The Classical Compact Groups and Gaussian Multiplicative Chaos
TThe Classical Compact Groups and Gaussian MultiplicativeChaos
Johannes Forkel ∗ and Jonathan P. Keating † August 19, 2020
Abstract
We consider powers of the absolute value of the characteristic polynomial of Haar distributedrandom orthogonal or symplectic matrices, as well as powers of the exponential of its argument, asa random measure on the unit circle minus small neighborhoods around ± . We show that forsmall enough powers and under suitable normalization, as the matrix size goes to infinity, theserandom measures converge in distribution to a Gaussian multiplicative chaos measure. Our resultis analogous to one on unitary matrices previously established by Christian Webb in [31]. We thuscomplete the connection between the classical compact groups and Gaussian multiplicative chaos.To prove this we establish appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankeldeterminants with merging singularities. Using a recent formula communicated to us by Claeys etal. , we are able to extend our result to the whole of the unit circle. Contents L -Limit 13 L -Limit on I (cid:15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Calculating the L -Limit on all of [0 , π ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ∗ [email protected] , Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom † [email protected] , Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom a r X i v : . [ m a t h - ph ] A ug Aymptotics of the Orthogonal Polynomials 31 ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.5 < t ≤ ω ( n ) /n . Local Parametrices near e ± ip . . . . . . . . . . . . . . . . . . . . . . . . 347.5.1 RHP for Φ ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5.2 < t ≤ ω ( n ) /n . Construction of a Local Parametrix near e ± ip in terms of Φ ± . . 367.5.3 < t ≤ ω ( n ) /n . Final Transformation . . . . . . . . . . . . . . . . . . . . . . . . 387.6 ω ( n ) /n < t < t . Local Parametrices near e ± ip . . . . . . . . . . . . . . . . . . . . . . . 397.6.1 ω ( n ) /n < t < t . Final Transformation . . . . . . . . . . . . . . . . . . . . . . . . 42 D n ( f p,t ) < t ≤ ω ( n ) /n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.1.2 ω ( n ) /n < t < t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.2 Integration of the Differential Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Φ n (0) , Φ n ( ± and D T + H,κn ( f p,t )
54A Convergence to the Gaussian Fields 55B Construction of the Gaussian Multiplicative Chaos Measure 56C Riemann-Hilbert Problem for Ψ
57D Riemann-Hilbert Problem for M In [22], Hughes, Keating and O’Connell proved that the real and the imaginary part of the logarithmof the characteristic polynomial of a random unitary matrix convergence jointly to a pair of Gaussianfields on the unit circle. Using this result Webb established in [31] and [28] a connection betweenrandom matrix theory and Gaussian multiplicative chaos (GMC), a theory developed first by Kahane inthe context of turbulence in [24] (see [30] for a review). Webb proved that powers of the characteristicpolynomial of a random unitary matrix converge, when suitably normalized, to Gaussian multiplicativechaos measures on the unit circle. A key technical input Webb used were results on Toeplitz determinantswith merging Fisher-Hartwig singularities due to Claeys and Krasovsky in [13].Since then the connection between the two fields has been extended to other random matrixensembles. In [9], Chhaibi and Najnudel proved convergence (in a different sense) of the characteristicpolynomials of matrices drawn from the Circular Beta Ensemble to a GMC measure on the unitcircle. In [7], Berestycki, Webb and Wong proved that powers of the absolute value of the logarithm ofthe characteristic polynomial of a matrix from the Gaussian Unitary Ensemble converge to Gaussianmultiplicative chaos measure on the real line. The connection with GMC is closely related to recentdevelopments concerning the extreme value statistics of the characteristic polynomials of randommatrices and the associated theory of moments of moments [1–4, 6, 8, 19–21, 29]. It also has interestingapplications to spectral statistics; for example, it implies strong rigidity estimates for the eigenvalues [10].Our purpose here it to extend Webb’s result to the other classical compact groups, i.e. to theorthogonal and symplectic groups. Our starting point is a theorem due to Assiotis and Keatingconcerning the convergence of the real and imaginary parts of the logarithm of the characteristicpolynomials of random orthogonal or symplectic matrices to a pair of Gaussian fields on the unit2ircle . This is the analogous result to the one for random matrices in [22]. We then complete theconnection between the classical compact groups and Gaussian multiplicative chaos, by showing thatfor the orthogonal and symplectic groups we get statements similar to the one Webb proved for theunitary group. We first prove convergence to the GMC measure after restricting all involved measuresto ( (cid:15), π − (cid:15) ) ∪ ( π + (cid:15), π − (cid:15) ) , i.e. we exclude small neighborhoods around ± . To prove convergenceaway from ± we computed the uniform asymptotics of Toeplitz and Toeplitz+Hankel determinantswith two pairs of merging singularities, which are all bounded away from ± . Our results on theseasymptotics are similar to those in [14] and [13], and the proof techniques we employ are stronglyinfluenced by these two papers.To prove convergence on the full unit circle we need also to know the uniform asymptotics ofToeplitz and Toeplitz+Hankel determinants with 3 or 5 singularities merging at ± . Claeys, Glesner,Minakov and Yang have recently communicated to us an expression for the uniform asymptotics ofToeplitz+Hankel determinants with arbitrarily many merging singularities. Using their formula allowsus to extend our analysis to around ± , and so to cover the full unit circle. Denote by O ( n ) the group of orthogonal n × n matrices, and by Sp (2 n ) the group of n × n symplecticmatrices, i.e. unitary n × n matrices that additionally satisfy U JU T = U T JU = J, (2.1)where J := (cid:18) I n − I n (cid:19) . (2.2)The characteristic polynomial p n ( θ ) = det (cid:0) I n − e − iθ U n (cid:1) = n (cid:89) k =1 (1 − e i ( θ k − θ ) ) (2.3)of U n in O ( n ) or Sp (2 n ) (then we have instead I n and the product is up to n ) is taken as a functionon the unit circle, where all its zeroes lie. Definition 1.
For n ∈ N , α ∈ R , β ∈ i R and θ ∈ [0 , π ) let f n,α,β ( θ ) = | p n ( θ ) | α e iβ (cid:61) ln p n ( θ ) , (2.4) where (with the sum being up to n for U n ∈ Sp (2 n ) ) (cid:61) ln p n ( θ ) := n (cid:88) k =1 ln (cid:61) (1 − e i ( θ k − θ ) ) , (2.5) with the branches on the RHS being the principal branches, such that (cid:61) ln(1 − e i ( θ l − θ ) ) ∈ (cid:16) − π , π (cid:105) , (2.6) where (cid:61) ln 0 := π/ . Further we define the random Radon measures µ n,α,β on S ∼ [0 , π ) by µ n,α,β ( d θ ) = f n,α,β ( θ ) E ( f n,α,β ( θ )) d θ. (2.7) This result has not previously been published. With the kind agreement of Dr. Assiotis, we set out the theorem andits proof in Appendix A ( N j ) j ∈ N be a sequence of independent standard (real) normal random variables and denote η j := 1 j is even . (2.8)We recall the following result from [15, 16]: Theorem 2. (Diaconis and Shahshahani, Diaconis and Evans) If U n is Haar distributed on O ( n ) wehave for any fixed k : (cid:18) Tr ( U n ) , √ Tr ( U n ) , ..., √ k Tr ( U kn ) (cid:19) d −−−−→ n →∞ (cid:18) N + η , N + η √ , ..., N k + η k √ k (cid:19) . (2.9) Similarly, if U n is Haar distributed on Sp (2 n ) , we have for any fixed k ∈ N : (cid:18) Tr ( U n ) , √ Tr ( U n ) , ..., √ k Tr ( U kn ) (cid:19) d −−−−→ n →∞ (cid:18) N − η , N − η √ , ..., N k − η k √ k (cid:19) . (2.10) Finally we have the bound E U n (cid:16)(cid:0) Tr ( U kn ) (cid:1) (cid:17) ≤ const min { k, n } , (2.11) where const is independent of k and n . Using this result, Assiotis and Keating have proved the following theorem : Theorem 3. (Assiotis, Keating) Let p n ( θ ) be the characteristic polynomial of a random U n ∈ O ( n ) ,w.r.t. Haar measure. Then for any (cid:15) > , ( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) converges in distribution in H − (cid:15) × H − (cid:15) to the pair of Gaussian fields (cid:16) X ( θ ) − x ( θ ) , ˆ X ( θ ) − ˆ x ( θ ) (cid:17) , where X ( θ ) = 12 ∞ (cid:88) j =1 √ j N j (cid:0) e − ijθ + e ijθ (cid:1) = ∞ (cid:88) j =1 √ j N j cos( jθ ) , ˆ X ( θ ) = 12 i ∞ (cid:88) j =1 √ j N j (cid:0) e − ijθ − e ijθ (cid:1) = − ∞ (cid:88) j =1 √ j N j sin( jθ ) ,x ( θ ) = 12 ∞ (cid:88) j =1 η j j (cid:0) e − ijθ + e ijθ (cid:1) = ∞ (cid:88) j =1 η j j cos( jθ ) , ˆ x ( θ ) = 12 i ∞ (cid:88) j =1 η j j (cid:0) e − ijθ − e ijθ (cid:1) = − ∞ (cid:88) j =1 η j j sin( jθ ) . (2.12) Similarly, for U n ∈ Sp (2 n ) and any (cid:15) > , ( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) converges in distribution in H − (cid:15) × H − (cid:15) to the pair of Gaussian fields (cid:16) X ( θ ) + x ( θ ) , ˆ X ( θ ) + ˆ x ( θ ) (cid:17) . The spaces H − (cid:15) are defined in Appendix A as certain closed subspaces of the negative Sobolev spaces H − (cid:15) . This result has not previously been published, and so with the kind agreement of Dr. Assiotis we set out the proofin Appendix A. emark 4. Formally one has E ( X ( θ ) X ( θ (cid:48) )) = ∞ (cid:88) j =1 cos( jθ ) cos( jθ (cid:48) ) j = 12 ∞ (cid:88) j =1 cos( j ( θ + θ (cid:48) )) j + ∞ (cid:88) j =1 cos( j ( θ − θ (cid:48) )) j = 14 ∞ (cid:88) j =1 j (cid:16) e ij ( θ + θ (cid:48) ) + e − ij ( θ + θ (cid:48) ) + e ij ( θ − θ (cid:48) ) + e − ij ( θ − θ (cid:48) ) (cid:17) = − (cid:16) ln | e iθ − e iθ (cid:48) | + ln | e iθ − e − iθ (cid:48) | (cid:17) . (2.13) Similarly one has that formally E (cid:16) ˆ X ( θ ) ˆ X ( θ (cid:48) ) (cid:17) = − (cid:16) ln | e iθ − e iθ (cid:48) ) | − ln | e iθ − e − iθ (cid:48) | (cid:17) , E (cid:16) X ( θ ) ˆ X ( θ (cid:48) ) (cid:17) = 12 (cid:16) (cid:61) ln(1 − e i ( θ + θ (cid:48) ) ) − (cid:61) ln(1 − e i ( θ − θ (cid:48) ) ) (cid:17) . (2.14)For α ∈ R and β ∈ i R define the field Y α,β ( θ ) = 2 α ( X ( θ ) ± x ( θ )) + 2 iβ (cid:16) ˆ X ( θ ) ± ˆ x ( θ ) (cid:17) . (2.15)Its covariance function is formally given byCov ( Y ( θ ) , Y ( θ (cid:48) )) = − α − β ) ln | e iθ − e iθ (cid:48) | − α + β ) ln | e iθ − e − iθ (cid:48) | + 4 iαβ (cid:61) ln(1 − e i ( θ + θ (cid:48) ) ) . (2.16)Motivated by Theorem 3 one expects that µ n,α,β behaves like e Y α,β for large n . Even though thecovariance function of Y α,β has logarithmic singularities, not only on the diagonal θ = θ (cid:48) but also onthe anti-diagonal θ = − θ (cid:48) , one can still construct a corresponding non-trivial Gaussian multiplicativechaos measure µ α,β , which can formally be written as µ α,β ( d θ ) = e Y α,β ( θ ) E ( e Y α,β ( θ ) ) d θ = e αX ( θ )+2 iβ ˆ X ( θ ) − Var ( Y α,β ( θ )) d θ. (2.17) µ α,β is properly defined in Appendix B as the almost sure limit in distribution of random Radonmeasures ( µ ( k ) α,β ) k ∈ N .For (cid:15) ∈ (0 , π/ define I (cid:15) := ( (cid:15), π − (cid:15) ) ∪ ( π + (cid:15), π − (cid:15) ) . Then our first main result is the following: Theorem 5.
Let α > − / and α − β < / . When restricting the random measures µ n,α,β and µ α,β to I (cid:15) , then as n → ∞ , for any fixed (cid:15) > , µ n,α,β converge weakly to µ α,β in the space of Radon measureson I (cid:15) equipped with the topology of weak convergence, i.e. for any F : { Radon measures on I (cid:15) } → R forwhich F ( µ n ) → F ( µ ) whenever µ n d −→ µ , it holds that E ( F ( µ n,α,β )) n →∞ −−−−→ E ( F ( µ α,β )) . (2.18)Claeys, Glesner, Minakov and Yang have recently communicated to us a new expression for theuniform asymptotics of Toeplitz+Hankel determinants. The specialisation of their formula to oursituation is stated in Theorem 18. Using this, we can prove our second main result, which extendsTheorem 5 to the full circle, but for a slightly smaller set of parameters α, β :5 heorem 6. Let α − β < / and additionally α ≥ , α < . Then the random measures µ n,α,β converge weakly to µ α,β in the space of Radon measure on [0 , π ) equipped with the topology of weakconvergence. Proof strategy:
Let I denote either I (cid:15) or [0 , π ) . We first remark that by Theorem 4.2. in [25],weak convergence of µ n,α,β to µ α,β in the space of Radon measures on I equipped with the topology ofweak convergence is equivalent to (cid:90) I g ( θ ) µ n,α,β ( d θ ) d −→ (cid:90) I g ( θ ) µ α,β ( d θ ) , (2.19)as n → ∞ , for any bounded continuous non-negative function g on I .Further we use Theorem 4.28 in [26]: Theorem 7.
For k, n ∈ N let ξ , ξ n , η k and η kn be random variables with values in a metric space ( S, ρ ) such that η kn d −→ η k as n → ∞ for any fixed k , and also η k d −→ η as k → ∞ . Then ξ n d −→ ξ holds underthe further condition lim k →∞ lim sup n →∞ E (cid:0) ρ ( η kn , ξ n ) ∧ (cid:1) = 0 . (2.20)Our setting corresponds to S = R , ρ = | · | , and ξ = (cid:90) I g ( θ ) µ α,β ( d θ ) , ξ n = (cid:90) I g ( θ ) µ n,α,β ( d θ ) ,η k = (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) , η kn = (cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) , (2.21)where µ ( k ) n,α,β will now be defined by truncating the Fourier series of ln f n,α,β . We have ln f n,α,β ( θ ) = − ∞ (cid:88) j =1 j (cid:0) ( α + β ) Tr ( U jn ) e − ijθ + ( α − β ) Tr ( U − jn ) e ijθ (cid:1) = − ∞ (cid:88) j =1 Tr ( U jn ) j (2 α cos( jθ ) − iβ sin( jθ )) , (2.22)where we used that for U ∈ O ( n ) or Sp (2 n ) we have Tr ( U − jn ) = Tr ( U jn ) . Definition 8.
