Y j =1 (cid:18) t j (cid:19) − α j e it j β j (cid:18) O (cid:18) N u − (cid:19)(cid:19) , (4.8) as N → ∞ , uniformly over the region M/N < t < . . . < t m < π − M/N with M > sufficientlylarge, and uniformly for α j and β j in compact subsets of [0 , + ∞ ) and i R respectively.Proof. We again follow the RH analysis from [24] to prove this, the main difference with the proofof Proposition 4.1 being that the RH solution at the points ± U ± be a disk with radius u ± , centered at ± 1. The RH analysis from [24] requires to constructa local parametrix in U ± . We now have, because of the explicit transformations Y T S R in [24], the identities Y ( ± 1) = T ( ± ± )( ± ± ) N = ( ± ± ) N S ( ± ± ) − g ( ± ± ) − S ( ± ± )= ( ± ± ) N ( RP ± ) ( ± ± ) − g ( ± − ( RP ± ) ( ± ± ) , (4.9)17here g ( ± 1) is the boundary value of g when coming from the region inside the lenses in the upperhalf plane, where P ± is the local parametrix defined in U ± , and where R is uniformly close to I as N → ∞ . In order to obtain large N asymptotics for Φ N ( ± P ± and the large N asymptotics for R . These computations have been done in [21,Section 7] (see in particular equations (7.23)–(7.26) in that paper, and note the different notations β j 7→ − β j and α α + , α m +1 α r +1 + ), for the convenience of the reader we sketch thesecomputations here, restricting ourselves to the situation in U + , as the case U − is similar, and alsorestricting ourselves for simplicity to α / ∈ Z . The local parametrix P + then takes the form P + ( z ) = E ( z )Ψ( ζ ( z )) g ( z ) − σ z − Nσ / , (4.10)where ζ ( z ) = N log z , where Ψ( ζ ) is the solution to a model RH problem (depending on α , see [21,Section 4.1]) whose solution can be constructed out of confluent hypergeometric functions which inthe case β = 0 at hand degenerate to Bessel functions, and where E is a function analytic at ± E ( z ) and Ψ( ζ ) can be found explicitly in formulas (4.25), (4.32) and (4.50) of [21]. We have E ( z ) = P ∞ ( z ) g ( z ) σ z Nσ / j (cid:18) e πiα j − e − πiα j (cid:19) , where P ∞ behaves close to 1, outside the unit circle, in the following way (see [24, formulas (79) and(72)]): P ∞ ( z ) ∼ − α m +1 ( z − − α σ e − P − k = −∞ V k σ m Y j =1 (cid:0) − e it j (cid:1) ( β j − α j ) σ (cid:0) − e − it j (cid:1) ( − β j − α j ) σ (4.11)as z → − π, π ).After a straightforward calculation we obtain, for z ∈ U + in the region outside the unit circle andoutside the lens, Y ( z ) = z N ( RP j ) ( z ) − g ( z ) − ( RP j ) ( z )= z N P ∞ ( z ) e πiα j (cid:0) Ψ ( ζ ( z )) + e πiα Ψ ( ζ ( z )) (cid:1) (cid:18) O (cid:18) N u + (cid:19)(cid:19) as N → ∞ , where Ψ ( ζ ) and Ψ ( ζ ) are entries of Ψ( ζ ) in a certain sector of the complex plane,given by Ψ ( ζ ) = − ζ − α e − πiα e − ζ/ ψ (1 − α , − α , ζ ) Γ(1 + α )Γ( α )and Ψ ( ζ ) = ζ − α e − πiα e ζ/ ψ ( − α , − α , e − πi ζ ) , where ψ ( a, c ; z ) is the confluent hypergeometric function of the second kind with, in the case where α / ∈ Z , the standard expansion of ψ ( a, c ; z ) as z → ψ ( a, c ; z ) = Γ(1 − c )Γ(1 + a − c ) (1 + O ( z )) + Γ( c − a ) z − c (1 + O ( z )) . Substituting these asymptotics, we obtain after a straightforward computation Y ( z ) = z N P ∞ ( z ) e πiα ζ ( z ) α Γ( − α )Γ( − α ) (cid:16) e ζ ( z ) / e − πiα + e − ζ ( z ) / (cid:17) (cid:18) O (cid:18) N u + (cid:19)(cid:19) . P ∞ and ζ ( z ) = N log z , we obtainΦ N (1) = e ( V − V (1)) N α m Y j =1 (cid:18) t j (cid:19) − α j e it j β j e − iπβ j cos πα α m +1 − Γ( − α )Γ( − α ) (cid:18) O (cid:18) N u + (cid:19)(cid:19) (4.12)as N → ∞ . Using the reflection formula and the doubling formula for the Gamma function, as wellas the relation Γ(1 + z ) = z Γ( z ), we