For k, n ∈ N , α ∈ R , β ∈ i R and θ ∈ I , let f ( k ) n,α,β ( θ ) = e − (cid:80) kj =1 Tr ( Ujn ) j (2 α cos( jθ ) − iβ sin( jθ )) , (2.23) and µ ( k ) n,α,β ( d θ ) = f ( k ) n,α,β ( θ ) E ( f ( k ) n,α,β ( θ )) d θ. (2.24)Thus to apply Theorem 7 we need to show that for any bounded continuous non-negative functionon I we have lim k →∞ (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) d = (cid:90) I g ( θ ) µ α,β ( d θ ) , (2.25) lim n →∞ (cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) d = (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) , (2.26)6nd lim k →∞ lim sup n →∞ E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) − (cid:90) I g ( θ ) µ n,α,β ( d θ ) (cid:12)(cid:12)(cid:12)(cid:12) ∧ (cid:19) = 0 . (2.27)The first limit (2.25) follows immediately since by definition almost surely lim k →∞ µ ( k ) α,β d = µ α,β , (2.28)so in particular almost surely (and thus also in distribution) lim k →∞ (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) = (cid:90) I g ( θ ) µ α,β ( d θ ) . (2.29)The second limit (2.26) will be proved in Section 5, using previously established results on the asymp-totics of Toeplitz+Hankel determinants.To show that the third limit (2.27) holds, we will prove in Section 4 that the following expectation goesto zero, as first n → ∞ and then k → ∞ , for any bounded continuous non-negative function g on I : E (cid:32)(cid:18)(cid:90) I g ( θ ) µ n,α,β ( d θ ) − (cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) (cid:19) (cid:33) = (cid:90) I (cid:90) I g ( θ ) g ( θ (cid:48) ) E (cid:16) f ( k ) n,α,β ( θ ) f ( k ) n,α,β ( θ (cid:48) ) (cid:17) E (cid:16) f ( k ) n,α,β ( θ ) (cid:17) E (cid:16) f ( k ) n,α,β ( θ (cid:48) ) (cid:17) d θ d θ (cid:48) − (cid:90) I (cid:90) I g ( θ ) g ( θ (cid:48) ) E (cid:16) f ( k ) n,α,β ( θ ) f n,α,β ( θ (cid:48) ) (cid:17) E (cid:16) f ( k ) n,α,β ( θ ) (cid:17) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) + (cid:90) I (cid:90) I g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) . (2.30)All the expectations can be expressed as (sums of) Toeplitz+Hankel determinants (see (3.8) andTheorem 19). To prove that (2.30) goes to zero as first n → ∞ and then k → ∞ , in the case I = I (cid:15) , α > − / and α − β < / , we needed to derive specific results on the asymptotics of Toeplitz+Hankeldeterminants of symbols with two pairs of merging singularities bounded away from ± , which arewritten down in the next section. To the best of our knowledge these results have not previously beenset out and are of independent interest. For the case I = [0 , π ) , α ≥ , α − β < / and α < ,we use recent results on uniform asymptotics of Toeplitz+Hankel determinants kindly communicatedto us by Claeys, Glesner, Minakov and Yang [11]. Definition 9. ( [14]) A function f : S → C is called a symbol with a fixed number of Fisher-Hartwigsingularities if it has the following form: f ( z ) = e V ( z ) z (cid:80) mj =0 β j m (cid:89) j =0 | z − z j | α j g z j ,β j ( z ) z − β j j , z = e iθ , θ ∈ [0 , π ) , (3.1)7 or some m = 0 , , ... , where z j = e iθ j , j = 0 , ..., m, θ < θ < ... < θ m < π ; (3.2) g z j ,β j ( z ) = (cid:40) e iπβ j ≤ arg z < θ j e − iπβ j θ j ≤ arg z < π , (3.3) (cid:60) α j > − / , β j ∈ C , j = 0 , ..., m, (3.4) and V ( z ) is analytic in a neighborhood of the unit circle. A point z j , j = 1 , ..., m , is included if andonly if either α j (cid:54) = 0 or β j (cid:54) = 0 , while always z = 1 , even if α = β = 0 . For (cid:15) ∈ (0 , π/ we consider the symbol f p,t ( z ) = e V ( z ) z (cid:80) j =0 β j (cid:89) j =0 | z − z j | α j g z j ,β j ( z ) z − β j j , z = e iφ , φ ∈ [0 , π ) , = e V ( z ) | z − | α | z + 1 | α (cid:89) j =1 | z − z j | α j | z − z j | α j g z j ,β j ( z ) g z j , − β j ( z ) z − β j j z jβ j , (3.5)where • z = 1 , z = e i ( p − t ) , z = e i ( p + t ) , z = − , z = z = e i (2 π − p − t ) , z = z = e i (2 π − p + t ) , with p ∈ ( (cid:15), π − (cid:15) ) , < t < (cid:15) , • α j ∈ ( − / , ∞ ) for j = 0 , ..., , and α = α , α = α , • β = β = 0 , β = − β ∈ i R , β = − β ∈ i R , • V ( z ) is real-valued on the unit circle, and satisfies V ( e iθ ) = V ( e − iθ ) . V has the Laurent series V ( z ) = ∞ (cid:88) k = −∞ V k z k , V k = 12 π (cid:90) π V ( e iθ ) e − ikθ d θ, (3.6)for which V k = V − k . The function e V ( z ) allows the standard Wiener-Hopf decomposition: e V ( z ) = b + ( z ) b b − ( z ) , b + ( z ) = e (cid:80) ∞ k =1 V k z k , b = e V , b − ( z ) = e (cid:80) − k = −∞ V k z k . (3.7)We compute the asymptotic behaviour as n → ∞ , uniformly in p ∈ ( (cid:15), π − (cid:15) ) and < t < t for asufficiently small t ∈ (0 , (cid:15) ) , of the Toeplitz+Hankel determinants D T + H, n ( f p,t ) = det ( f j − k + f j + k ) n − j,k =0 ,D T + H, n ( f p,t ) = det ( f j − k − f j + k +2 ) n − j,k =0 ,D T + H, n ( f p,t ) = det ( f j − k − f j + k +1 ) n − j,k =0 ,D T + H, n ( f p,t ) = det ( f j − k + f j + k +1 ) n − j,k =0 , (3.8)where f j = f p,t,j = 12 π (cid:90) π f p,t ( e iθ ) e − ijθ d θ. (3.9)The asymptotics of D T + H,κn , κ = 1 , , , , were computed from the asymptotics of the Toeplitzdeterminants D n ( f p,t ) = det ( f p,t,j − k ) n − j,k =0 , (3.10)8nd the asymptotics of Φ n (0) , Φ n ( ± , where Φ n ( z ) = z n + ... are the monic orthogonal polynomialsw.r.t. f p,t , using Theorem 2.6 and Lemma 2.7 from [14].To compute the asymptotics of D n ( f p,t ) we followed the approach in [13], while the asymptotics of Φ n (0) , Φ n ( ± will be taken from [14] and it will be argued that they also hold uniformly in p , t . Beforestating the results on Toeplitz and Toeplitz + Hankel determinants we state the following theoremfrom [13] (not in the most general version) which describes the relevant Painlevé transcendents. Theorem 10. (Claeys, Krasovsky) Let α , α , α + α > − , β , β ∈ i R and consider the σ -form ofthe Painleve V equation s σ ss = ( σ − sσ s + 2 σ s ) − σ s − θ )( σ s − θ )( σ s − θ )( σ s − θ ) , (3.11) where the parameters θ , θ , θ , θ are given by θ = − α + β + β , θ = α + β + β ,θ = α − β − β , θ = − α − β + β . (3.12) Then there exists a solution σ ( s ) to (3.11) which is real and free of poles for s ∈ − i R + , and which hasthe following asymptotic behavior along the negative imaginary axis: σ ( s ) =2 α α − ( β + β ) O ( | s | δ ) , s → − i + ,σ ( s ) = β − β s − ( β − β ) O ( | s | − δ ) , s → − i ∞ , (3.13) for some δ > . Our result on the uniform asymptotics of D n ( f p,t ) and D T + H,κn ( f p,t ) is then the following. Theorem 11.
Let f p,t be as in (3.5) with α + α > − / , and let σ satisfy the conditions of Theorem10. Then we have the following large n asymptotics, uniformly for p ∈ ( (cid:15), π − (cid:15) ) and < t < t , for asufficiently small t ∈ (0 , (cid:15) ) : ln D n ( f p,t ) =2 int ( β − β ) + nV + ∞ (cid:88) k =1 kV k + ln( n ) (cid:88) j =0 ( α j − β j ) − (cid:88) j =0 ( α j − β j ) (cid:32) ∞ (cid:88) k =1 V k z kj (cid:33) + ( α j + β j ) (cid:32) ∞ (cid:88) k =1 V k z jk (cid:33) + (cid:88) ≤ j One can probably get similar results if more generally one chooses complex α j , β j with (cid:60) ( α j ) > − / , but to prove our main result Theorem 5 this isn’t necessary. Remark 13. The requirements p ∈ ( (cid:15), π − (cid:15) ) , t ∈ (0 , (cid:15) ) are necessary for us to be able to apply theproof techniques in [13]. The results there only hold for two merging singularities, while if p → , π wehave 5 singularities merging at ± , and if t → (cid:15) we can have p ± t → , π which means 3 singularitiesare merging at ± . For our second result, which corresponds to Theorem 1.11 in [13], we first state Theorems 1.1(proven in [17]) and 1.25 from [14]: Theorem 14. (Ehrhardt) Let f be as in Definition 9 with max j,k |(cid:60) β j − (cid:60) β k | < , and α j ± β j (cid:54) = , − , ... for j, k = 0 , ..., m . Then as n → ∞ , ln D n ( f p,t ) = nV + ∞ (cid:88) k =0 kV k V − k + (ln n ) m (cid:88) j =0 ( α j − β j ) − m (cid:88) j =0 ( α j − β j ) (cid:32) ∞ (cid:88) k =1 V k z kj (cid:33) + ( α j + β j ) (cid:32) ∞ (cid:88) k =1 V − k z jk (cid:33) + (cid:88) ≤ j Comparing the uniform asymptotics of D n ( f p,t ) in Theorem 11 with the non-uniformasymptotics one gets from Theorem 14, one can see that the different expansions are related in the ollowing way: (cid:88) j =1 ln G (1 + α j + β j ) G (1 + α j − β j ) G (1 + 2 α j ) − iπ ( α β − α β ) + o (1) non − uniform =2 int ( β − β ) + 2 (cid:90) − int s (cid:18) σ ( s ) − α α + 12 ( β + β ) (cid:19) d s + 4( β β − α α ) ln 12 nt + 2 ln G (1 + α + α + β + β ) G (1 + α + α − β − β ) G (1 + 2 α + 2 α ) + o (1) uniform . (3.23) This is exactly the same relationship as the one between the non-uniform and uniform expansions of D n ( f t ) in [13] (see their (1.8), (1.24) and (1.26)). Since the uniform asymptotics of D T + H,κn ( f p,t ) were derived from the uniform asymptotics of D n ( f p,t ) / and Φ n ( ± / , Φ n (0) , using Theorem 2.6and Lemma 2.7 in [14], the relationship between the uniform asymptotics of D T + H,κn ( f p,t ) and thenon-uniform asymptotics one gets from Theorem 15 is given by (3.23), with both sides divided by and n replaced by n . Our second result extends Theorem 14 for the symbol f p,t and Theorem 15 in the case r = 2 : Theorem 17. Let ω ( x ) be a positive, smooth function for x sufficiently large, s.t. ω ( n ) → ∞ , ω ( n ) = o ( n − ) , as n → ∞ . (3.24) Then for any t ∈ (0 , (cid:15) ) the expansion of D n ( f p,t ) one gets from Theorem 14 holds uniformly in p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t . For r = 2 , the expansion of Theorem 15 holds uniformly in θ , θ ∈ ( (cid:15), π − (cid:15) ) for which ω ( n ) /n < | θ − θ | . To prove Theorem 6 we make use of a recent result on uniform Toeplitz+Hankel asymptotics up toa multiplicative constant, kindly communicated to us by Claeys, Glesner, Minakov and Yang [11]. Theorem 18. (Claeys, Glesner, Minakov, Yang) Let f be as in (3.21) with r ∈ N , α = α r +1 = 0 and α j ≥ , j = 1 , ..., r . Then we have uniformly over the entire region < θ < ... < θ r < π , as n → ∞ , D T + H, n ( f ) = F e nV r (cid:89) j =1 n α j − β j (cid:18) sin θ j + 1 n (cid:19) α j − α j − β j × e O (1) ,D T + H, n ( f ) = F e nV r (cid:89) j =1 n α j − β j (cid:18) sin θ j + 1 n (cid:19) − α j − α j − β j × e O (1) ,D T + H, n ( f ) = F e nV r (cid:89) j =1 n α j − β j (cid:18) sin θ j n (cid:19) − α j − α j − β j (cid:18) cos θ j n (cid:19) α j − α j − β j × e O (1) ,D T + H, n ( f ) = F e nV r (cid:89) j =1 n α j − β j (cid:18) sin θ j n (cid:19) α j − α j − β j (cid:18) cos θ j n (cid:19) − α j − α j − β j × e O (1) , (3.25) where F = (cid:89) ≤ j 1) = e − (cid:80) kj =1 ( ± jj (cid:60) (cid:16) ( α − β ) (cid:16) e ijθ + e ijθ (cid:48) (cid:17)(cid:17) , ˆ σ ,θ,θ (cid:48) ( ± 1) = e − (cid:80) kj =1 ( ± jj (cid:60) ( ( α − β ) e ijθ ) | ∓ e iθ (cid:48) | α e iβ ( π − θ (cid:48) ) g e iθ (cid:48) ,β ( ± , ˆ σ ,θ,θ (cid:48) (1) = | − e iθ | α | − e iθ (cid:48) | α e iβ ( π − θ ) e iβ ( π − θ (cid:48) ) , ˆ σ ,θ,θ (cid:48) ( − 1) = | e iθ | α | e iθ (cid:48) | α e iβ ( π − θ ) e iβ ( π − θ (cid:48) ) e − iπ ( β + β ) , ˆ σ ,θ ( ± 1) = e − (cid:80) kj =1 ( ± jj (cid:60) ( ( α − β ) e ijθ ) , ˆ σ ,θ (cid:48) (1) = | − e iθ (cid:48) | α e iβ ( π − θ (cid:48) ) , ˆ σ ,θ (cid:48) ( − 1) = | e iθ (cid:48) | α e iβ ( π − θ (cid:48) ) e − iπβ , (4.29)19here β , β are chosen as in (4.19) for ˆ σ ,θ,θ (cid:48) , and as in (4.17) for ˆ σ ,θ (cid:48) . Thus by (4.6), (4.11) and(4.29) we get E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) f ( k )2 n +1 ,α,β ( θ (cid:48) ) (cid:17) = 12 ˆ σ ,θ,θ (cid:48) (1) D T + H, n ( σ ,θ,θ (cid:48) ) + 12 ˆ σ ,θ,θ (cid:48) ( − D T + H, n ( σ ,θ,θ (cid:48) )= 12 e (cid:80) kj =1 2 j (cid:60) (cid:16) ( α − β )( e ijθ + e ijθ (cid:48) ) (cid:17) × (cid:32) e − (cid:80) kj =1 1 j (cid:60) (cid:16) ( α − β ) (cid:16) e ijθ + e ijθ (cid:48) (cid:17)(cid:17) e (cid:80) kj =1 1 − ( − jj (cid:60) (cid:16) ( α − β )( e ijθ + e ijθ (cid:48) ) (cid:17) + e − (cid:80) kj =1 ( − jj (cid:60) (cid:16) ( α − β ) (cid:16) e ijθ + e ijθ (cid:48) (cid:17)(cid:17) e − (cid:80) kj =1 1 − ( − jj (cid:60) (cid:16) ( α − β )( e ijθ + e ijθ (cid:48) ) (cid:17) (cid:33) (1 + o (1))= e (cid:80) kj =1 2 j (cid:60) (cid:16) ( α − β )( e ijθ + e ijθ (cid:48) ) (cid:17) e − (cid:80) kj =1 1+( − jj (cid:60) (cid:16) ( α − β )( e ijθ + e ijθ (cid:48) ) (cid:17) (1 + o (1)) , (4.30)uniformly in θ, θ (cid:48) ∈ [0 , π ) .Similarly we obtain from (4.6), (4.14, (4.16), (4.18) and (4.29), that E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) (cid:17) = e (cid:80) kj =1 2 j (cid:60) ( ( α − β ) e ijθ ) e − (cid:80) kj =1 1+( − jj (cid:60) ( ( α − β ) e ijθ )(1 + o (1)) , (4.31)uniformly in θ, θ (cid:48) ∈ [0 , π ) , and E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) ) (cid:17) = e (cid:80) kj =1 2 j (cid:60) ( ( α − β ) e ijθ ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) × e − iπαβ z αβ (2 n ) ( α − β ) | − e iθ (cid:48) | − ( α + β ) × G (1 + α + β ) G (1 + α − β ) G (1 + 2 α ) × e − (cid:80) kj =1 1+( − jj (cid:60) ( ( α − β ) e ijθ ) e iβ ( π − θ (cid:48) ) e − iπ β | − e iθ (cid:48) | α | e iθ (cid:48) | α (1 + o (1)) , E O (2 n +1) ( f n +1 ,α,β ( θ (cid:48) ))= e − iπαβ z αβ (2 n ) ( α − β ) | − e iθ (cid:48) | − ( α + β ) × G (1 + α + β ) G (1 + α − β ) G (1 + 2 α ) × e iβ ( π − θ (cid:48) ) e − iπ β | − e iθ (cid:48) | α | e iθ (cid:48) | α (1 + o (1)) , (4.32)uniformly in θ, θ (cid:48) ∈ I (cid:15) , where z = (cid:40) e iθ (cid:48) < θ (cid:48) < πe i (2 π − θ (cid:48) ) π < θ (cid:48) < π , β = (cid:40) β < θ (cid:48) < π − β π < θ (cid:48) < π . (4.33)20rom (4.6), (4.20) and (4.29) we obtain E O (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) ))= e − iπα ( β + β +2 β ) × (2 n ) α − β ) × | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) × z β α z β α | − e iθ | − ( α + β ) | − e iθ | − ( α + β ) × G (1 + α + β ) G (1 + α − β ) G (1 + 2 α ) × e iβ ( π − θ ) e iβ ( π − θ (cid:48) ) e − iπ ( β + β ) × | − e iθ | α | e iθ | α | − e iθ (cid:48) | α | e iθ (cid:48) | α (1 + o (1)) , (4.34)uniformly in θ, θ (cid:48) ∈ I (cid:15) for which ln( n ) /n < min {| θ − θ (cid:48) | , | θ + θ (cid:48) − π |} , and where z , z , β , β arechosen as in (4.19).From (4.6), (4.23) and (4.29) we obtain E O (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) ))= e int ( β − β ) e − iπα ( β + β ) × (2 n ) α − β ) × | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) × (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) | p | − α + β β ) × e (cid:82) − int s ( σ ( s ) − α + ( β + β ) ) d s × G (1 + 2 α + β + β ) G (1 + 2 α − β − β ) G (1 + 4 α ) × e iβ ( π − θ ) e iβ ( π − θ (cid:48) ) e − iπ ( β + β ) × | − e iθ | α | e iθ | α | − e iθ (cid:48) | α | e iθ (cid:48) | α (1 + o (1)) , (4.35)uniformly in θ, θ (cid:48) ∈ J t j where ( p, t ) = ψ j ( θ, θ (cid:48) ) .Combining (4.30), (4.31) and (4.32), we obtain E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) f ( k )2 n +1 ,α,β ( θ (cid:48) ) (cid:17) E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) (cid:17) E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ (cid:48) ) (cid:17) = e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) (1 + o (1)) , (4.36)uniformly in θ, θ (cid:48) ∈ [0 , π ) , and E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) ) (cid:17) E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) (cid:17) E O (2 n +1) ( f n +1 ,α,β ( θ (cid:48) ))= e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) (1 + o (1)) , (4.37)21niformly in θ, θ (cid:48) ∈ I (cid:15) .By (4.32) and (4.34) we obtain E O (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) )) E O (2 n +1) ( f n +1 ,α,β ( θ )) E O (2 n +1) ( f n +1 ,α,β ( θ (cid:48) ))= | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) z αβ z αβ e − πiαβ (1 + o (1)) , = e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) (1 + o (1)) , (4.38)uniformly in θ, θ (cid:48) ∈ I (cid:15) for which ln( n ) /n < min {| θ − θ (cid:48) | , | θ + θ (cid:48) − π |} , where z , z , β , β are chosenas in (4.19).Finally, by (4.32), (4.35), with β , β chosen as in (4.19), we obtain E SO (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) )) E SO (2 n +1) ( f n +1 ,α,β ( θ )) E SO (2 n +1) ( f n +1 ,α,β ( θ (cid:48) ))= (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) | p | − α + β β ) × e (cid:82) − int s ( σ ( s ) − α + ( β + β ) ) d s × e iαβ ( p − t ) e iαβ ( p + t ) e − iπαβ e πi ( αβ − αβ ) e int ( β − β ) × G (1 + 2 α + β + β ) G (1 + 2 α − β − β ) G (1 + 4 α ) × G (1 + 2 α ) G (1 + α + β ) G (1 + α − β ) G (1 + α + β ) G (1 + α − β ) × (1 + o (1)) , (4.39)uniformly in ( θ, θ (cid:48) ) ∈ J t j where ( p, t ) = ψ j ( θ, θ (cid:48) ) . In the same way as in the odd orthogonal case one can use (4.6), (4.11) - (4.23) and (4.29) to obtain E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) f ( k )2 n,α,β ( θ (cid:48) ) (cid:17) E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) (cid:17) E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ (cid:48) ) (cid:17) = e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) (1 + o (1)) , (4.40)uniformly in θ, θ ∈ [0 , π ) , and E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) f n,α,β ( θ (cid:48) ) (cid:17) E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) (cid:17) E O (2 n ) ( f n,α,β ( θ (cid:48) )) = e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) (1 + o (1)) , (4.41)uniformly in θ, θ (cid:48) ∈ I (cid:15) , and E O (2 n ) ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E O (2 n ) ( f n,α,β ( θ )) E O (2 n ) ( f n,α,β ( θ (cid:48) ))= e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) (1 + o (1)) , (4.42)22niformly in θ, θ (cid:48) ∈ I (cid:15) for which ln( n ) /n < min {| θ − θ (cid:48) | , | θ + θ (cid:48) − π |} , and finally, with β , β chosenas in (4.19), E O (2 n ) ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E O (2 n ) ( f n,α,β ( θ )) E O (2 n ) ( f n,α,β ( θ (cid:48) ))= (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) | p | − α + β β ) × e (cid:82) − int s ( σ ( s ) − α + ( β + β ) ) d s × e iαβ ( p − t ) e iαβ ( p + t ) e − iπαβ e πi ( αβ − αβ ) e int ( β − β ) × G (1 + 2 α + β + β ) G (1 + 2 α − β − β ) G (1 + 4 α ) × G (1 + 2 α ) G (1 + α + β ) G (1 + α − β ) G (1 + α + β ) G (1 + α − β ) × (1 + o (1)) , (4.43)uniformly in ( θ, θ (cid:48) ) ∈ J t j for which ln( n ) /n ≥ min {| θ − θ (cid:48) | , | θ + θ (cid:48) − π |} , where ( p, t ) = ψ j ( θ, θ (cid:48) ) .In Section 5 we will also need that E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) (cid:17) = e − (cid:80) kj =1 ηjj (cid:60) ( ( α − β ) e ijθ ) + (cid:80) kj =1 2 j (cid:60) ( ( α − β ) e ijθ ) , (4.44)uniformly in θ ∈ [0 , π ) . L -Limit on I (cid:15) In this section all the expectations are either over O ( n ) or Sp (2 n ) . By (2.30), (4.24), (4.25), (4.36),(4.37), (4.40) and (4.41) we obtain lim n →∞ E (cid:32)(cid:18)(cid:90) I (cid:15) g ( θ ) µ n,α,β ( d θ ) − (cid:90) I (cid:15) g ( θ ) µ ( k ) n,α,β ( d θ ) (cid:19) (cid:33) = lim n →∞ (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) − (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) d θ d θ (cid:48) . (4.45)Let A n := { ( θ, θ (cid:48) ) ∈ I (cid:15) × I (cid:15) : ln( n ) /n < min {| θ − θ (cid:48) | , | θ + θ (cid:48) − π |}} , then by (4.26), (4.38) and(4.42) we have lim n →∞ (cid:90) (cid:90) A n g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) = (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) d θ d θ (cid:48) ≤ const (cid:90) I (cid:15) (cid:90) I (cid:15) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) d θ d θ (cid:48) < ∞ , (4.46)since α + β ≤ α − β < / . 23or the integral over the rest of I (cid:15) × I (cid:15) we follow the proofs of Theorem 1.15 in [13] and Corollary20 in [31]. By (4.27), (4.39), (4.43) we have (cid:90) (cid:90) J t j \ A n g ( θ ) g ( θ ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) ≤ const n α − β β ) (cid:90) /n e int ( β − β ) (cid:12)(cid:12)(cid:12)(cid:12) sin t t (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) e (cid:82) − int s ( σ ( s ) − α + ( β + β ) ) d s d t + const (cid:90) (ln n ) /n /n e int ( β − β ) (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) e (cid:82) − int s ( σ ( s ) − α + ( β + β ) ) d s d t. (4.47)for j = 1 , ..., . For t ∈ (0 , /n ] the integrand is bounded, thus the first summand is of order O ( n α − β β ) − ) = o (1) , as α − β β ≤ α − β < / . The second integral equals e (cid:82) − i s ( σ ( s ) − α + ( β + β ) ) d s (cid:90) (ln n ) /n /n e int ( β − β ) (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) e (cid:82) − int − i s ( σ ( s ) − α + ( β + β ) ) d s d t ≤ const (cid:90) (ln n ) /n /n e int ( β − β ) (cid:12)(cid:12)(cid:12)(cid:12) sin t nt (cid:12)(cid:12)(cid:12)(cid:12) − α − β β ) e (cid:82) − int − i β − β − s ( α − β β )+ O ( | s | − − δ ) d s d t = const (cid:90) (ln n ) /n /n | t | − α − β β ) e (cid:82) − int − i O ( | s | − − δ ) d s d t = o (1) , (4.48)where in the inequality we used the small and large s asymptotics of σ ( s ) along the negative imaginaryaxis, see Theorem 10, and where the last equality follows since α − β β ) ≤ α − β ) < .Thus we obtain lim n →∞ E (cid:32)(cid:18)(cid:90) I (cid:15) g ( θ ) µ n,α,β ( d θ ) − (cid:90) I (cid:15) g ( θ ) µ ( k ) n,α,β ( d θ ) (cid:19) (cid:33) = (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) d θ d θ (cid:48) − (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) d θ d θ (cid:48) . (4.49)We have (cid:60) (cid:0) ( α − β ) e ijθ (cid:1) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) =( α − β ) e ij ( θ + θ (cid:48) ) + ( α + β ) e − ij ( θ + θ (cid:48) ) + ( α − β )( e ij ( θ − θ (cid:48) ) + e − ij ( θ − θ (cid:48) ) )=2( α − β ) cos( j ( θ − θ (cid:48) )) + 2( α + β ) cos( j ( θ + θ (cid:48) )) − αβ (cid:16) e ij ( θ + θ (cid:48) ) − e − ij ( θ + θ (cid:48) ) (cid:17) . (4.50)Since ln | e iθ − e iθ (cid:48) | = − ∞ (cid:88) j =1 j cos( j ( θ − θ (cid:48) )) , ln | e iθ − e − iθ (cid:48) | = − ∞ (cid:88) j =1 j cos( j ( θ + θ (cid:48) )) , (4.51)24nd − ∞ (cid:88) j =1 j ( e ij ( θ + θ (cid:48) ) − e − ij ( θ + θ (cid:48) ) ) = ln(1 − e i ( θ + θ (cid:48) ) ) − ln(1 − e − i ( θ + θ (cid:48) ) )=2 i (cid:61) ln(1 − e i ( θ + θ (cid:48) ) )= i (cid:40) θ + θ (cid:48) − π ≤ θ + θ (cid:48) ≤ πθ + θ (cid:48) − π π < θ + θ (cid:48) < π = ln e i ( θ + θ (cid:48) − π ) , (4.52)we see that e (cid:80) ∞ j =1 1 j (cid:60) ( ( α + iβ ) e ijθ ) (cid:60) (cid:16) ( α + iβ ) e ijθ (cid:48) (cid:17) = | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) e αβ ln e i ( θ + θ (cid:48)− π ) . (4.53)Because E (cid:0) ( ... ) (cid:1) ≥ we have (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) e αβ ln e i ( θ + θ (cid:48)− π ) d θ d θ (cid:48) ≥ lim sup k →∞ (cid:90) I (cid:15) (cid:90) I (cid:15) g ( θ ) g ( θ (cid:48) ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) d θ d θ (cid:48) . (4.54)Now we use that g is non-negative to apply Fatou to get the other inequality, thus we have proven that lim k →∞ lim n →∞ E (cid:32)(cid:18)(cid:90) I (cid:15) g ( θ ) µ ( k ) n,α,β ( d θ ) − (cid:90) I (cid:15) g ( θ ) µ n,α,β ( d θ ) (cid:19) (cid:33) = 0 . (4.55) L -Limit on all of [0 , π ) In this section we let as before α − β < / , and additionally α ≥ and α < . Using Theorem 18we will show that under these additional assumptions it holds that lim n →∞ (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) = (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) d θ d θ (cid:48) , (4.56)and lim n →∞ (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) E (cid:16) f ( k ) n,α,β ( θ ) f n,α,β ( θ (cid:48) ) (cid:17) E (cid:16) f ( k ) n,α,β ( θ ) (cid:17) E ( f n,α,β ( θ (cid:48) )) d θ d θ (cid:48) = (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) d θ d θ (cid:48) , (4.57)25here the expectations are over O ( n ) or Sp (2 n ) , by following the proof of Corollary 2.1 in [18]. Then itfollows together with (2.30), (4.24), (4.36) and (4.40) that lim n →∞ E (cid:32)(cid:18)(cid:90) π g ( θ ) µ n,α,β ( d θ ) − (cid:90) π g ( θ ) µ ( k ) n,α,β ( d θ ) (cid:19) (cid:33) = (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) e αβ ln e i ( θ + θ (cid:48)− π ) | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) d θ d θ (cid:48) − (cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) e (cid:80) kj =1 4 j (cid:60) ( ( α − β ) e ijθ ) (cid:60) (cid:16) ( α − β ) e ijθ (cid:48) (cid:17) d θ d θ (cid:48) ≥ . (4.58)This again goes to zero if we next let k → ∞ , by applying Fatou and (4.53).By Theorems 18 and 19 we have E Sp (2 n ) ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E Sp (2 n ) ( f n,α,β ( θ )) E Sp (2 n ) ( f n,α,β ( θ (cid:48) )) = D T + H, n ( σ ,θ,θ (cid:48) ) D T + H, n ( σ ,θ ) D T + H, n ( σ ,θ (cid:48) )= e O (1) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ − θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α − β ) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ + θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α + β ) , (4.59)as n → ∞ , uniformly in (Lebesgue almost all) ( θ, θ (cid:48) ) ∈ [0 , π ) . By the same theorems we get E O (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) )) = 12 (cid:0) ˆ σ ,θ,θ (cid:48) (1) D T + H, n ( σ ,θ,θ (cid:48) ) + ˆ σ ,θ,θ (cid:48) ( − D T + H, n ( σ ,θ,θ (cid:48) ) (cid:1) = 12 F σ ,θ,θ (cid:48) n α − β ) e O (1) 2 (cid:89) j =1 sin | θ j | + n sin | θ j | (cid:16) cos | θ j | + n (cid:17) − α + e O (1) 2 (cid:89) j =1 cos | θ j | + n cos | θ j | (cid:16) sin | θ j | + n (cid:17) − α , (4.60)and E O (2 n +1) ( f n +1 ,α,β ( θ ))= 12 F σ ,θ n α − β e O (1) sin | θ | + n sin | θ | (cid:16) cos | θ | + n (cid:17) − α + e O (1) cos | θ | + n cos | θ | (cid:16) sin | θ | + n (cid:17) − α , (4.61)as n → ∞ , uniformly for (Lebesgue almost all) ( θ, θ (cid:48) ) ∈ [0 , π ) , where ˆ σ ,θ,θ (cid:48) ,..., ˆ σ ,θ are defined in(4.3). Thus we can see that also for the expectations over O (2 n + 