obtain the statement of the proposition. The other cases, namelythe asymptotics for Φ N (1) for α ∈ Z and the asymptotics for Φ N ( − 1) can be obtained in a similarway, we refer the reader to [21, Section 7] for details.0 z z z z Figure 2: Opening of lenses in the case of 4 singularities z , z , z , z partitioned into three clusters. Proposition 4.3. We have Φ N (1) = m Y j =1 (cid:18) sin t j n (cid:19) − α j × e O (1) , Φ N ( − 1) = m Y j =1 (cid:18) cos t j n (cid:19) − α j × e O (1) , (4.13) as N → ∞ , uniformly over the entire region < t < . . . < t m < π , and uniformly for α j and β j incompact subsets of [0 , + ∞ ) and i R respectively.Proof. We again follow the RH analysis from [24] to prove this. We restrict to the computationof Φ N (+1), as the computation of Φ N ( − 1) is similar, or can be derived from log Φ N (+1) aftertransforming the symbol by a rotation. Also, we can restrict to the case t ≤ M/N for some large M > 0, since the case t > M/N was handled in Proposition 4.1 and this implies the weaker result(4.2). 19et us take M > M , such that 2 t ≤ M /N , and define the clusters A , . . . , A ℓ , depending on M and on N , as before. The points ± t will then belong to the same cluster, which we label as A . By restricting to a subsequence of the positive integers N , we can assume that the numbers ofpoints in each cluster are independent of N . We write 2 µ for the number of points in A , such that A = { e ± it k } µ k =1 , and we observe that the average of the points in A is equal to ˆ t = 0. Next, wewrite U for the disk with radius µ ( M , M , N ) / µ ( M , M , N ) given by (4.3),and we use the local transformation ζ ( z ) = N log z for z ∈ U . We have ζ (1) = 0 and we define w k,N = − iζ ( e it k ) = N t k for 1 ≤ k ≤ µ . Note that w k,N ≤ M / k ≤ µ because of theclustering condition.The RH analysis from [24] requires us to construct a local parametrix in U . We now have,because of the explicit transformations Y T S R in [24] (see Figure 2 for the shape of thejump contour for S in this case), the identities Y (1) = (1 + ) N T (1 + ) = S (1 + ) = ( RP ) (1 + ) , where R (1) is bounded as N → ∞ , uniformly under clustering condition ( M , M , N ) for sufficientlylarge M (it is in fact close to I , but we will not need this). The corresponding lenses are described inFigure 2. Moreover, P is the local parametrix defined in U . The construction of this local parametrixis explained in detail in [24, Section 6.3]. We omit the technical details of this construction, andrestrict ourselves to the elements from it that we need for our purposes. As z → + , we have P (1 + ) = E (1)Φ(0; w ,N , . . . , w µ ,N ) (cid:18) g (1 + ) g (1 + ) − (cid:19) , (4.14)where Φ( ζ ; w , . . . , w µ ) is the solution to a model RH problem depending on parameters w , . . . , w µ , E (1) is given by E (1) = P ∞ (1 + ) (cid:18) (cid:19) µ Y ν =1 ( − iw ν,N ) β ν σ exp[ πi ( α ν − β ν ) σ ] g (1 + ) − σ , (4.15)with P ∞ (1 + ) the global parametrix given by (4.6). It is easily seen from this expression that E (1)is bounded as N → ∞ , uniformly in the parameters t , . . . , t m .Since P ∞ (1 + ) is diagonal, E (1) is off-diagonal and after a straightforward calculation we obtain Y (1) = ( RP ) (1 + )= E (1)Φ (0; w ,N , . . . , w µ ,N ) g (1 + ) − + O g (1 + ) − Φ(0; w ,N , . . . , w µ ,N ) N µ ( M , M , N ) ! , (4.16)as N → ∞ , uniformly under clustering condition ( M , M , N ) for M large enough.The matrix Φ(0; w , . . . , w µ ) is continuous as a function of w , . . . , w µ > ǫ for any ǫ , see [24,Section 5.3], and this implies that Y (1) = O ( g (1 + ) − / ) , N → ∞ , uniformly under clustering condition ( M , M , N ) for M large enough and with w ,N , . . . , w µ ,N > ǫ for some ǫ > 0, which implies the result in this case by (2.1).In order to evaluate the asymptotics of Φ(0; w , . . . , w µ ) when some of the w j ’s, say w , . . . , w k ,tend to 0 as N → ∞ , we need to follow the construction of another local parametrix Q in [24, Section5.3]. We again omit the details of this construction and refer the interested reader to [24]. The resultfrom this construction is thatΦ(0; w , . . . , w µ ) = F N k Y ν =1 ( − iw ν ) α ν σ U N (cid:18) − 11 0 (cid:19) D, F N is uniformly bounded as N → ∞ , U N is upper-triangular, and D is a diagonal matrixindependent of N , and the determinants of F N , D, U N are all equal to 1. It follows thatlog Φ (0; w , . . . , w µ ) = k X ν =1 α ν log | w ν | + O (1) . Substituting this in (4.16) and recalling that E (1) is uniformly bounded as N → ∞ , we getlog Y (1) = − 12 log g (1 + )+ k X ν =1 α ν log | w ν,N | + O (1) = − m X j =1 α j log | − e it j | +2 k X ν =1 α ν log( N t j )+ O (1) . It is straightforward to derive the result from this estimate. Under the assumptions of Theorem 2.1, we have by Proposition 1.1 and Propositions 4.1–4.2 that E (0 , +) n [ f ] = (cid:20) E U n [ g ] − Φ n − (1)Φ n − ( − (cid:21) / = C n (cid:2) E U n [ g ] (cid:3) / (1 + o (1)) , as n → ∞ , where C n is as in (2.6). The asymptotics for E (2 , − ) n [ f ] and E (1 , ± ) n [ f ] follow in a similarfashion. This ends the proof of Theorem 2.1.Under the assumptions of Theorem 2.2, we use Proposition 1.1 and Proposition 4.3 to obtain theuniform large N asymptotics E (0 , +) n [ f ] = (cid:0) E U n [ g ] (cid:1) / [ − Φ n − (1)Φ n − ( − − = (cid:0) E U n [ g ] (cid:1) m Y j =1 (cid:18) sin t j n (cid:19) α j (cid:18) cos t j n (cid:19) α j × e O (1) = (cid:0) E U n [ g ] (cid:1) m Y j =1 (cid:18) sin t j + 1 n (cid:19) α j × e O (1) . By similar computations, we obtain the required asympotics for E (2 , − ) n [ f ] and E (1 , ± ) n [ f ]. This endsthe proof of Theorem 2.2. In this section, we assume that g ( e it ) defined by (1.3) is of the form (2.11), i.e. g ( e it ) = e V ( e it ) × ( ≤ | t | ≤ t ,s for t < | t | ≤ π, for some real-valued function V analytic in a neighborhood of the unit circle, and with s ∈ [0 , .1 Asymptotics for Φ N ( ± Let Φ N be the monic polynomial of degree N , orthogonal with the weight g on the unit circle,characterized by the orthogonality conditions (1.8). The proof of the following result is based onthe RH representation for Φ N ( z ) , see Section 4.1, and on the large N asymptotic analysis of theRH problem in spirit of the analysis performed in [17]. We do not follow exactly the steps oftransformations from [17], but introduce a slightly different sequence of transformations. The mostsignificant differences of our analysis from the one done in [17] is that, first, during the step Y T we make a cosmetic transformation inside the unit disk, | z | < , and second, the function φ used inStep 3 is different from the one in [17]: they coincide up to a constant for | z | > | z | < . Proposition 5.1. Let V be as in Theorem 2.5. As N → ∞ with s = 0 , or as N → ∞ and at thesame time s → in such a way that s ≤ (cid:0) tan t (cid:1) N , we have the large N asymptotics Φ N (1) = √ t − π + ( − N π (cid:18) sin t (cid:19) N e − V (1)2 δ ( ∞ ) − (1 + o (1)) , Φ N ( − 1) = ( − N cos t (cid:18) t (cid:19) N δ − ( − δ ( ∞ ) − (1 + o (1)) . These asymptotics are also valid as t → π , as long as N ( π − t ) → ∞ . The o (1) terms can bewritten as O (cid:18) N ( π − t ) + s (tan t ) − N √ N ( π − t ) (cid:19) . Remark 5.2. Note that when s = 0 or s ≪ (tan t ) N , the first error term N ( π − t ) dominates thesecond. On the other hand, when s is close