1) we get E O (2 n +1) ( f n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) )) E O (2 n +1) ( f n +1 ,α,β ( θ )) E O (2 n +1) ( f n +1 ,α,β ( θ (cid:48) ))= e O (1) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ − θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α − β ) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ + θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α + β ) , (4.62)as n → ∞ , uniformly in (Lebesgue almost all) ( θ, θ (cid:48) ) ∈ [0 , π ) . Similarly, Theorem 18 gives E O (2 n ) ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E O (2 n ) ( f n,α,β ( θ )) E O (2 n ) ( f n,α,β ( θ (cid:48) ))= e O (1) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ − θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α − β ) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ + θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:19) − α + β ) , (4.63)26s well as E Sp (2 n ) (cid:16) f ( k ) n,α,β ( θ ) f n,α,β ( θ (cid:48) ) (cid:17) E Sp (2 n ) (cid:16) f ( k ) n,α,β ( θ ) (cid:17) E Sp (2 n ) ( f n,α,β ( θ (cid:48) )) = e O (1) , E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) f n +1 ,α,β ( θ (cid:48) ) (cid:17) E O (2 n +1) (cid:16) f ( k )2 n +1 ,α,β ( θ ) (cid:17) E O (2 n +1) ( f n +1 ,α,β ( θ (cid:48) )) = e O (1) , E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) f n,α,β ( θ (cid:48) ) (cid:17) E O (2 n ) (cid:16) f ( k )2 n,α,β ( θ ) (cid:17) E O (2 n ) ( f n,α,β ( θ (cid:48) )) = e O (1) , (4.64)as n → ∞ , uniformly in (Lebesgue almost all) ( θ, θ (cid:48) ) ∈ [0 , π ) .Now, for a given measureable subset R ⊂ [0 , π ) , we denote L (cid:15) ( R ) = (cid:90) R (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ − θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) (cid:19) − α − β ) (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ + θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) (cid:19) − α + β ) d θ d θ (cid:48) ,K (cid:15) ( R ) = (cid:90) R (cid:18) sin (cid:12)(cid:12)(cid:12)(cid:12) θ − θ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) (cid:19) − α − β ) d θ d θ (cid:48) . (4.65)In the case α + β > we have L (cid:15) ( R ) < L ( R ) < ∞ for any (cid:15) > (since α − β ) , α + β ) , α < ),while in the case α + β ≤ we have K (cid:15) ( R ) < K ( R ) < ∞ for any (cid:15) > . For η > we define R ( η ) = (cid:26) ( θ, θ (cid:48) ) ∈ [0 , π ) : sin | θ − θ (cid:48) | , sin | θ + θ (cid:48) | > µ (cid:27) R ( η ) = R ( η ) c . (4.66)It follows by (4.59), (4.62) and (4.63) that for any η > there exists a C > and N ∈ N such that (cid:90) R ( η ) g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) ≤ (cid:40) CL ( R ( η )) , α + β > ,CK ( R ( η )) , α + β ≤ , (4.67)for n > N . Fix δ > . Since L ( R ( η )) , K ( R ( η ) → as η → , it follows that there exists an η > and an N ∈ N such that (cid:90) R ( η ) g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) < δ/ (4.68)for n > N and η < η .Using (4.26), (4.38) and (4.42), we get that for any fixed η > it holds that (cid:90) R ( η ) g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) = (1 + o (1)) L ( R ( η ))=(1 + o (1)) L ([0 , π ) ) − (1 + o (1)) L ( R ( η )) . (4.69)We pick η < η such that I ( R ( η )) < δ/ , then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ( η ) g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) − L ([0 , π ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | L ( R ( η )) + o (1) | < δ/ o (1) . (4.70)27ogether with (4.68) we obtain that there exists an N ∈ N such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) π (cid:90) π g ( θ ) g ( θ (cid:48) ) E ( f n,α,β ( θ ) f n,α,β ( θ (cid:48) )) E ( f n,α,β ( θ )) E ( f n,α,β ( θ (cid:48) )) − L ([0 , π ) ) (cid:12)(cid:12)(cid:12)(cid:12) < δ (4.71)for all n > N . Since δ > is arbitrary, this shows the first part of (4.56). (4.57) follows from (4.64) ina similar way. In this section we prove (2.26), i.e. that for any fixed k ∈ N and bounded continuous function g : I → R it holds that (cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) d −→ (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) , (5.1)as n → ∞ , where µ ( k ) α,β is defined in Definition 8.We consider the function F : R k → R , F ( z , ..., z k ) = (cid:90) I g ( θ ) e − (cid:80) kj =1 zj √ j ( α cos( jθ ) − iβ sin( jθ )) e ± (cid:80) kj =1 ηjj (cid:60) (( α − β ) e ijθ )+ (cid:80) kj =1 2 j (cid:60) (( α − β ) e ijθ ) d θ, (5.2)which is continuous since the integrator is continuous in z , ..., z n and θ , and bounded in θ for any fixed z , ..., z n . Then we have, with ± corresponding to symplectic/orthogonal: (cid:90) I g ( θ ) µ ( k ) n,α,β ( d θ ) = (cid:90) I g ( θ ) e − (cid:80) kj =1 Tr ( Ujn ) √ j ( α cos( jθ ) − iβ sin( jθ )) E ( f ( k ) n,α,β ( θ )) d θ = 11 + o (1) (cid:90) I g ( θ ) e − (cid:80) kj =1 Tr ( Ujn ) √ j ( α cos( jθ ) − iβ sin( jθ )) e ± (cid:80) kj =1 ηjj (cid:60) (( α − β ) e ijθ )+ (cid:80) kj =1 2 j (cid:60) (( α − β ) e ijθ ) d θ d −→ (cid:90) I g ( θ ) e − (cid:80) kj =1 N j ∓ ηj √ j √ j ( α cos( jθ ) − iβ sin( jθ )) e ± (cid:80) kj =1 ηjj (cid:60) (( α − β ) e ijθ )+ (cid:80) kj =1 2 j (cid:60) (( α − β ) e ijθ ) d θ d = (cid:90) I g ( θ ) e (cid:80) kj =1 N j √ j ( α cos( jθ ) − iβ sin( jθ )) − (cid:80) kj =1 2 j (cid:60) ( ( α − β ) e ijθ ) d θ = (cid:90) I g ( θ ) µ ( k ) α,β ( d θ ) , (5.3)where in the second equality we used (4.16), (4.31), and (4.44), where the convergence in distributionfollows from Theorem 2 and the continuous mapping theorem, and where the penultimate equalityfollows from the fact that −N j d = N j . 28 Riemann-Hilbert Problem for a System of Orthogonal Poly-nomials and a Differential Identity By the integral representation for a Toeplitz-determinant and since f p,t > except at z , ..., z , it holdsthat D n ( f p,t ) ∈ (0 , ∞ ) for all n ∈ N . Thus we can define the polynomials φ n ( z ) = 1 (cid:112) D n ( f p,t ) D n +1 ( f p,t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f p,t, f p,t, − . . . f p,t, − n f p,t, f p,t, . . . f p,t, − n +1 . . . . . . . . .f p,t,n − f p,t,n − . . . f p,t, − z . . . z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = χ n z n + ..., ˆ φ n ( z ) = 1 (cid:112) D n ( f p,t ) D n +1 ( f p,t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f p,t, f p,t, − . . . f p,t, − n +1 f p,t, f p,t, . . . f p,t, − n +2 z. . . . . . . . . . . .f p,t,n f p,t,n − . . . f p,t, z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = χ n z n + ..., (6.1)where the leading coefficient χ n is given by χ n = (cid:115) D n ( f p,t ) D n +1 ( f p,t, ) . (6.2)The above polynomials satisfy the orthogonality relations π (cid:90) π φ n ( e iθ ) e − ikθ f p,t ( e iθ ) d θ = χ − n δ nk , π (cid:90) π ˆ φ n ( e − iθ ) e ikθ f p,t ( e iθ ) d θ = χ − n δ nk , (6.3)for k = 0 , , ..., 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 matrix-valued function Y ( z ) = Y ( z ; n, p, t ) given by Y ( z ) = (cid:32) χ − n φ n ( z ) χ − n (cid:82) C φ n ( ξ ) ξ − z f p,t ( ξ ) d ξ πiξ n − χ n − z n − ˆ φ n − ( z − ) − χ n − (cid:82) C ˆ φ n − ( ξ − ) ξ − z f p,t ( ξ ) d ξ πiξ (cid:33) (6.4)is the unique solution of the following Riemann-Hilbert problem: RHP for Y (a) Y : C \ C → C × is analytic.(b) The continuous boundary values of Y from inside the unit circle, denoted Y + , and from outside,denoted Y − , exist on C \ { z , ..., z } , and are related by the jump condition Y + ( z ) = Y − ( z ) (cid:18) z − n f p,t ( z )0 1 (cid:19) , z ∈ C \ { z , ..., z } . (6.5)(c) Y ( z ) = ( I + O (1 /z )) (cid:18) z n z − n (cid:19) , as z → ∞ .29d) As z → z k , z ∈ C \ C , k = 0 , ..., , we have Y ( z ) = (cid:18) O (1) O (1) + O ( | z − z k | α k ) O (1) O (1) + O ( | z − z k | α k ) (cid:19) , if α k (cid:54) = 0 , (6.6)and Y ( z ) = (cid:18) O (1) O (1) + O (ln | z − z k | ) O (1) O (1) + O (ln | z − z k | ) (cid:19) , if α k = 0 . (6.7)From the RHP and Liouville’s theorem it follows that det Y ( z ) = 1 for all z ∈ C \ C . Using this, onecan see quickly that the solution is unique.We have Y ( z ; n, p, t ) (0) = χ n − and Y ( z ; n, p, t ) ( z ) = χ − n φ n ( z ) = Φ n ( z ) , thus if we know theasymptotics of Y , we know the asymptotics of Φ n , φ n and χ n . The Fourier coefficients are differentiable in t , thus ln D n ( f p,t ) is differentiable in t for all p ∈ ( (cid:15), π − (cid:15) ) and n ∈ N . We calculate: ∂∂t ln (cid:12)(cid:12)(cid:12) z − e i ( p − t ) (cid:12)(cid:12)(cid:12) α = ∂∂t ln (cid:12)(cid:12)(cid:12)(cid:12) θ − ( p − t )2 (cid:12)(cid:12)(cid:12)(cid:12) α = α cot θ − ( p − t )2 = iα z + e i ( p − t ) z − e i ( p − t ) . (6.8)Similarly we obtain ∂∂t ln (cid:12)(cid:12)(cid:12) z − e i ( p + t ) (cid:12)(cid:12)(cid:12) α = − iα z + e i ( p + t ) z − e i ( p + t ) ,∂∂t ln (cid:12)(cid:12)(cid:12) z − e i (2 π − ( p + t )) (cid:12)(cid:12)(cid:12) α = iα z + e i (2 π − ( p + t )) z − e i (2 π − ( p + t )) ,∂∂t ln (cid:12)(cid:12)(cid:12) z − e i (2 π − ( p − t ) (cid:12)(cid:12)(cid:12) α = − iα z + e i (2 π − ( p − t )) z − e i (2 π − ( p − t )) . (6.9)Therefore we get ∂f ( z ) ∂t = if ( z ) (cid:88) k =1 , , , q k (cid:18) α k z + z k z − z k + β k (cid:19) = if ( z ) (cid:88) k =1 , , , q k (cid:18) α k + β k + 2 α k z k z − z k (cid:19) = if ( z ) (cid:88) k =1 , , , q k (cid:18) β k + 2 α k z k z − z k (cid:19) (6.10)where q k = 1 for k = 1 , and q k = − for k = 2 , . In the last line we used that (cid:80) k =1 , , , q k α k = 0 .Set ˜ Y ( z ) = Y ( z ) in a neighborhood of z k if α k ≥ . If α k < the second column of Y has a term oforder ( z − z k ) α k , which explodes as z → z k . We set ˜ Y j = Y j , j = 1 , , ˜ Y j = Y j − c j ( z − z k ) α k in aneighborhood of z k , with c j such that ˜ Y is bounded in that neighborhood. Then we have Proposition 20. Let n ∈ N . Then the following differential identity holds: i dd t ln D n ( f p,t ) = (cid:88) k =1 , , , q k (cid:18) nβ k − α k z k (cid:18) d Y − d z ˜ Y (cid:19) ( z k ) (cid:19) , (6.11) with q k as above and (cid:16) d Y − d z ˜ Y (cid:17) ( z k ) = lim z → z k (cid:16) d Y − d z ˜ Y (cid:17) ( z ) with z → z k non-tangentially to theunit circle. roof. The proof for α k (cid:54) = 0 , k = 1 , , , works exactly like the proof of Proposition 2.1 in [13]. Wehave to modify their (2.16), which we replace with our (6.10). The singularities at ± are independentof p and t and thus always stay within f p,t . As in their Remark 2.2 we then get the identity for α k = 0 for some k ∈ { , , , } by letting that α k go to zero in (6.10), which is continuous in α k on bothsides. Set T ( z ) = Y ( z ) (cid:32) z − n z n (cid:33) , | z | > ,Y ( z ) , | z | < . (7.1)Then by the RH conditions for Y , we obtain the following RH condition for T : RHP for T (a) T : C \ C → C × is analytic.(b) The continuous boundary values of T from the inside, T + , and from outside, T − , of the unit circleexist on C \ { z , ..., z } , and are related by the jump condition T + ( z ) = T − ( z ) (cid:18) z n f p,t ( z )0 z − n (cid:19) , z ∈ C \ { z , ..., z } . (7.2)(c) T ( z ) = I + O (1 /z ) , as z → ∞ .(d) As z → z k , z ∈ C \ C , k = 0 , ..., , we have T ( z ) = (cid:18) O (1) O (1) + O ( | z − z k | α k ) O (1) O (1) + O ( | z − z k | α k ) (cid:19) , if α k (cid:54) = 0 , (7.3)and T ( z ) = (cid:18) O (1) O (1) + O (ln | z − z k | ) O (1) O (1) + O (ln | z − z k | ) (cid:19) , if α k = 0 . (7.4) Define the Szegö function D ( z ) = exp (cid:18) πi (cid:90) C ln f p,t ( ξ ) ξ − z d ξ (cid:19) , (7.5)which is analytic inside and outside of C and satisfies D + ( z ) = D − ( z ) f p,t ( z ) , z ∈ C \ { z , ..., z } . (7.6)We have (see (4.9)-(4.10) in [14]): D ( z ) = e (cid:80) ∞ V j z j (cid:89) k =0 (cid:18) z − z k z k e iπ (cid:19) α k + β k =: D in ,p,t ( z ) , | z | < , (7.7)and D ( z ) = e − (cid:80) − −∞ V j z j (cid:89) k =0 (cid:18) z − z k z (cid:19) − α k + β k =: D out ,p,t ( z ) , | z | > , (7.8)31nd thus D out ,p,t ( z ) − = e (cid:80) − −∞ V j z j (cid:89) k =0 (cid:18) z − z k z (cid:19) α k − β k . (7.9)The branch of ( z − z k ) α k ± β k is fixed by the condition that arg( z − z k ) = 2 π 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 toinfinity. 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 + 2 π . By (7.6) we have that f p,t ( e iθ ) = D in ,p,t ( e iθ ) D out ,p,t ( e iθ ) − , (7.10)and this function extends analytically to the complex plane with the 6 branch cuts z k R + , k = 0 , ... ,which we orient away from zero. Then we obtain for the jumps of f p,t : f p,t + = f p,t − e πi ( α j − β j ) , on z j (0 , ,f p,t + = f p,t − e − πi ( α j + β j ) , on z j (1 , ∞ ) . (7.11) z z z z Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in + − + − + − Σ ,out Σ Σ ,in Σ + − Σ Figure 1: The contour Σ S for S We factorize the jump matrix of T as follows: (cid:18) z n f p,t ( z )0 z − n (cid:19) = (cid:18) z − n f p,t ( z ) − (cid:19) (cid:18) f p,t ( z ) − f p,t ( z ) − (cid:19) (cid:18) z n f p,t ( z ) − (cid:19) . (7.12)We then fix a lens-shaped region as in Figure 1 and define S ( z ) = T ( z ) , outside the lens T ( z ) (cid:32) z − n f p,t ( z ) − (cid:33) , in the parts of the lenses outside the unit circle ,T ( z ) (cid:32) − z n f p,t ( z ) − (cid:33) , in the parts of the lenses inside the unit circle . (7.13)32he following RH conditions for S can be verified directly: RHP for S (a) S : C \ Σ S → C × is analytic.(b) S + ( z ) = S − ( z ) J S ( z ) for z ∈ Σ S \ { z , ..., z } , where J S is given by J S ( z ) = (cid:32) z − n f p,t ( z ) − (cid:33) , on Σ ,out ∪ Σ ,out ∪ Σ ,out ∪ Σ ,out , (cid:32) f p,t ( z ) − f p,t ( z ) − (cid:33) , on Σ ∪ Σ ∪ Σ ∪ Σ , (cid:32) z n f p,t ( z ) − (cid:33) , on Σ ,in ∪ Σ ,in ∪ Σ ,in ∪ Σ ,in , (cid:32) z n f p,t ( z )0 z − n (cid:33) , on Σ ∪ Σ . (7.14)(c) S ( z ) = I + O (1 /z ) , as z → ∞ .(d) As z → z k from outside the lenses, k = 0 , ..., , we have S ( z ) = (cid:18) O (1) O (1) + O ( | z − z k | α k ) O (1) O (1) + O ( | z − z k | α k ) (cid:19) , if α k (cid:54) = 0 , (7.15)and S ( z ) = (cid:18) O (1) O (1) + O (ln | z − z k | ) O (1) O (1) + O (ln | z − z k | ) (cid:19) , if α k = 0 . (7.16)The behaviour of S ( z ) as z → z k from the other regions is obtained from these expressions byapplication of the appropriate jump conditions.Fix δ , δ > such that the discs U ± := { z : | z − ± | < δ } , U ± := { z : | z − e ± ip | < δ } (7.17)are disjoint for any p ∈ ( (cid:15), π − (cid:15) ) . Let t ∈ (0 , (cid:15) ) such that e i ( p ± t ) ∈ U + and e i (2 π − ( p ± t )) ∈ U − for oneand hence for all p ∈ ( (cid:15), π − (cid:15) ) . Then one observes that on the inner and out jump contours and outsideof U ∪ U − ∪ U + ∪ U − the jump matrix J S ( z ) converges to the identity matrix as n → ∞ , uniformlyin z , t < t and p ∈ ( (cid:15), π − (cid:15) ) . Define the function N ( z ) = (cid:32) D in ,p,t ( z ) 00 D in ,p,t ( z ) − (cid:33) (cid:32) − (cid:33) , | z | < , (cid:32) D out ,p,t ( z ) 00 D out ,p,t ( z ) − (cid:33) , | z | > . (7.18)One can easily verify that N satisfies the following RH conditions: RH problem for N N : C \ C → C × is analytic.(b) N + ( z ) = N − ( z ) (cid:18) f p,t ( z ) − f p,t ( z ) − (cid:19) for z ∈ C \ { z , ..., z } .(c) N ( z ) = I + O (1 /z ) as z → ∞ . ± The local parametrix near ± are constructed in exactly the same way as in [14]. We are looking for asolution of the following RHP: RH problem for P ± ( z ) (a) P ± : U ± \ Σ S → C × is analytic.(b) P ± ( z ) + = P ± ( z ) − J S ( z ) for z ∈ U ± ∩ Σ S .(c) As z → ± , S ( z ) P ± ( z ) − = O (1) .(d) P ± satisfies the matching condition P ± ( z ) N − ( z ) = I + o (1) as n → ∞ , uniformly in z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and < t < t . P ± is given by (4.15), (4.23), (4.24), (4.47)-(4.50) in [14] and one can see from their constructionthat when all the other singularities are bounded away from ± , then the matching condition is uniformin the location of the other singularities, i.e. holds uniformly in p ∈ ( (cid:15), π − (cid:15) ) and < t < t . < t ≤ ω ( n ) /n . Local Parametrices near e ± ip Let ω ( x ) be a positive, smooth function for x sufficiently large, s.t. ω ( n ) → ∞ , ω ( n ) = o ( n − ) , as n → ∞ . (7.19)For < t ≤ /n and /n < t ≤ ω ( n ) /n we will construct local parametrices in U ± which satisfy thesame 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 × is analytic.(b) P ± ( z ) + = P ± ( z ) − J S ( z ) for z ∈ U ± ∩ Σ S .(c) As n → ∞ , we have P ± ( z ) N − ( z ) = ( I + o (1)) , z ∈ ∂U ± , (7.20)uniformly for p ∈ ( (cid:15), π − (cid:15) ) and < t < t .(d) As z → z k , S ( z ) P ± ( z ) − = O (1) , k = 1 , for + and k = 4 , for − .34 .5.1 RHP for Φ ± Define Φ + ( ζ, s ) =Ψ + ( ζ, s ) , − < (cid:61) < ,e πi ( α − β ) σ , (cid:61) ζ > ,e − πi ( α − β ) σ , (cid:61) ζ < − , Φ − ( ζ, s ) =Ψ + ( ζ, s ) , − < (cid:61) < ,e πi ( α + β ) σ , (cid:61) ζ > ,e − πi ( α + β ) σ , (cid:61) ζ < − , (7.21)where Ψ + ( ζ, s ) equals Ψ( ζ, s ) , defined in Appendix C, and Ψ − ( ζ, s ) equals Ψ( ζ, s ) with ( α , α , β , β ) in the appendix changed to ( α , α , β , β ) = ( α , α , − β , − β ) . The RH conditions for Φ ± followdirectly from the RHP for Ψ . + i − i (cid:18) − (cid:19) (cid:18) (cid:19)(cid:18) − (cid:19) e πi ( α + β ) σ e πi ( α − β ) σ e πi ( α + β ) σ e πi ( α − β ) σ (cid:18) (cid:19)(cid:18) − (cid:19) Figure 2: The jump contour Σ and the jump matrices for Φ + . Φ − has the same jump contour, whilethe jump matrices of Φ − are given by replacing ( α , α , β , β ) with ( α , α , − β , − β ) in the jumpmatrices of Φ + . RH Problem for Φ ± (a) Φ ± : C \ Σ → C × is analytic, where Σ = ∪ k =1 Σ k , Σ = i + e iπ R + , Σ = i + e iπ R + Σ = i − R + , Σ = − i − R + , Σ = − i + e − iπ R + , Σ = − i + e iπ R + , Σ = − i + R + , Σ = i + R + , Σ =[ − i, i ] , with the orientation chosen as in Figure 2 ("-" is always on the RHS of the contour).35b) Φ + satisfies the jump conditions Φ + ( ζ ) + = Φ + ( ζ ) − V k , ζ ∈ Σ k , (7.22)where V = (cid:18) (cid:19) , V = (cid:18) − (cid:19) , (7.23) V = e πi ( α − β ) σ , V = e πi ( α + β ) σ , (7.24) V = (cid:18) − 10 1 (cid:19) , V = (cid:18) (cid:19) , (7.25) V = e πi ( α + β ) σ , V = e πi ( α + β ) σ , (7.26) V = (cid:18) − (cid:19) . (7.27)The jump conditions of Φ − are given by replacing ( α , α , β , β ) with ( α , α , − β , − β ) in thejump matrices of Φ + .(c) We have in all regions: Φ ± ( ζ ) = (cid:18) I + Ψ ± , ζ + Ψ ± , ζ + O ( ζ − ) (cid:19) ˆ P ( ∞ ) ± ( ζ ) e − is ζσ , as ζ → ∞ , (7.28)where ˆ P ( ∞ )+ ( ζ, s ) = P ( ∞ )+ ( ζ, s ) , − < (cid:61) < ,e πi ( α − β ) σ , (cid:61) ζ > ,e − πi ( α − β ) σ , (cid:61) ζ < − , ˆ P ( ∞ ) − ( ζ, s ) = P ( ∞ )+ ( ζ, s ) , − < (cid:61) < ,e πi ( α + β ) σ , (cid:61) ζ > ,e − πi ( α + β ) σ , (cid:61) ζ < − , (7.29)with P ( ∞ )+ ( ζ, s ) = (cid:18) is (cid:19) − ( β + β ) σ ( ζ − i ) − β σ ( ζ + i ) − β σ ,P ( ∞ ) − ( ζ, s ) = (cid:18) is (cid:19) ( β + β ) σ ( ζ − i ) β σ ( ζ + i ) β σ , (7.30)with the branches corresponding to the arguments between and π , and where s ∈ − i R + .(d) Φ ± has singular behaviour near ± i which is inherited from Ψ . The precise conditions follow from(7.21), (C.8), (C.10), (C.12) and (C.14). < t ≤ ω ( n ) /n . Construction of a Local Parametrix near e ± ip in terms of Φ ± We choose P ± as in (7.21) in [13], i.e. P ± ( z ) = E ± ( z )Φ ± ( ζ ; s ) W ± ( z ) , ζ = 1 t ln ze ± ip , s = − int, (7.31) • where (cid:61) ln takes values in ( − σ, σ ) for some σ > ,36 where E ± is an analytic matrix-valued function in U ± , • and where W is given by W ± ( z ) = (cid:40) − z n σ f p,t ( z ) − σ σ , for | z | < ,z n σ f p,t ( z ) σ σ , for | z | > , σ = (cid:18) (cid:19) , σ = (cid:18) − (cid:19) . (7.32)The singularities z = e i ( p ± t ) for + and z = e i (2 π − ( p ∓ t )) for − correspond to the values ζ = ± i . Thejumps of W ± follow from (7.11): W ± ( z ) + = W ± ( z ) − (cid:18) f p,t ( z ) − f − p,t ( z ) 0 (cid:19) , z ∈ C,W ± ( z ) + = W ± ( z ) − e − πi ( α j − β j ) σ , z ∈ z j (0 , ,W ± ( z ) + = W ± ( z ) − e πi ( α j + β j ) σ , z ∈ z j (1 , ∞ ) . (7.33)Choose Σ S such that t ln (cid:16) Σ S ∩ U ± e ± ip (cid:17) ⊂ Σ ∪ i R , where Σ is the contour of the RHP for Φ ± , as shownin Figure 2. Inside U ± the combinded jumps of W ( z ) and Φ are the same as the jumps of S : P ± ( z ) + = E ± ( z )Ψ ± ( z ) − e πi ( α j − β j ) σ W ± ( z ) − e − πi ( α j − β j ) σ = P ± ( z ) − , z ∈ z j (0 , ,P ± ( z ) + = E ± ( z )Ψ ± ( z ) − e πi ( α j + β j ) σ W ± ( z ) − e πi ( α j + β j ) σ = P ± ( z ) − , z ∈ z j (1 , ∞ ) ,P ± ( z ) + = E ± ( z )Ψ ± ( z ) W ± ( z ) − (cid:18) f p,t ( z ) − f − p,t ( z ) 0 (cid:19) = P ± ( z ) − (cid:18) f p,t ( z ) − f − p,t ( z ) 0 (cid:19) , z ∈ Σ k ∩ U ± , k = 0 , , , ,P ± ( z ) + = E ± ( z )Ψ ± ( z ) − (cid:18) − (cid:19) W ± ( z ) − (cid:18) f p,t ( z ) − f − p,t ( z ) 0 (cid:19) = P ± ( z ) − (cid:18) z n f p,t ( z )0 z − n (cid:19) , z ∈ Σ k ∩ U ± , k = 1 , ,P ± ( z ) + = E ± ( z )Ψ ± ( z ) − (cid:18) − (cid:19) W ± ( z )= P ± ( z ) − (cid:18) z n f p,t ( z ) − (cid:19) , z ∈ Σ k,in ∩ U ± , k = 0 , , , ,P ± ( z ) + = E ± ( z )Ψ ± ( z ) − (cid:18) (cid:19) W ± ( z )= P ± ( z ) − (cid:18) z − n f p,t ( z ) − (cid:19) , z ∈ Σ k,out ∩ U ± , k = 0 , , , . (7.34)By the condition (d) of the RHP for S , the singular behaviour of W near z k , k = 1 , , , and condition(d) of the RHP for Φ ± , the singularities of S ( z ) P ± ( z ) − at z , z for + , and at z , z for − , are removable.What remains is to choose E ± such that the matching condition (c) of P ± holds. Define E ± ( z ) = σ ( D in,p,t ( z ) D out,p,t ( z )) − σ e ∓ ip n σ ˆ P ( ∞ ) ± ( ζ, s ) − , (7.35)From (7.7) and (7.8) one quickly sees that the branch cuts and singularities of ( D in,p,t ( z ) D out,p,t ( z )) − σ cancel out with those of ˆ P ( ∞ ) ± ( z ) − , so that E ± is analytic in U ± .In exactly the same way as in the proof of Proposition 7.1 in [13] one can see that the matchingcondition (c) is satisfied, i.e. we get: 37 roposition 21. As n → ∞ we have P ± ( z ) N ( z ) − =( I + O ( n − )) , (7.36) uniformly for z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and < t < t with t sufficiently small.Proof. Consider first the case where c ≤ nt ≤ C , with some c > small and some C > large,which will be fixed below. Then | ζ | = | t ln ze ± ip | > δn for z ∈ ∂U ± , and s = − int remains boundedand bounded away from zero. Thus by (7.21), (7.31) and (C.6) we have P ( z ) N ( z ) − = E ± ( z )( I + O ( n − )) ˆ P ( ∞ ) ± ( ζ, s ) (cid:16) ze ± ip (cid:17) − n σ W ± ( z ) N ( z ) − , z ∈ ∂U ± , n → ∞ . (7.37)Since the RHP for Ψ is solvable for c ≤ nt ≤ C , general properties of Painlevé RHPs imply that theerror term is valid uniformly for c ≤ nt ≤ C . By (7.18) and (7.32) we obtain z − n σ W ± ( z ) N ( z ) − = ( D in,p,t ( z ) D out,p,t ( z )) σ σ , z ∈ U ± . (7.38)Thus we have P ± ( z ) N ( z ) − = E ± ( z )( I + ( O ( n − )) E ± ( z ) − , (7.39)and since one can quickly see that E ± ( z ) is bounded uniformly for z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and c ≤ nt ≤ C , we get that (7.36) holds uniformly z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and c ≤ nt ≤ C .Now consider the case C < nt < ω ( n ) . In this case we cannot use the expansion (C.6) since theargument s of Ψ is not bounded. Instead we need to use the large | s | = 2 nt asymptotics for Ψ , whichwere computed in [13][Section 5]. As is apparent from their (7.30) - (7.33), we have for C sufficientlylarge P ± ( z ) N ( z ) − = ( I + ( O ( n − )) , n → ∞ , z ∈ ∂U ± , (7.40)uniformly for C /n < t < t , z ∈ ∂U ± and p ∈ ( (cid:15), π − (cid:15) ) .If nt < c , we can use the small | s | asymptotics for Ψ( ζ ; s ) for large values of ζ = t ln ze ± ip , ascalculated in Section 6 of [13]. From their (7.34) and (7.35) we see that P ± ( z ) N ( z ) − = ( I + ( O ( n − )) , n → ∞ , z ∈ ∂U ± , (7.41)uniformly for < t < C /n , z ∈ ∂U ± and p ∈ ( (cid:15), π − (cid:15) ) . < t ≤ ω ( n ) /n . Final Transformation Define R ( z ) = S ( z ) N − ( z ) z ∈ U ∞ \ Σ S , U ∞ := C \ local parametrices ,S ( z ) P − ± ( z ) , z ∈ U ± \ Σ S ,S ( z ) P − ± ( z ) , z ∈ U ± \ Σ S . (7.42)Then R solves the following RHP: RH problem for R (a) R : C \ Σ R → C × is analytic, where Σ R is shown in Figure 3(b) R ( z ) has the following jumps: R + ( z ) = R − ( Z ) N ( z ) (cid:18) f p,t ( z ) − z − n (cid:19) N ( z ) − , z ∈ Σ j,out , j = 0 , , , ,R + ( z ) = R − ( Z ) N ( z ) (cid:18) f p,t ( z ) − z n (cid:19) N ( z ) − , z ∈ Σ j,in , j = 0 , , , ,R + ( z ) = R − ( Z ) P ± ( z ) N ( z ) − , z ∈ ∂U ± \ intersection points ,R + ( z ) = R − ( Z ) P ± ( z ) N ( z ) − , z ∈ ∂U ± \ intersection points . (7.43)38 U − U + − + U − Σ ,out Σ ,in Σ ,out Σ ,in Σ ,out Σ ,in + − + − Σ ,out Σ ,in Figure 3: Contour Σ R for the RHP of R .