to (tan t ) N , the second error term becomes dominant,and is O ( √ N ( π − t ) ) . Furthermore, when t is not approaching π , the factor ( π − t ) in the errorterms can be omitted, but as t → π, the error term becomes larger due to it. Denote γ = { z : | z | = 1 , arg z ∈ ( − t , t ) } , γ c = { z : | z | = 1 , arg z ∈ ( t , π ) ∪ ( − π, − t ) } , both oriented in the counter-clockwise direction. Proof. The asymptotic analysis of the RH problem from Section 4.1 can be done using the followingsteps of transformations, Y T b T e T S R. Here the transformation Y T normalizes the asymptotics at infinity, while the transformations T b T e T are preparatory transformations before opening of the lenses. Then, e T S consists ofopening of the lenses, and S R is the final transformation to pass to a small-norm RH problem;this step involves construction of parametrices. We start by giving some more details about each ofthese transformations. Step 1. Define T ( z ) = Y ( z ) z − Nσ , | z | > ,Y ( z ) − 11 0 ! , | z | < . Here the transformation for | z | > z asymptotics of Y , while the trans-formation for | z | < ( z ) has the asymptotics T ( z ) → I as z → ∞ and satisfies the jump T + ( z ) = T − ( z ) (cid:18) g ( z ) − z N z − N (cid:19) for z on the unit circle C . Step 2. The jump for T ( z ) is highly oscillating for z ∈ C , and the next step is to factorize it intoproduct of two matrix functions, which can then be moved respectively inside or outside the unitdisk where they would be exponentially small. This is done differently for z ∈ γ and for z ∈ γ c , andwe start with γ. The idea is to exchange the term g ( z ) in the (1 , 1) entry of the jump for the T with1; an appropriate factorization will then easily follow. This is achieved with the help of the followingfunction δ ( z ) ,δ ( z ) = exp h ( z )2 πi Z γ V ( ζ ) dζ ( ζ − z ) h ( ζ ) , where the function h ( ζ ) = (( ζ − z )( ζ − z )) / is analytic in ζ ∈ C \ γ c and asymptotic to ζ as ζ → ∞ . The function δ is analytic in C \ C , has afinite non-zero limit as ζ → ∞ , and its boundary values satisfy the following conjugation conditionson the circle C : δ + ( z ) δ − ( z ) = 1 , z ∈ γ c , δ + ( z ) δ − ( z ) = e V ( z ) , z ∈ γ. Using the properties V ( z ) = V ( z − ) for | z | = 1 and h ( ζ ) = ζh ( ζ − ) , one can check that for all z wehave δ ( z ) δ ( z − ) = 1 and δ ( z ) = δ ( z ) . Let b T ( z ) = δ ( ∞ ) σ T ( z ) δ ( z ) − σ . b T tends to I as z → ∞ and satisfies the following jumps: b T + ( z ) = b T − ( z ) − z N δ + ( z ) e V ( z ) z N δ − ( z ) e V ( z ) ! , z ∈ γ, b T + ( z ) = b T − ( z ) s e V ( z ) δ − ( z ) δ + ( z ) − z N z − N ! , z ∈ γ c , We see that the jump matrix on γ can be factorized into a product of a lower-triangular and anupper-triangular matrix with ones on the diagonals, and this allows to “open lenses” around γ, inother words allows to get rid of oscillating entries on γ by transforming them into exponentiallysmall ones on lenses. However, we still have oscillating entries on γ c , and we cannot follow the samestrategy as for γ (i.e., to transform the (1 , 1) entry in the jump matrix to 1). Instead, we transformoff-diagonal entries into constant ones, by introducing the following function φ ( z ) , which is to replacethe function log z in z N = e N log z , and thus to transform the entries z ± N into 1 . Step 3. Define φ ( z ) = z Z z ( ζ + 1)d ζζh ( ζ ) + πi, ℓ = − t > , where the path of integration should not cross ( −∞ , ∪ γ c . Then one can check that φ ( z ) − log z = ℓ + O ( z − ) as z → ∞ , and φ ( z ) = φ ( z ) for all z, and φ ( z ) − log z is analytic in C \ γ c , where the principalbranch of the logarithm is taken. The function φ − ( z ) − φ + ( z ) is continuous and real-valued on γ c , and its maximum over γ c is attained at the point − , with φ − ( − − φ + ( − 1) = − t > . Let e T ( z ) = e N ( ℓ − πi ) σ b T ( z ) e − N ( φ ( z ) − πi − log z ) σ , z Ω out Ω in γ out γ c γγ in − z z Ω out Ω in γ out γ c γγ in − − z z S (on the left), and for R (on the right).then e T ( z ) → I as z → ∞ and e T satisfies the following jumps: e T + ( z ) = e T − ( z ) e N ( φ ( z ) − πi ) δ + ( z ) − e V ( z ) e − N ( φ ( z ) − πi ) δ − ( z ) e V ( z ) ! = e − N ( φ ( z ) − πi ) δ − ( z ) e V ( z ) ! e N ( φ ( z ) − πi ) δ + ( z ) − e V ( z ) ! , z ∈ γ, e T + ( z ) = e T − ( z ) s e N ( φ − ( z ) − φ + ( z )) e V ( z ) δ − ( z ) δ + ( z ) − z N z − N ! , z ∈ γ c . Step 4. The next step is the opening of lenses around γ. Consider the regions as indicated in theleft part of Figure 3, and define S ( z ) = e T ( z ) δ ( z ) − e − V ( z ) ( − N e − N ( φ ( z )) ! , z ∈ Ω out , e T ( z ) δ ( z ) e − V ( z ) ( − N e Nφ ( z ) ! , z ∈ Ω in , e T ( z ) , elsewhere . Step 5a. Now, we take r > S as follows. Let U z , U z , U − be (non-intersecting) disks centered at z , z , − r cos t ; their boundaries ∂U z , ∂U z , ∂U − are oriented in the counter-clockwise direction. Define (we use the letters u (up), d (down), l (left) to distinguish between theparametrices at the points z , z , − , respectively; see also the right part of Figure 3) P ( z ) = P ∞ ( z ) , z ∈ C \ ( U z ∪ U z ∪ U − ) ,P u ( z ) , z ∈ U z ,P d ( z ) , z ∈ U z ,P l ( z ) , z ∈ U − , 24e see that the radius r cos t of the disks shrinks as t approaches π. For us, the explicit expressionsfor the local parametrices P u and P d will be unimportant because we only need to evaluate Φ N atthe points ± 1; however, we will still need them in order to estimate the error term. The form of theouter parametrix P ∞ on the other hand is more important: it is given by P ∞ ( z ) = (cid:18) ( κ ( z ) + κ ( z ) − ) i ( κ ( z ) − κ ( z ) − ) − i ( κ ( z ) − κ ( z ) − ) ( κ ( z ) + κ ( z ) − ) (cid:19) , where κ ( z ) = (cid:18) z − z z − z (cid:19) / , analytic in z ∈ C \ γ c and asymptotic to 1 at infinity. Note that κ (1) = e i ( π − t ) / , κ − ( − 1) = e − it / . Step 5b: Local parametrix at z . Change of variable. First of all, the linear fractional change of variable k = k ( z ) = − z z z − z mapsthe points of the unit circle to the real line as follows: z , − 7→ − , z → ∞ , , and thus allows to separate the points z , − , z , which might be merging as t → π. Next, using thevariable k the function φ ( z ) can be written as φ ( z ) = πi − i cos t Z k ( z )0 ( k + 1)d k ( k + z )( k + z ) √ k , (5.1)where the path of integration does not intersect ( −∞ , , and the principal branch of the squareroot is taken. This prompts to introduce a local variable ζ = ζ ( z ; t ) in the disk U z as follows: φ ( z ) =: πi − i cos t √ ζ, so that ζ = k (1 + O ( k )) , k → , and the branch cut for √ ζ, i.e. thehalf-line ζ < , corresponds to z ∈ γ c . Introduce also the new large parameter τ := 2 N cos t , then N ( φ ( z ) − πi ) = − iτ √ ζ. Bessel parametrix. Similarly as e.g. in [34, Section 6] (but note the different sign of the off-diagonalentries of the jump matrices), we construct a function which solves exactly the same jumps as S ina small neighborhood of the point z . DefineΨ( ζ ) = √ π e − πi √ ζI ( − i √ ζ ) − √ π e πi √ ζK ( − i √ ζ ) −√ π e πi I ( − i √ ζ ) √ π e − πi K ( − i √ ζ ) ! , arg ζ ∈ ( π − α, π ) , = √ π e πi √ ζK ( i √ ζ ) − √ π e πi √ ζK ( − i √ ζ ) √ π e − πi K ( i √ ζ ) √ π e − πi K ( − i √ ζ ) ! , arg ζ ∈ ( − π + α, π − α ) , = √ π e πi √ ζK ( i √ ζ ) √ π e − πi √ ζI ( i √ ζ ) √ π e − πi K ( i √ ζ ) √ π e πi I ( i √ ζ ) ! , arg