(c) R ( z ) = I + O (1 /z ) as z → ∞ .One quickly sees that uniformly in z ∈ Σ j,out ∪ Σ j,in \ U ∞ we have R + ( z ) = R − ( z )( I + O ( e − δn )) (7.44)for some δ > and uniformly in p ∈ ( (cid:15), π − (cid:15) ) , < t < t . By Proposition 21 we have that R + ( z ) = R − ( z ) P ± ( z ) N ( z ) − = R − ( z )( I + O ( n − )) , (7.45)uniformly for z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and < t < t . Because of the matching condition (d) of P ± wehave R + ( z ) = R − ( z ) P ± ( z ) N ( z ) − = R − ( z )( I + O ( n − )) , (7.46)uniformly for z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and < t < t .We see that we have a normalized RHP with small jumps, which by the standard theory on RHPimplies that R ( z ) = I + O ( n − ) , d R ( z ) d z = O ( n − ) , (7.47)as n → ∞ , uniformly for z off the jump contour and uniformly in p ∈ ( (cid:15), π − (cid:15) ) , < t < t . ω ( n ) /n < t < t . Local Parametrices near e ± ip We now transfer the construction from Section 7.5 in [13] to our setting in a completely straightforwardmanner. Although the parametrices P ± from the previous section are valid for the whole region < t < t we need to construct more explicit parametrices for the case ω ( n ) /n < t < t to get a simplerlarge n expansion for Y , which is needed for the analysis in the next section.39n the case ω ( n ) /n < t < t ζ = t ln ze ± ip is not necessarily large on ∂U ± . But we can construct alarge s = − int expansion for Y , as | s | = nt is large.We modify the S -RHP in the following way: We choose the contour Σ S as in Figure 4. z z z z Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in + − + − + − Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in Σ ,out Σ Σ ,in Figure 4: The contour Σ S for S in the case ω ( n ) /n < t < t .The points z = 1 , z = − we surround with the same small neighborhoods U , U and take thesame local parametrices P ± as in the last section.Let U , U be small non-intersecting disks around ± i , those are the same neighborhoods as in Section5 of [13]. We surround the points z = e i ( p − t ) , z = e i ( p + t ) by small neighborhoods ˜ U , ˜ U , with ˜ U being the image of U under the inverse of the map ζ = t ln ze ip , and ˜ U being the image of U underthe same map. Similarly we surround z = e i (2 π − ( p + t )) by a small neighborhood ˜ U , which is the imageof U under the inverse of the map ζ = t ln ze − ip , and z = e i (2 π − ( p − t )) we surround by ˜ U which is theimage of U under the same map. Since the disks U , U are fixed in the ζ -plane, the neighborhoods ˜ U , ˜ U , ˜ U , ˜ U contract in the z -plane if t decreases with n .As global parametrix outside these neighborhoods we choose N ( z ) as in the previous section. For k = 1 , , , we choose the local parametrices in ˜ U k as follows: ˜ P ( z ) = ˜ E ( z ) M ( α ,β ) ( nt ( ζ ( z ) + i ))Ω ( z ) W + ( z ) , ζ = 1 t ln ze ip , ˜ P ( z ) = ˜ E ( z ) M ( α ,β ) ( nt ( ζ ( z ) − i ))Ω ( z ) W + ( z ) , ζ = 1 t ln ze ip , ˜ P ( z ) = ˜ E ( z ) M ( α ,β ) ( nt ( ζ ( z ) + i ))Ω ( z ) W − ( z ) , ζ = 1 t ln ze − ip , ˜ P ( z ) = ˜ E ( z ) M ( α ,β ) ( nt ( ζ ( z ) − i ))Ω ( z ) W − ( z ) , ζ = 1 t ln ze − ip , (7.48)40here Ω k ( z ) = (cid:40) e i π ( α k − β k ) σ , (cid:61) ζ > e − i π ( α k − β k ) σ , (cid:61) ζ < , (7.49) W ± ( z ) is given in (7.32), M ( α k ,β k ) ( λ ) is given in Appendix D, with α = α k , β = β k , and ˜ E k ( z ) = σ ( D in,p,t ( z ) D out,p,t ( z )) − σ / Ω k ( z )( nt ( ζ ± i )) β k σ z − n σ k , (7.50)with + for k = 1 , and − for k = 2 , . By (7.18) and (7.32) we obtain z − n σ W ± ( z ) N ( z ) − = ( D in,p,t ( z ) D out,p,t ( z )) σ σ , z ∈ U ± . (7.51)Using the large argument expansion (D.6) for M ( α ,β ) ( nt ( ζ + i )) for z ∈ ∂ ˜ U , we see that ˜ P ( z ) N ( z ) − = ˜ E ( z ) (cid:18) I + M ( α ,β ) nt ( ζ + i ) + O (( nt ) − (cid:19) × ( nt ( ζ + i )) − β σ (cid:16) ze i ( p + t ) (cid:17) − n σ Ω ( z ) W + ( z ) N ( z ) − = ˜ E ( z ) (cid:18) I + M ( α ,β ) nt ( ζ + i ) + O (( nt ) − (cid:19) × ( nt ( ζ + i )) − β σ z n σ Ω ( z ) ( D in,p,t ( z ) D out,p,t ( z )) σ σ = ˜ E ( z ) (cid:0) I + O (( nt ) − ) (cid:1) ˜ E ( z ) − = (cid:0) I + O (( nt ) − ) (cid:1) , (7.52)uniformly in z ∈ ∂ ˜ U , p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t , since ˜ E ( z ) is uniformly bounded for ω ( n ) /n < t < t , p ∈ ( (cid:15), π − (cid:15) ) and z ∈ ˜ U . Similarly one obtains that for k = 2 , , P k ( z ) N ( z ) − = (cid:0) I + O (( nt ) − ) (cid:1) , (7.53)uniformly in z ∈ ∂ ˜ U k , p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t .Choose Σ S such that t ln (cid:16) Σ S z k (cid:17) ⊂ (cid:16) e ± πi R ∪ i R ∪ R (cid:17) in ˜ U k . Then one can easily verify, as in (7.34),that ˜ P k has the same jumps as S in ˜ U k , so that S ( z ) ˜ P k ( z ) − is meromorphic in ˜ U k , with at most anisolated singulary at z k . The singular behaviour of S and W ± near z k , and of M ( α k ,β k ) near (given in(D.8) and (D.12)), imply that S ( z ) ˜ P − k ( z ) is bounded at z k , which shows that that ˜ P k is a parametrixfor S in ˜ U k with the matching condition (7.52) with N ( z ) at ∂ ˜ U k .41 .6.1 ω ( n ) /n < t < t . Final Transformation U U − ˜ U − +˜ U ˜ U ˜ U Σ ,out Σ ,in Σ ,out Σ ,in Σ ,out Σ ,in + − + − Σ ,out Σ ,in Figure 5: Contour Σ ˜ R for the RHP of ˜ R in the case ω ( n ) /n < t < t We transfer Section 7.5.1 in [13] to our case. Figure 5 shows the contour chosen for the RHP of ˜ R ,which we define as follows: ˜ R ( z ) = S ( z ) ˜ P k ( z ) − , z ∈ ˜ U k ,S ( z ) P ± ( z ) − , z ∈ U ± ,S ( z ) N ( z ) − , z ∈ C \ (cid:16) ˜ U ∪ ˜ U ∪ ˜ U ∪ ˜ U ∪ U ∪ U − (cid:17) . (7.54)Then ˜ R is analytic, in particular has no jumps inside any of the local parametrices ˜ U k , k = 1 , , , , U ± , or on the unit circle. On the rest of the lenses we can see that the jump matrix is I + O ( e − δnt ) forsome δ > , uniformly in p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t . Because of the matching condition (d) of P ± we have as in the case < t < ω ( n ) /n that ˜ R + ( z ) = ˜ R − ( z ) P ± ( z ) N ( z ) − = ˜ R − ( z )( I + O ( n − )) , (7.55)uniformly for z ∈ ∂U ± , p ∈ ( (cid:15), π − (cid:15) ) and < t < t . Using (7.52), we get that ˜ R + ( z ) = ˜ R − ( z ) ˜ P k ( z ) N ( z ) − = ˜ R − ( z )( I + O (( nt ) − ) , (7.56)uniformly for z ∈ ∂ ˜ U k , p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t . 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 ˜ R ( z ) = I + O (( nt ) − ) , d ˜ R ( z ) d z = O (( nt ) − ) , (7.57)uniformly for z off the jump contour of ˜ R , and uniformly in p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t .42 Asymptotics of D n ( f p,t ) This section is a transfer of Section 8 in [13] to our case. Proposition 22. Let α , α , α + α > − , let σ ( s ) be the solution to (3.11) and let ω ( x ) be a positive,smooth function for x sufficiently large, s.t. ω ( n ) → ∞ , ω ( n ) = o ( n − ) , as n → ∞ . (8.1) Then the following asymptotic expansion holds: i dd t ln D n ( f p,t ) =2 n ( β − β ) + d ( p, t ; α , α , β , α , β , α )+ d ( p, t ; α , α , β , α , β , α ) + d ( p, t ; α , α , β , α , β , α ) + (cid:15) n,p,t , (8.2) where for the error term (cid:15) n,p,t (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t (cid:15) n,p,τ d τ (cid:12)(cid:12)(cid:12)(cid:12) = O ( ω ( n ) − δ ) = o (1) , (8.3) for some δ > , uniformly in p ∈ ( (cid:15), π − (cid:15) ) and < t < t , and where d ( p, t ; α , α , β , α , β , α ) =2 α z V (cid:48) ( z ) − α z V (cid:48) ( z ) + 2( β + β ) ∞ (cid:88) j =1 jV j (cos j ( p + t ) − cos j ( p − t )) − i (2 α + β + β β ) cos p − t sin p − t + i (2 α + β + β β ) cos p + t sin p + t + i ( β − β ) cos p sin p − iα α cos p − t sin p − t + 2 iα α sin p − t cos p − t + 2 iα α cos p + t sin p + t − iα α sin p + t cos p + t d ( p, t ; α , α , β , α , β , α ) = 2 it σ ( s ) + i (4 α α − ( β + β ) ) cos t sin td ( p, t ; α , α , β , α , β , α ) =2 σ s (cid:32) − ∞ (cid:88) j =1 jV j (cos j ( p − t ) + cos j ( p + t )) − (cid:88) j =0 α j − iβ cos p − t sin p − t − iβ cos p + t sin p + t + i ( β − β ) (cid:18) cos t sin t − t (cid:19) − i ( β + β ) cos p sin p (cid:33) . (8.4) Proof. The proof is analogous to the proof of Proposition 8.1 in [13]. As is done there, we assume belowthat α k > , k = 1 , , , , for simplicity of notation. Once (8.2) is proven under this assumption, thecase where α k = 0 for some k then follows from the uniformity of the error terms in α k , k = 1 , , , .Extending to the case where α k < for some k is straightforward.We prove the proposition first in the regime < t ≤ ω ( n ) /n and then in the regime ω ( n ) /n < t < t .Using the transformation Y → T → S inside the unit circle, outside the lenses, we can rewrite thedifferential identity (6.11) in the form i dd t ln D n ( f p,t ) = (cid:88) k =1 , , , q k (cid:32) nβ k + 2 α k z k (cid:18) S − d S d z (cid:19) + , ( z k ) (cid:33) , (8.5)with q k = 1 for k = 1 , and q k = − for k = 2 , , and where the limit z → z k is taken from the insideof the unit circle and outside the lenses. 43 .1.1 < t ≤ ω ( n ) /n By (7.42) we get S ( z ) = R ( z ) P ± ( z ) , z ∈ U ± , (8.6)and thus (cid:18) S − d S d z (cid:19) ( z ) = (cid:18) P − ± d P ± d z (cid:19) ( z ) + A n,p,t ( z ) , z ∈ U e ± ip ,A n,p,t ( z ) = (cid:18) P − ± ( z ) R − ( z ) dRdz ( z ) P ± ( z ) (cid:19) . (8.7)Following exactly the same approach as on pages 60, 61 in [13], we can use (C.6) and the small andlarge | s | asymptotics from Sections 5, 6 of [13] to obtain that for k = 1 , , , (cid:90) t | A n,p,t ( z k ) | d t = o ( ω ( n ) − ) , n → ∞ , (8.8)uniformly in < t ≤ ω ( n ) /n and p ∈ ( (cid:15), π − (cid:15) ) , and thus also ˜ (cid:15) n,p,t := 2 (cid:88) k =1 , , , q k α k z k A n,p,t ( z k ) = O ( ω ( n ) − ) , (8.9)uniformly in < t ≤ ω ( n ) /n and p ∈ ( (cid:15), π − (cid:15) ) .By (7.31) we have (cid:18) P − ± d P ± d z (cid:19) ( z ) = − n z + 12 f (cid:48) p,t f p,t ( z ) + (cid:18) Φ − ± d Φ ± d z (cid:19) ( z ) + (cid:18) Φ − ± E − ± d E ± d z Φ ± (cid:19) ( z ) , (8.10)with z inside the unit circle and outside of the lenses of Σ S . By (7.35) we have for z near e ip : E ± ( z ) − d E ± ( z ) d z = h ± ( z ) σ , (8.11)where h ± ( z ) = ± β z ln ze ± ip ± itz ± β z ln ze ± ip ∓ itz − ∞ (cid:88) j =1 jV j z j − + 12 −∞ (cid:88) j = − jV j z j − − (cid:88) j =0 β j z − z j − z (cid:88) j =0 α j . (8.12)In the following equation we need the fact that z ) = 1 z + 12 + O ( z ) , z → , z − ln z k = 1ln zz k = z k z − z k + 12 + O ( | z − z k | ) , z → z k . (8.13)Let k = 1 , and denote k (cid:48) = 1 for k = 2 , k (cid:48) = 2 for k = 1 . Putting together (8.10) and (8.11), we44btain for k = 1 , (as in (8.33) in [13]) (cid:18) P − d P + d z (cid:19) , + ( z k ) = − n z k + lim z → z k (cid:18) f (cid:48) p,t f p,t ( z ) − α k z ln z − z ln z (cid:19) + 1 tz k (cid:18) F − ,k (cid:48) d F + ,k (cid:48) d z (cid:19) (cid:18) t ln z k e ip (cid:19) + h + ( z k )( F − ,k (cid:48) σ F + ,k (cid:48) ) (cid:18) t ln z k e ip (cid:19) = − n z k + 12 V (cid:48) ( z k ) + (cid:88) j =0 β j z k + (cid:88) j,j (cid:54) = k α j z k − z j − α j z k + 1 tz k (cid:18) F − ,k (cid:48) d F + ,k (cid:48) d z (cid:19) (cid:18) t ln z k e ip (cid:19) + h + ( z k )( F − ,k (cid:48) σ F + ,k (cid:48) ) (cid:18) t ln z k e ip (cid:19) , (cid:18) P − − d P − d z (cid:19) , + ( z k ) = − n z k + lim z → z k (cid:18) f (cid:48) p,t f p,t ( z ) − α k z ln z − z ln z k (cid:19) + 1 tz k (cid:18) F − − ,k d F − ,k d z (cid:19) (cid:18) t ln z k e − ip (cid:19) + h − ( z )( F − − ,k σ F − ,k ) (cid:18) t ln z k e − ip (cid:19) = − n z k + 12 V (cid:48) ( z k ) + (cid:88) j =0 β j z k + (cid:88) j,z j (cid:54) = z k α j z k − z j − α j z k + 1 tz k (cid:18) F − − ,k d F − ,k d z (cid:19) (cid:18) t ln z k e − ip (cid:19) + h − ( z k )( F − − ,k σ F − ,k ) (cid:18) t ln z k e − ip (cid:19) , (8.14)where in the second equalities we used (8.13), and where F ± ,k equal the functions F k defined in (C.8),(C.10), (C.12) and (C.14), with ( α , α , β , β ) in the appendix replaced by ( α , α , − β , − β ) in the − case. When replacing ( α , α , β , β ) in the Painlevé equation (3.52) in [13] with ( α , α , β , β ) or ( α , α , − β , − β ) , we get the same Painlevé equation as in our Theorem 10. Thus we can see thatPropositions 3.1 and 3.2 in [13] become in our case: Proposition 23. We have the identities α ( F + , ( i ; s ) − σ F + , ( i ; s )) = − σ s ( s ) + β + β ,α ( F + , ( − i ; s ) − σ F + , ( − i ; s )) = σ s ( s ) + β + β ,α ( F − , ( i ; s ) − σ F − , ( i ; s )) = − σ s ( s ) − β + β ,α ( F − , ( − i ; s ) − σ F − , ( − i ; s )) = σ s ( s ) − β + β , (8.15) and α (cid:0) F + , ( i ; s ) − F + , ,ζ ( i ; s ) (cid:1) = i σ ( s ) − i β + β ) s + i β + β ) ,α (cid:0) F + , ( − i ; s ) − F + , ,ζ ( − i ; s ) (cid:1) = − i σ ( s ) − i β + β ) s − i β + β ) ,α (cid:0) F − , ( i ; s ) − F − , ,ζ ( i ; s ) (cid:1) = i σ ( s ) + i β + β ) s + i β + β ) ,α (cid:0) F − , ( − i ; s ) − F − , ,ζ ( − i ; s ) (cid:1) = − i σ ( s ) + i β + β ) s − i β + β ) . (8.16)45utting together (8.5), (8.7) and (8.14), we obtain: i dd t ln D n ( f p,t )= (cid:88) k =1 , , , q k nβ k + α k z k V (cid:48) ( z k ) + α k (cid:88) j =0 β j + 2 α k (cid:88) j (cid:54) = k α j z k z k − z j − α k (cid:88) j (cid:54) = k α j + (cid:88) k =1 ( − k +1 (cid:18) α k z k h + ( z k ) (cid:16) F − ,k (cid:48) σ F + ,k (cid:48) (cid:17) (( − k i ) + 2 t α k (cid:16) F − ,k (cid:48) F (cid:48) + ,k (cid:48) (cid:17) (( − k i ) (cid:19) + (cid:88) k =1 ( − k (cid:18) α k z k h − ( z k ) (cid:16) F − − ,k σ F − ,k (cid:17) (( − k +1 i ) + 2 t α k (cid:16) F − − ,k F (cid:48)− ,k (cid:17) (( − k +1 i ) (cid:19) + ˜ (cid:15) n,p,t , =2 n ( β − β ) + 2 α z V (cid:48) ( z ) − α z V (cid:48) ( z )+ 2 (cid:88) k =1 , , , q k α k (cid:88) j (cid:54) = k α j z k z k − z j + (cid:88) k =1 ( − k +1 (cid:18) α k z k h + ( z k ) (cid:16) F − ,k (cid:48) σ F + ,k (cid:48) (cid:17) (( − k i ) + 2 t α k (cid:16) F − ,k (cid:48) F (cid:48) + ,k (cid:48) (cid:17) (( − k i ) (cid:19) + (cid:88) k =1 ( − k (cid:18) α k z k h − ( z k ) (cid:16) F − − ,k σ F − ,k (cid:17) (( − k +1 i ) + 2 t α k (cid:16) F − − ,k F (cid:48)− ,k (cid:17) (( − k +1 i ) (cid:19) + ˜ (cid:15) n,p,t , (8.17)where we used that since β = − β , β = − β , α = α , α = α and V ( z ) = V ( z ) , it holds that: (cid:88) k =1 , , , q k α k (cid:88) j (cid:54) = k α j =0 , (cid:88) j =0 β j = 0 , zV (cid:48) ( z ) = − zV (cid:48) ( z ) . (8.18)By Proposition 23 we get (cid:88) k =1 ( − k +1 t α k (cid:16) F − ,k (cid:48) F (cid:48) + ,k (cid:48) (cid:17) (( − k i ) + (cid:88) k =1 ( − k t α k (cid:16) F − − ,k F (cid:48)− ,k (cid:17) (( − k +1 i )= 2 it σ ( s ) − it ( β + β ) , (8.19)and (cid:88) k =1 ( − k +1 α k z k h + ( z k ) (cid:16) F − ,k (cid:48) σ F + ,k (cid:48) (cid:17) (( − k i )+ (cid:88) k =1 ( − k α k z k h − ( z k ) (cid:16) F − − ,k σ F − ,k (cid:17) (( − k +1 i )= (2 σ s ( s ) + β + β ) ( z h + ( z ) + z h − ( z )) + (2 σ s ( s ) − β − β ) ( z h + ( z ) + z h − ( z ))=2 σ s ( z h + ( z ) + z h − ( z ) + z h + ( z ) + z h − ( z ))+ ( β + β ) ( z h + ( z ) + z h − ( z ) − z h + ( z ) − z h − ( z )) . (8.20)46hen (8.17), (8.19) and (8.20) result in i dd t ln D n ( f p,t ) =2 n ( β − β ) + 2 α z V (cid:48) ( z ) − α z V (cid:48) ( z )+ 2 (cid:88) k =1 , , , q k α k (cid:88) j (cid:54) = k α j z k z k − z j + 2 it σ ( s ) − it ( β + β ) + 2 σ s ( z h + ( z ) + z h − ( z ) + z h + ( z ) + z h − ( z ))+ ( β + β ) ( z h + ( z ) + z h − ( z ) − z h + ( z ) − z h − ( z ))+ ˜ (cid:15) n,p,t . (8.21)With h + ( z ) and h − ( z ) given in (8.12), and using (8.13), we obtain: h + ( z ) z = − ∞ (cid:88) j =1 jV j ( z j + z j ) − (cid:88) j (cid:54) =1 β j z z − z j − (cid:88) j =0 α j + β − β it ,h + ( z ) z = − ∞ (cid:88) j =1 jV j ( z j + z j ) − (cid:88) j (cid:54) =2 β j z z − z j − (cid:88) j =0 α j + β β it ,h − ( z ) z = − ∞ (cid:88) j =1 jV j ( z j + z j ) − (cid:88) j (cid:54) =5 β j z z − z j − (cid:88) j =0 α j − β − β it ,h − ( z ) z = − ∞ (cid:88) j =1 jV j ( z j + z j ) − (cid:88) j (cid:54) =4 β j z z − z j − (cid:88) j =0 α j − β β it . (8.22)We thus have h + ( z ) z + z h − ( z ) = − ∞ (cid:88) j =1 jV j cos j ( p − t ) − (cid:88) j =0 α j − β it + β (cid:18) z + z z − z (cid:19) + β (cid:18) − z z − z + z z − z − z z − z + z z − z (cid:19) = − ∞ (cid:88) j =1 jV j cos j ( p − t ) − (cid:88) j =0 α j − β it − iβ cos p − t sin p − t − iβ cos t sin t − iβ cos p sin p ,h + ( z ) z + z h − ( z ) = − ∞ (cid:88) j =1 jV j cos j ( p + t ) − (cid:88) j =0 α j + β it + β (cid:18) z + z z − z (cid:19) + β (cid:18) − z z − z + z z − z − z z − z + z z − z (cid:19) = − ∞ (cid:88) j =1 jV j cos( p + t ) − (cid:88) j =0 α j + β it − iβ cos p + t sin p + t + iβ cos t sin t − iβ cos p sin p . (8.23)47urther we calculate (cid:88) k =1 , q k α k (cid:88) j (cid:54) = k α j z k z k − z j = α α (cid:18) z z − − z z − (cid:19) + α (cid:18) z z − z − z z − z (cid:19) + α α (cid:18) z z − z + z z − z − z z − z − z z − z (cid:19) + α α (cid:18) z z + 1 − z z + 1 (cid:19) , − (cid:88) k =2 , q k α k (cid:88) j (cid:54) = k α j z k z k − z j = α α (cid:18) z z − − z z − (cid:19) + α α (cid:18) z z − z + z z − z − z z − z − z z − z (cid:19) + α (cid:18) z z − z − z z − z (cid:19) + α α (cid:18) z z + 1 − z z + 1 (cid:19) , (cid:88) k =1 , , , q k α k (cid:88) j (cid:54) = k α j z k z k − z j = − iα cos p − t sin p − t + iα cos p + t sin p + t + 2 iα α cos t sin t + α α z + 1 z − α α z − z + 1 − α α z + 1 z − − α α z − z + 1= − iα cos p − t sin p − t + iα cos p + t sin p + t + 2 iα α cos t sin t − iα α cos p − t sin p − t + iα α sin p − t cos p − t + iα α cos p + t sin p + t − iα α sin p + t cos p + t . (8.24)Putting together (8.21), (8.23) and (8.24) we get that uniformly in p ∈ ( (cid:15), π − (cid:15) ) and < t ≤ ω ( n ) /n : i dd t ln D n ( f p,t ) =2 n ( β − β ) + 2 α z V (cid:48) ( z ) − α z V (cid:48) ( z ) − iα cos p − t sin p − t + 2 iα cos p + t sin p + t + 4 iα α cos t sin t − iα α cos p − t sin p − t + 2 iα α sin p − t cos p − t + 2 iα α cos p + t sin p + t − iα α sin p + t cos p + t + 2 it σ ( s ) − it ( β + β ) + 2 σ s ( s ) (cid:32) − ∞ (cid:88) j =1 jV j (cos j ( p − t ) + cos j ( p + t )) − (cid:88) j =0 α j + β − β it − iβ cos p − t sin p − t − iβ cos p + t sin p + t + i ( β − β ) cos t sin t − i ( β + β ) cos p sin p (cid:33) + ( β + β ) (cid:32) ∞ (cid:88) j =1 jV j (cos j ( p + t ) − cos j ( p − t )) − β + β it − iβ cos p − t sin p − t + iβ cos p + t sin p + t − i ( β + β ) cos t sin t + i ( β − β ) cos p sin p (cid:33) + ˜ (cid:15) n,p,t . (8.25)Simplifying further and setting (cid:15) n,p,t = ˜ (cid:15) n,p,t for < t ≤ ω ( n ) /n we obtain (8.2) for < t ≤ ω ( n ) /n .48 .1.2 ω ( n ) /n < t < t For ω ( n ) /n < t < t , z ∈ ˜ U k , we obtain instead of (8.7): (cid:18) S − ( z ) d S ( z ) d z (cid:19) = (cid:32) ˜ P k ( z ) − d ˜ P k ( z ) d z (cid:33) + A n,p,t ,A n,p,t ( z ) = (cid:32) ˜ P k ( z ) − ˜ R − ( z ) d ˜ R ( z ) d z ˜ P k ( z ) (cid:33) , (8.26)with ˜ R ( z ) given in (7.54). From (7.48) and (7.32) it follows that for | z | < (cid:32) ˜ P k ( z ) − d ˜ P k ( z ) d z (cid:33) = − n z + 12 f (cid:48) p,t ( z ) f p,t ( z ) + (cid:18) M k ( z ) − d M k ( z ) d z (cid:19) + ˜ h ( z ) (cid:0) M k ( z ) − σ M k ( z ) (cid:1) ,A n,p,t ( z k ) = lim z → z k A n,p,t ( z )= lim z → z k (cid:32) M − k ˜ E − k ˜ R − d ˜ R d z ˜ E k M k (cid:33) ( z ) , (8.27)where M ( z ) = M ( α ,β ) ( nt ( 1 t ln ze ip + i )) ,M ( z ) = M ( α ,β ) ( nt ( 1 t ln ze ip − i )) ,M ( z ) = M ( α ,β ) ( nt ( 1 t ln ze − ip + i )) ,M ( z ) = M ( α ,β ) ( nt ( 1 t ln ze − ip − i )) , (8.28)and where ˜ h ( z ) σ = ˜ E ( z ) − d ˜ E ( z ) d z , ˜ h ( z ) = h + ( z ) − β z ln ze ip − itz , ˜ h ( z ) σ = ˜ E ( z ) − d ˜ E ( z ) d z , ˜ h ( z ) = h + ( z ) − β z ln ze ip + itz , ˜ h ( z ) σ = ˜ E ( z ) − d ˜ E ( z ) d z , ˜ h ( z ) = h − ( z ) + β z ln ze − ip − itz , ˜ h ( z ) σ = ˜ E ( z ) − d ˜ E ( z ) d z , ˜ h ( z ) = h − ( z ) + β z ln ze ip + itz , (8.29)with h + ( z ) , h − ( z ) given in (8.12). By (7.57) and the fact that ˜ E k is uniformly bounded for p ∈ ( (cid:15), π − (cid:15) ) , ω ( n ) /n < t < t , and z ∈ ˜ U k , we see that A n,p,t ( z k ) = O (( nt ) − ) = O ( ω ( n ) − ) (8.30)uniformly in p ∈ ( (cid:15), π − (cid:15) ) , ω ( n ) /n < t < t , and thus also ˜ (cid:15) n,p,t := 2 (cid:88) k =1 , , , q k α k z k A n,p,t ( z k ) = O ( ω ( n ) − ) , (8.31)49niformly in < t < ω ( n ) /n and p ∈ ( (cid:15), π − (cid:15) ) . This implies that as n → ∞ (cid:90) t ω ( n ) /n | ˜ (cid:15) n,p,t | d t = o ( ω ( n ) − ) , (8.32)uniformly in p ∈ ( (cid:15), π − (cid:15) ) .From (8.41) and (8.42) in [13] we can see that for z → z k inside the unit circle and outside of thelenses of Σ S we have (cid:18) M − d M d z (cid:19) ( z ) = − (cid:18) β α + α n ln ze ip + int (cid:19) nz + o (1) , (cid:18) M − d M d z (cid:19) ( z ) = − (cid:18) β α + α n ln ze ip − int (cid:19) nz + o (1) , (cid:18) M − d M d z (cid:19) ( z ) = − (cid:18) − β α + α n ln ze − ip + int (cid:19) nz + o (1) , (cid:18) M − d M d z (cid:19) ( z ) = − (cid:18) − β α + α n ln ze − ip − int (cid:19) nz + o (1) , (8.33)and in the same limit, ( M k σ M ) ( z k ) = β k α k . (8.34)Together with (8.22) and (8.29) we obtain (again in the same limit) z k (cid:18) M − k d M k d z (cid:19) ( z ) + z k ˜ h k ( z k ) (cid:0) M − k σ M k (cid:1) ( z k )= (cid:32) β k α k + α k n ln zz k (cid:33) n + β k α k − ∞ (cid:88) j =1 jV j ( z jk + z kj ) − (cid:88) j,j (cid:54) = k β j z k z k − z j − (cid:88) j =0 α j + β k + o (1) . (8.35)Combining this with (8.13) and (8.27) we get α k z k (cid:32) ˜ P − k d ˜ P k d z (cid:33) + , ( z k )= − n ( α k + β k ) + α k z k V (cid:48) ( z k ) + 2 α k (cid:88) j,j (cid:54) = k α j z k z k − z j − α k (cid:88) j,j (cid:54) = k α j + 2 β k − ∞ (cid:88) j =1 jV j ( z jk + z kj ) − (cid:88) j,j (cid:54) = k β j z k z k − z j − (cid:88) j =0 α j + β k . (8.36)Together with (8.5), (8.26) and (8.31) we obtain i dd t ln D n ( f p,t ) = S ( p, t ; α , α , β , α , β , α ) + ˜ (cid:15) n,p,t , (8.37)50here S ( p, t ; α , α , β , α , β , α )=2 (cid:88) k =1 ( − k +1 ( α k − β k ) ∞ (cid:88) j =1 jV j z jk − ( α k + β k ) ∞ (cid:88) j =1 jV j z kj − β − β ) (cid:88) k =0 α k − i ( α + β ) cos( p − t )sin( p − t ) + 2 i ( α + β ) cos( p + t )sin( p + t ) + 4 i ( α α − β β ) cos t sin t − iα α cos p − t sin p − t + 2 iα α sin p − t cos p − t + 2 iα α cos p + t sin p + t − iα α sin p + t cos p + t . (8.38)Now we compare this expression to (8.25), obtained for < t ≤ ω ( n ) /n . Consider (8.21) for large s = − int and without the error term. Substituting there the asymptotics of σ ( s ) from Theorem 10and using (8.24) we see that i dd t ln D n ( f p,t ) =2 α z V (cid:48) ( z ) − α z V (cid:48) ( z ) − iα cos p − t sin p − t + 2 iα cos p + t sin p + t + 4 iα α cos t sin t − iα α cos p − t sin p − t + 2 iα α sin p − t cos p − t + 2 iα α cos p + t sin p + t − iα α sin p + t cos p + t + 1 it β β + 2 β ( h + ( z ) z + h − ( z ) z ) − β ( h + ( z ) z + h − ( z ) z )+ Θ n,p,t , (8.39)where Θ n,p,t arises from the error term in the asymptotics of σ ( s ) , and becomes of order ω ( n ) − δ afterintegration w.r.t. t , i.e. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) tω ( n ) /n Θ n,p,τ d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( ω ( n ) − δ ) , (8.40)uniformly in p ∈ ( (cid:15), π − (cid:15) ) and ω ( n ) /n < t < t . Using (8.23) now we see that n ( β − β ) + d + d + d = S + Θ n,p,t , ω ( n ) /n < t < t . (8.41)Thus when setting (cid:15) n,p,t =˜ (cid:15) n,p,t + Θ n,p,t , ω ( n ) /n < t < t , (8.42)we see that (8.2) remains valid also in the region ω ( n ) /n < t < t , where the smallness of the errorterms follows from (8.31). Remark 24. Integrating (8.37) from t to t with ω ( n ) /n < t < t and using Theorem 14 for theexpansion of D n ( f p,t ) , we get the same expansion for D n ( f p,t ) that Theorem 14 gives. The errorterm is then O ( ω ( n ) − ) and uniform in p ∈ ( (cid:15), π − (cid:15) ) for ω ( n ) /n < t < t . Thus we have proven thestatement on Toeplitz determinants in Theorem 17. .2 Integration of the Differential Identity We now integrate (8.2), where we use exactly the same approach as in Section 8.2 of [13]. We obtain (cid:90) t d ( p, τ ; α , α , β , α , β , α ) d τ = + 2 iα (cid:16) V ( e i ( p − t ) ) − V ( e ip ) (cid:17) + 2 iα (cid:16) V ( e i ( p + t ) ) − V ( e ip ) (cid:17) + 2( β + β ) ∞ (cid:88) j =1 V j (sin j ( p + t ) + sin j ( p − t ) − jp )+ i (cid:0) α + β + β β (cid:1) ln sin p − t sin p + i (cid:0) α + β + β β (cid:1) ln sin p + t sin p + it ( β − β ) cos p sin p + 4 iα α ln sin p − t sin p − iα α ln cos p − t cos p + 4 iα α ln sin p + t sin p − iα α ln cos p + t cos p , (8.43)and (cid:90) t d ( p, τ ; α , α , β , α , β , α ) d τ = − i (cid:90) − int s (cid:18) σ ( s ) − α α + 12 ( β + β ) (cid:19) d s + i (cid:0) ( β + β ) − α α (cid:1) ln sin tt , (8.44)and (cid:90) t d ( p, τ ; α , α , β , α , β , α ) d τ =( β − β ) (cid:34) ∞ (cid:88) j =1 V j (sin j ( p − t ) − sin j ( p + t )) − t (cid:88) j =0 α j + iβ ln sin p − t sin p − iβ ln sin p + t sin p + i ( β − β ) ln sin tt − it ( β + β ) cos p sin p (cid:35) . (8.45)Putting things together we see that ln D n ( f p,t ) = ln D n ( f p, ) + 2 int ( β − β ) − α (cid:16) V ( e i ( p − t ) ) − V ( e ip ) (cid:17) − α (cid:16) V ( e i ( p + t ) ) − V ( e ip ) (cid:17) + 4 i ∞ (cid:88) j =1 V j ( β sin j ( p − t ) + β sin j ( p + t ) − ( β + β ) sin jp ) − (cid:0) α + β (cid:1) ln sin p − t sin p − (cid:0) α + β (cid:1) ln sin p + t sin p − α α ln sin p − t sin p − α α ln cos p − t cos p − α α ln sin p + t sin p − iα α ln cos p + t cos p − i (cid:90) − int s (cid:18) σ ( s ) − α α + 12 ( β + β ) (cid:19) d s + 4 ( β β − α α ) ln sin tt − it ( β − β ) (cid:88) j =0 α j + o (1) , (8.46)52niformly in p ∈ ( (cid:15), π − (cid:15) ) and < t < t . To calculate the asymptotics of D n ( f p, ) we use Theorem 14and get ln D n ( f p, ) = nV + ∞ (cid:88) k =1 kV k + ln n (cid:0) α + α + 2( α + α ) − β + β ) (cid:1) − α ln ( b + (1) b − (1)) − α ln ( b + ( − b − ( − − α + α − β − β ) ln b + ( e ip ) − α + α + β + β ) ln b + ( e − ip ) − α α ln 2 − α ( α + α ) ln 2 sin p − α ( α + α ) ln 2 cos p − β + β ) + ( α + α ) ) ln 2 sin p + 2 iα ( β + β )( p − π ) + 2 iα ( β + β ) p + 2 i ( α + α )( β + β )(2 p − π )+ ln G (1 + α ) G (1 + 2 α ) + ln G (1 + α ) G (1 + 2 α )+ ln G (1 + α + α + β + β ) G (1 + α + α − β − β ) G (1 + 2 α + 2 α ) + o (1) , (8.47)uniformly in p ∈ ( (cid:15), π − (cid:15) ) . Combining the 2 last equations we obtain ln D n ( f p,t ) =2 int ( β − β ) + nV + ∞ (cid:88) k =1 kV k + ln n (cid:88) j =0 ( α j − β j ) − (cid:88) j =0 ( α j − β j ) (cid:32) ∞ (cid:88) k =1 V k z kj (cid:33) + ( α j + β j ) (cid:32) ∞ (cid:88) k =1 V k z jk (cid:33) − (cid:0) α + β (cid:1) ln 2 sin( p − t ) − (cid:0) α + β (cid:1) ln 2 sin( p + t ) − α α + β β ) ln 2 sin p − α α ln 2 − α α ln 2 sin p − t − α α ln 2 cos p − t − α α ln 2 sin p + t − α α ln 2 cos p + t 2+ 2 (cid:90) − int s (cid:18) σ ( s ) − α α + 12 ( β + β ) (cid:19) d s + 4 ( β β − α α ) ln sin tnt + 2 iα β ( p − t − π ) + 2 iα β ( p + t − π ) + 2 iα β ( p − t ) + 2 iα β ( p + t )+ 2 iα β (2 p − t − π ) + 2 iα β (2 p − t − π ) + 2 iα β (2 p + 2 t − π ) + 2 iα β (2 p + 2 t − π )+ ln G (1 + α ) G (1 + 2 α ) + ln