ζ ∈ ( − π, − π + α ) , where α ∈ (0 , π ) and I j , K j , j = 0 , + ( ζ ) = Ψ − ( ζ ) (cid:20) (cid:21) , ζ ∈ ( ∞ e i ( π − α ) , + ( ζ ) = Ψ − ( ζ ) (cid:20) − 10 1 (cid:21) , ζ ∈ ( ∞ e − i ( π + α ) , + ( ζ ) = Ψ − ( ζ ) (cid:20) − 11 0 (cid:21) , ζ ∈ (0 , −∞ ) , where the orientation of the segments is fromthe first mentioned point to the last one, and ( ∞ e iβ , 0) denotes the ray coming from infinity to the25rigin at an angle β ∈ R (see the left part of Figure 4). Besides, the function Ψ satisfies the uniformin arg ζ ∈ [ − π, π ] asymptoticsΨ( ζ ) = ζ σ √ (cid:18) − i − i (cid:19) E ( ζ )e − i √ ζσ , E ( ζ ) = I + O ( 1 √ ζ ) , ζ → ∞ . We will also need the function b Ψ( ζ ) := Ψ( ζ ) G ( ζ ) , where G ( ζ ) := I − s πi log ζ ! , arg ζ ∈ ( π − α, π ) ,I − s πi log ζ − − ! , arg ζ ∈ ( − π + α, π − α ) ,I + s πi log ζ ! , arg ζ ∈ ( − π, − π + α ) . The function b Ψ satisfies the jumps as in the right part of Figure 4.For z : | z − z | < r cos t , define P u ( z ) = B u ( z ) b Ψ( τ ζ ) δ ( z ) − σ e − V ( z )sgn(log | z | ) σ e − N ( φ ( z ) − πi ) σ , where B u ( z ) = P ( ∞ ) ( z ) δ ( z ) σ e V ( z )sgn(log | z | ) σ √ (cid:20) ii (cid:21) ( τ ζ ) − σ and B u ( z ) is analytic in U z (i.e.,does not have jumps across C ). Here sgn( x ) = x | x | is the signum function, so that e − V ( z )sgn(log | z | ) σ equals e V ( z ) σ for | z | < − V ( z ) σ for | z | > . The function P u ( z ) satisfies the samejumps as S ( z ) inside U z , and on the boundary ∂U z we have the following matching condition: P ( z ) P ( ∞ ) ( z ) − = P ( ∞ ) ( z ) δ ( z ) σ e V ( z )sgn(log | z | ) σ ·E ( τ ζ )e − iτ √ ζσ G ( τ ζ ) · e − N ( φ ( z ) − π ) σ δ ( z ) − σ e − V ( z )sgn(log | z | ) σ P ( ∞ ) ( z ) − = I + O ( 1 τ √ ζ ) , as τ ζ → ∞ . Here we used that P ∞ is bounded on ∂U z uniformly in t . Step 5c: Local parametrix at z . For z inside U z we define P ( z ) := σP ( z ) σ, where σ = (cid:20) (cid:21) . Step 5d: Local parametrix at − . For z ∈ U − , define P l ( z ) = P ( ∞ ) ( z ) G l ( z ) , where G l ( z ) = (cid:20) − sf ( z ) 1 (cid:21) for | z | < G l ( z ) = (cid:20) sf ( z )0 1 (cid:21) for | z | > , with f ( z ) = 12 πi Z γ c e N ( φ − ( ξ ) − φ + ( ξ )) e V ( ξ ) δ − ( ξ ) δ + ( ξ ) d ξξ − z . Note that φ − − φ + has a double zero at the point z = − , and hence large N asymptotics of f ( z )can be obtained by classical saddle point methods. Using (5.1), we see that the large parameter is N cos t rather than N, and for z ∈ ∂D − we have | f ( z ) | = O ( p N cos t )e Nφ − ( − . The matchingcondition on the circle | z + 1 | = r cos t is P l ( z ) P ( ∞ ) ( z ) − = P ( ∞ ) ( z ) G ( z ) P ( ∞ ) ( z ) − = I + O q N cos t s (tan t − N , − 11 0 (cid:21) (cid:20) (cid:21)(cid:20) − 10 1 (cid:21) − + −− + (cid:20) s − 11 0 (cid:21) (cid:20) (cid:21)(cid:20) − 10 1 (cid:21) − + −− +Figure 4: Jumps for the functions Ψ( ζ ) (on the left) and b Ψ( ζ ) ( on the right).as N cos t → ∞ . Here we used that P ∞ is bounded on ∂U − uniformly in t . Step 6. Define the error function R by the formula S ( z ) = R ( z ) P ( z ) , where as before, P means P u , P d , P l in the relevant disks, and P means P ∞ elsewhere. The jumpconditions for R on the disks U z , U z , U − allow to conclude that R ( z ) = I + O (( N cos t ) − +( N cos t ) − / s (tan t ) − N ) uniformly in z, as N cos t → ∞ , under the conditions of Theorem 2.5(note that this is consistent with the results of the RH analysis from [17]). Tracing back the chainof transformations from R to Y, we find that (as N cos t → ∞ ) Y ( z ) = (cid:18) 12 ( κ ( z ) + κ ( z ) − ) R ( z ) − i κ ( z ) − κ ( z ) − ) R ( z ) − ( κ ( z ) + κ ( z ) − ) R ( z ) + i ( κ ( z ) − κ ( z ) − ) R ( z ) δ ( z ) e V ( z ) e N ( φ ( z ) − πi ) ! e N ( φ ( z )+log z − ℓ ) δ ( z ) δ ( ∞ ) , for z ∈ Ω out , and that Y ( z ) = (cid:18) 12 ( κ ( z ) + κ ( z ) − ) R ( z ) − i 12 ( κ ( z ) − κ ( z ) − ) R ( z ) (cid:19) e N ( φ ( z )+log z − ℓ ) δ ( z ) δ ( ∞ ) , for z ∈ { z : | z | > } \ Ω out . From here, using δ − (1) = e − V (1) , φ (1) = 0 , we obtain Y ( − 1) = (cid:18) cos t R ( − − − sin t R ( − − (cid:19) (cid:18) t (cid:19) N ( − N δ − ( − δ ( ∞ ) ,Y (1) = (cid:18)(cid:18) cos π − t − N sin π − t (cid:19) R (1) + (cid:18) sin π − t − ( − N cos π − t (cid:19) R (1) (cid:19) · (cid:18) sin t (cid:19) N e − V (1) / δ ( ∞ ) . Subtituting the asymptotics R ( z ) = I + o (1) for R , we obtain the result. From Proposition 5.1, we obtainΦ N (1)Φ N ( − 1) = ( − N C N (1 + o (1)) , Φ N (1)Φ N ( − 1) = ( − N e C N (1 + o (1)) , N → ∞ , where C N = √ t t − π + ( − N π (cid:18) sin t (cid:19) N (cid:18) t (cid:19) N e − V (1) δ − ( − δ ( ∞ ) , e C N = √ t − π +( − N π cos t (cid:18) sin t t (cid:19) N e − V (1) δ − ( − . Substituting this in Proposition 1.1, we obtain (2.15). The goal is to apply Theorem 2.5 to compute the averages in (2.20), but this requires certain adapta-tions. One needs to make the change of variables θ k π − θ k for k = 1 , ..., n in the averages (2.20),which given (1.1) yields E +2 n ( t ; 0) = E (0 , +) n [ f ] ,E − n +2 ( t ; 0) = E (2 , − ) n [ f ] ,E ± n +1 ( t ; 0) = E (1 , ∓ ) n [ f ] , where f is related to g in (2.11) with V = 0, s = 0 and with the change of parameter t π − t .One may therefore compute the right-hand side of the above equalities using Theorem 2.5, and thisyields E +2 n ( t ; 0) = 2 e ζ ′ ( − (cid:18) cos t (cid:19) − n (cid:18) t (cid:19) − n − (cid:0) cos t (cid:1) n (2 n sin t ) (1 + o (1)) ,E − n +2 ( t ; 0) = 2 e ζ ′ ( − − (cid:18) cos t (cid:19) n (cid:18) t (cid:19) n +12 (cid:0) cos t (cid:1) n (2 n sin t ) (1 + o (1)) ,E ± n +1 ( t ; 0) = 2 e ζ ′ ( − " (1 + sin t ) n (cid:0) cos t (cid:1) n ± (cid:0) cos t (cid:1) n (cid:0) n sin t (cid:1) (1 + o (1)) , as n → ∞ , and this is equivalent to the desired result. One then applies the interrelation (2.21) toobtain the asymptotics for the C β E ensembles with β = 1 , The symbol f t ,s in (2.19) is associated to g t ,s in (2.18) through equation (1.3). One then noticesthe relation g t ,s = s t π g, where g is defined by (2.1) with V = 0, m = 1, t = t , α = α = α m +1 = 0 and β = log s πi .Applying Theorem 2.1, we get E +2 n ( t ; s ) = s nt π C E U n [ g ] (1 + o (1)) ,E − n +2 ( t ; s ) = s nt π C − E U n [ g ] (1 + o (1)) ,E ± n +1 ( t ; s ) = s nt π ˜ C ± E U n [ g ] (1 + o (1)) , C = e log s e − t s π , ˜ C = e − log s . But now from [20, Theorem 1.11], for t fixed or when t → nt → + ∞ one knows that E U n [ g ] = (4 n sin t ) log2 s π (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) s πi (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (1 + o (1)) , from which the result follows. One then applies the interrelation (2.21) to obtain the asymptotics inthe C β E ensembles. In a similar fashion, to prove Corollary 2.10, one uses Theorem 2.2. Let n be a positive integer and consider the n free eigenangles θ ≤ . . . ≤ θ n in O ± N . Define thecounting measure N (0 ,t ) = P nk =1 χ (0 ,t ) ( θ k ) as the number of eigenangles in (0 , t ), for 0 < t ≤ π . Forlater convenience, let us also write θ = 0 and θ n +1 = π .We first use a discretization of the supremum of the counting function to bound the two quantitiesof interest in Theorem 2.12. Lemma 6.1. In O +2 n , O − n +2 , and O ± n +1 , we have almost surely max k =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) θ k − πkn (cid:12)(cid:12)(cid:12)(cid:12) ≤ πn (cid:18) k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12)(cid:19) , sup t ∈ (0 ,π ) (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) . Proof. Since [0 , π ) = ⊔ n − j =0 [ πjn , π ( j +1) n ), for each k = 1 , ..., n there exists a unique j ∈ { , ..., n − } such that πjn ≤ θ k < π ( j +1) n . Given that N (0 ,t ) is a non-decreasing function of t , we find the followingestimates, N (0 , πjn ) − ( j + 1) ≤ N (0 ,θ k ) − nθ k π ≤ N (0 , π ( j +1) n ) − j. Because of the ordering of the eigenangles, N (0 ,θ k ) = k − 1, so that (cid:16) N (0 , πjn ) − ( j − (cid:17) − ≤ nπ (cid:18) πkn − θ k (cid:19) ≤ (cid:16) N (0 , π ( j +1) n ) − j (cid:17) + 1 , and it then suffices to take the maximum or minimum over k and j to obtain the first estimate.Using a similar partitioning argument, one hassup t ∈ (0 ,π ) (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) = max k =0: n sup t ∈ ( θ k ,θ k +1 ] (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) . Now as a function of t , N (0 ,t ) is left-continuous, has a jump of size 1 at each θ k , is constant andequals k on ( θ k , θ k +1 ], thereforesup t ∈ ( θ k ,θ k +1 ] N (0 ,t ) − ntπ = k − nθ k π = nπ (cid:18) πkn − θ k (cid:19) , inf t ∈ ( θ k ,θ k +1 ] N (0 ,t ) − ntπ = k − nθ k +1 π = nπ (cid:18) π ( k + 1) n − θ k +1 (cid:19) − . t ∈ (0 ,π ) (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) ≤ nπ max k =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) θ k − πkn (cid:12)(cid:12)(cid:12)(cid:12) , and it then suffices to use the previous estimate to conclude. Lemma 6.2. In O +2 n , O − n +2 , and O ± n +1 , for any α > , γ > there exists C γ > such that P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) > α (cid:19) ≤ C γ e − γα n γ π +1 . Proof. By definition and Boole’s inequality one has P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) > α (cid:19) ≤ P (cid:18) max k =1 ,...,n N (0 , πkn ) − ( k − > α (cid:19) + P (cid:18) min k =1 ,...,n N (0 , πkn ) − ( k − < − α (cid:19) , as well as (the last term of the sum always vanishes) P (cid:18) max k =1 ,...,n N (0 , πkn ) − ( k − > α (cid:19) ≤ n − X k =1 P (cid:16) N (0 , πkn ) > k − α (cid:17) , P (cid:18) min k =1 ,...,n N (0 , πkn ) − ( k − < − α (cid:19) ≤ n − X k =1 P (cid:16) − N (0 , πkn ) > − k + 1 + α (cid:17) . Applying Chernoff’s bound for γ > t ∈ (0 , π ) P (cid:18) N (0 ,t ) > ntπ + α (cid:19) ≤ e − γα e − γntπ E ( j, ± ) n h e γχ [ − t,t ] (arg z ) i , P (cid:18) − N (0 ,t ) > − ntπ + α (cid:19) ≤ e − γα e γntπ E ( j, ± ) n h e − γχ [ − t,t ] (arg z ) i , Therefore, for any δ ∈ R \ { } , t ∈ [ n , π − n ], one may write, using Corollary 2.8, for some C δ > E ( j, ± ) n h e δχ [ − t,t ] (arg z ) i ≤ C δ e δntπ ( n sin t + 1) δ π . This leads to the following estimate for some C γ > P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) > α (cid:19) ≤ C γ e − γα n γ π n − X k =1 (cid:18) sin πkn (cid:19) γ π , and since as n → + ∞ n − X k =1 (cid:18) sin πkn (cid:19) γ π ∼ n Z (sin πt ) γ π dt, this ends the proof. 30n order to prove Theorem 2.12, we use on one hand Lemma 6.1, which implies P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) θ k − πkn (cid:12)(cid:12)(cid:12)(cid:12) > (1 + ǫ ) log nn (cid:19) + P sup t ∈ (0 ,π ) (cid:12)(cid:12)(cid:12)(cid:12) N (0 ,t ) − ntπ (cid:12)(cid:12)(cid:12)(cid:12) > (cid:18) π + ǫ (cid:19) log nπ ! ≤ P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) > (1 + ǫ ) log nπ − (cid:19) , while on the other it follows from Lemma 6.2 that for any γ > C γ > P (cid:18) max k =1 ,...,n (cid:12)(cid:12)(cid:12) N (0 , πkn ) − ( k − (cid:12)(cid:12)(cid:12) > (1 + ǫ ) log nπ − (cid:19) ≤ C γ n γ π − (1+ ǫ ) γπ +1 . 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