G (1 + α ) G (1 + 2 α )+ ln G (1 + α + α + β + β ) G (1 + α + α − β − β ) G (1 + 2 α + 2 α ) + o (1) , (8.48)uniformly in p ∈ ( (cid:15), π − (cid:15) ) and < t < t . We note that (cid:88) ≤ j Our work was supported by ERC Advanced Grant 740900 (LogCorRM). We are most grateful to TheoAssiotis for his kind permission to state Theorem 3 here and to set out its proof in Appendix A, aswell as for many extremely helpful discussions. We are also most grateful to Tom Claeys, Gabriel54lesner, Alexander Minakov and Meng Yang for kindly sharing with us their work in progress and forcommunicating to us one of their results, which we quote with their permission as Theorem 18, prior topublication. We make use of this result to prove our Theorem 6. Finally, we thank Mo Dick Wong forhelpful comments and suggestions. A Convergence to the Gaussian Fields In this appendix we set out the proof of Theorem 3. As stated in the introduction, this theorem wasestablished by Assiotis & Keating and we are most grateful to Dr Assiotis for permitting us to use itand to give its proof here.We first need the following definition: Definition 25 (The Sobolev spaces H − (cid:15) ) . For s ∈ R , consider the space of formal Fourier series H s = f ∼ (cid:88) j ∈ Z f k e ijθ : (cid:88) j ∈ Z (1 + j ) s | f j | < ∞ (A.1) with inner product (cid:104) f, g (cid:105) s = (cid:88) j ∈ Z (1 + j ) s f j g ∗ j . (A.2) The closed subspace { f ∈ H s : f = 0 } will be denoted by H s . Remark 26. ( H s , (cid:104)· , ·(cid:105) s ) is a Hilbert space for all s ∈ R . For s ≥ H s is a subspace of H , i.e. thespace of square-integrable functions on the unit circle. For s < , H s can be interpreted as the dualspace of H − s , and as a space of generalized functions. Proof of Theorem 3 . We first recall that ln(1 − z ) = − ∞ (cid:88) k =1 z k k (A.3)for | z | ≤ , where for z = 1 both side equal −∞ . Thus by using the identity ln det = Tr ln we see thatfor the characteristic polynomial of U n ∈ O ( n ) ∪ Sp (2 n ) we have ln p n ( θ ) = − ∞ (cid:88) k =1 Tr ( U kn ) k e − ikθ . (A.4)This expansion, Theorem 2 and the fact that ( N j ) j ∈ N d = − ( N j ) j ∈ N then imply convergence of ( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) on cylinder sets.It thus remains to show tightness of ( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) n ∈ N in H − (cid:15) × H − (cid:15) , i.e. for every δ > we have to find a compact K δ ⊂ H − (cid:15) × H − (cid:15) for which sup n ∈ N P (( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) ∈ K cδ ) < δ. (A.5)By the Rellich-Kondrachov theorem we have that for any s ∈ ( − (cid:15), the closed ball B (0 , R ) H s of radius R > in H s is compact in H − (cid:15) . Thus when fixing s , choosing K δ = B (0 , R ( δ )) H s × B (0 , R ( δ )) H s andusing Chebyshev’s inequality, we get sup n ∈ N P (( (cid:60) ln p n ( θ ) , (cid:61) ln p n ( θ )) ∈ K cδ ) = sup n ∈ N P (cid:0) max (cid:8) ||(cid:60) ln p n ( θ ) || H s , ||(cid:61) ln p n ( θ ) || H s (cid:9) > R ( δ ) (cid:1) ≤ R ( δ ) sup n ∈ N E (cid:16) max (cid:110) ||(cid:60) ln p n ( θ ) || H s , ||(cid:61) ln p n ( θ ) || H s (cid:111)(cid:17) . (A.6)55e have sup n ∈ N E (cid:16) ||(cid:60) ln p n ( θ ) || H s (cid:17) = sup n ∈ N E (cid:16) ||(cid:61) ln p n ( θ ) || H s (cid:17) = sup n ∈ N E (cid:32) ∞ (cid:88) k =1 (1 + k ) s Tr ( U kn ) k (cid:33) = sup n ∈ N ∞ (cid:88) k =1 (1 + k ) s E (cid:0) Tr ( U kn ) (cid:1) k ≤ const sup n ∈ N ∞ (cid:88) k =1 (1 + k ) s min { k, n } k ≤ const ∞ (cid:88) k =1 (1 + k ) s k < ∞ , (A.7)where in the first inequality we used (2.11). Thus choosing R ( δ ) big enough we get (A.5). This finishesthe proof. B Construction of the Gaussian Multiplicative Chaos Measure In this section we "exponentiate" the field Y α,β in (2.15) to obtain the non-trivial Gaussian multiplicativechaos measure µ α,β . We follow the approach of Kahane in [24], with the only difference being that inour case the Gaussian field’s covariance function (2.16) has singularities not just on the diagonal, butalso on the antidiagonal.For k ∈ N define the truncated Gaussian fields Y ( k ) α,β ( θ ) = k (cid:88) j =1 √ j (cid:18) α cos( jθ ) (cid:18) N j ± η j √ j (cid:19) − iβ sin( jθ ) (cid:18) N j ± η j √ j (cid:19)(cid:19) = k (cid:88) j =1 √ j (cid:18) N j (2 α cos( jθ ) − iβ sin( jθ )) ± η j √ j (2 α cos( jθ ) − iβ sin( jθ )) (cid:19) (B.1)whose covariance functions are given byCov ( Y ( k ) α,β ( θ ) , Y ( k ) α,β ( θ (cid:48) ))= k (cid:88) j =1 j (cid:0) α − β ) cos( j ( θ − θ (cid:48) )) + 2( α + β ) cos( j ( θ + θ (cid:48) )) − iαβ sin( j ( θ + θ (cid:48) )) (cid:1) . (B.2)Define the measures µ ( k ) α,β on S ∼ [0 , π ) by µ ( k ) α,β ( dθ ) = e Y ( k ) α,β ( θ ) E (cid:16) e Y ( k ) α,β ( θ ) (cid:17) d θ = e αX ( k ) ( θ )+2 iβ ˆ X ( k ) ( θ ) − Var ( Y ( k ) α,β ( θ )) d θ. (B.3)For any measurable A ⊂ [0 , π ) and k ∈ N it holds by Fubini that E (cid:16) µ ( k ) α,β ( A ) (cid:17) = (cid:90) π A d θ, (B.4)56nd E (cid:16) µ ( k ) α,β ( A ) | σ ( N , ..., N k − ) (cid:17) = µ ( k − α,β ( A ) . (B.5)Thus, being a non-negative martingale, µ ( k ) α,β ( A ) converges a.s. to a random variable which will bedenoted by µ α,β ( A ) . One can show that a.s. the map A (cid:55)→ µ α,β ( A ) is a measure and we have thata.s. µ kα,β d −→ µ α,β in the space of Radon measures on S ∼ [0 , π ) , equipped with the topology of weakconvergence.Recall that I denotes either I (cid:15) = ( (cid:15), π − (cid:15) ) ∪ ( π + (cid:15), π − (cid:15) ) or [0 , π ) . For any measurable A ⊂ I the martingale (cid:16) µ ( k ) α,β ( A ) (cid:17) k ∈ N is bounded in L (and thus uniformly integrable), since by (4.54) and(4.58) we see that lim sup k →∞ E (cid:16) µ ( k ) α,β ( A ) (cid:17) = lim sup k →∞ (cid:90) A (cid:90) A e Cov ( Y ( k ) α,β ( θ ) ,Y ( k ) α,β ( θ (cid:48) )) d θ d θ (cid:48) ≤ (cid:90) A (cid:90) A | e iθ − e iθ (cid:48) | − α − β ) | e iθ − e − iθ (cid:48) | − α + β ) e αβ ln e i ( θ + θ (cid:48)− π ) d θ d θ (cid:48) < ∞ , (B.6)for α > − / and α − β < / in the case I = I (cid:15) , and α ≥ , α − β < / and α < in the case I = [0 , π ) . Thus, by (B.4), we get that E ( µ α,β ( A )) = lim k →∞ E (cid:16) µ ( k ) α,β ( A ) (cid:17) = (cid:90) π A d θ, (B.7)which implies that the event { µ α,β is the measure } does not have probability . Since that event isindependent of any finite number of the N k , k ∈ N , Kolmogorov’s zero-one law implies that this eventthen has probability . Thus µ α,β is a.s. non-trivial for α ≥ , α − β < / and α < , while whenrestricted to I (cid:15) the measure µ α,β is almost surely non-trivial for α > − / and α − β < / . In bothcases 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 [13]. We include it hereto make our account self-contained. We use Ψ to construct local parametrices for the RHP for theorthogonal polynomials in Section 7.5. We always assume that α , α > − and β , β ∈ i R (in [13]also the more general case of α , α , β , β ∈ C was considered). RH Problem for Ψ (a) Ψ : C \ Γ → C × is analytic, where Γ = ∪ k =1 Γ k , Γ = i + e iπ R + , Γ = i + e iπ R + Γ = − i + e iπ R + , Γ = − i + e iπ R + , Γ = − i + R + , Γ = i + R + , Γ = [ − i, i ] , with the orientation chosen as in Figure 6 ("-" is always on the RHS of the contour).(b) Ψ satisfies the jump conditions Ψ( ζ ) + = Ψ( ζ ) − J k , ζ ∈ Γ k , (C.1)57here J = (cid:18) e πi ( α − β ) (cid:19) , J = (cid:18) − e − πi ( α − β ) (cid:19) , (C.2) J = (cid:18) − e πi ( α − β ) (cid:19) , J = (cid:18) e − πi ( α − β ) (cid:19) (C.3) J = e πiβ σ , J = e πiβ σ , (C.4) J = (cid:18) − (cid:19) . (C.5)(c) We have in all regions: Ψ( ζ ) = (cid:18) I + Ψ ζ + Ψ ζ + O ( ζ − ) (cid:19) P ( ∞ ) ( ζ ) e − is ζσ , as ζ → ∞ , (C.6)where P ( ∞ ) ( ζ ) = (cid:18) is (cid:19) − ( β + β ) σ ( ζ − i ) − β σ ( ζ + i ) − β σ , (C.7)with the branches corresponding to the arguments between and π , and where s ∈ − i R + .(d) The functions F and F defined in (C.8), (C.10), (C.12) and (C.14) below are analytic functionsof ζ at i and − i respectively. + i − i (cid:18) − (cid:19) (cid:18) e πi ( α − β ) (cid:19)(cid:18) − e − πi ( α − β ) (cid:19) e πiβ σ e πiβ σ (cid:18) e − πi ( α − β ) (cid:19)(cid:18) − e πi ( α − β ) (cid:19) IIIV VIIII IVFigure 6: The jump contour and jump matrices for Ψ .The solution Ψ = Ψ( ζ ; s ) to this RHP not only depends on the complex variable ζ , but also on thecomplex parameter s ∈ − i R + . Without the additional condition (d) on the behaviour of Ψ near thepoints ± i , the RHP wouldn’t have a unique solution. If α / ∈ N ∪ { } , define F ( ζ, s ) by the equations Ψ( ζ ; s ) = F ( ζ, s )( ζ − i ) α σ G j , ζ ∈ region j, (C.8)58here j ∈ { I, II, III, V I } , and where ( ζ − i ) α σ is taken with the branch cut on i + e πi R + , withthe argument of ζ − i between − π/ and π/ . The matrices G j are piecewise constant matricesconsistent with the jump relations; they are given by G III = (cid:18) g (cid:19) , g = − i sin(2 α ) ( e πiα − e − πiβ ) ,G V I = G III J − , G I = G V I J , G II = G I J . (C.9)It is straighforward to verify that F has no jumps near i , and it is thus meromorphic in a neighborhoodof i , with possibly an isolated singularity at i .Similarly, for ζ near − i , if α / ∈ N ∪ { } , we define F by the equations Ψ( ζ ; s ) = F ( ζ, s )( ζ + i ) α σ H j , ζ ∈ region j, (C.10)where j ∈ { III, , IV, V, V I } , where ( ζ + i ) α σ is taken with the branch cut on − i + e πi R + , with theargument of ζ + i between − π/ and π/ , and where the matrices H j are piecewise constant matricesconsistent with the jump relations; they are given by H III = (cid:18) h (cid:19) , h = − i sin(2 α ) ( e πiβ − e − πiα ) ,H IV = G III J − , H V = G IV J − , H V I = H V J . (C.11)Similarly as at i , one shows using the jump conditions for Ψ that F is meromorphic near − i with apossible singularity at − i .If α ∈ N ∪{ } , the constant g and the matrices G j are ill-defined, and we need a different definitionof F : Ψ( ζ ; s ) = F ( ζ ; s )( ζ − i ) α σ (cid:18) g int ln( ζ − i )0 1 (cid:19) G j , ζ ∈ region j, (C.12)where g int = e − πiβ − e πiα πie πiα , (C.13)and G III = I , and the other G j ’s are defined as above by applying the appropriate jump conditions.Thus defined, F has no jumps in a neighborhood of i . Similarly, if α ∈ N ∪ { } , we define F by theexpression: Ψ( ζ ; s ) = F ( ζ ; s )( ζ + i ) α σ (cid:18) e − πiα − e πiβ πie πiα (cid:19) H j , ζ ∈ region j, (C.14)with H III = I , and the other H j ’s expressed via H III as in (C.11). Then F has no jumps near − i .Given parameters s, α , α , β , β , the uniqueness of the function Ψ which satisfies RH conditions(a) - (d) can be proven by standard arguments.In Section 3 of [13] it was shown that for α , α , α + α > − and β , β ∈ i R , the RHP is solvablefor any s ∈ − i R + . Furthermore they analyzed the RHP asymptotically as s → − i ∞ and s → − i + . Remark 27. We have to be careful what their ( α , α , β , β ) correspond to in our case, when using Ψ from [13]. In their paper α , β correspond to the singularity left of the merging point and α , β correspond to the singularity to the right of the merging point, while for us in the case + for example, α , β are right and α , β are left. Riemann-Hilbert Problem for M This appendix is a mostly verbatim transfer of Section 4 of [13]. We include it here to make ouraccount self-contained. Let α > − and β ∈ i R . In Section 4.2.1 of [12], see also [14, 23, 27], a function M = M ( α,β ) was constructed explicitly in terms of the confluent hypergeometric function, which solvesthe following RH problem: RH Problem for M (a) M : C \ (cid:16) e ± πi R ∪ R + (cid:17) → C × is analytic,(b) M has continuous boundary values on e ± πi R ∪ R + \ { } related by the conditions: M ( λ ) + = M ( λ ) − (cid:18) e πi ( α − β ) (cid:19) , λ ∈ e iπ R + , (D.1) M ( λ ) + = M ( λ ) − (cid:18) − e − πi ( α − β ) (cid:19) , λ ∈ e iπ R + , (D.2) M ( λ ) + = M ( λ ) − (cid:18) e πi ( α − β ) (cid:19) , λ ∈ e iπ R + , (D.3) M ( λ ) + = M ( λ ) − (cid:18) − e − πi ( α − β ) (cid:19) , λ ∈ e iπ R + , (D.4) M ( λ ) + = M ( λ ) − ( λ ) e πiβσ , λ ∈ R + , (D.5)where all the rays of the jump contour are oriented away from the origin.(c) Furthermore, in all sectors, M ( λ ) = ( I + M λ − + O ( λ − )) λ − βσ e − λσ , as λ → ∞ , (D.6)where < arg λ < π , and M = M ( α,β )1 = (cid:32) α − β − e − πiβ Γ(1+ α − β )Γ( α + β ) e πiβ Γ(1+ α + β )Γ( α − β ) − α + β (cid:33) . (D.7) (cid:18) e πi ( α − β ) (cid:19)(cid:18) − e − πi ( α − β ) (cid:19) e πiβσ (cid:18) − e − πi ( α − β ) (cid:19)(cid:18) e πi ( α − β ) (cid:19) 243 15Figure 7: The jump contour and jump matrices for M.We use M to construct local parametrices for the RHP of the orthogonal polynomials in Section7.6. For that we also need the local behaviour of M at zero in region 3, i.e. the region between the60ines e πi R + and e πi R + . Write M ≡ M (3) in this region. It is known (see Section 4.2.1 of [12]) that M (3) can be written in the form M (3) ( λ ) = L ( λ ) λ ασ ˜ G , α / ∈ N ∪ { } , (D.8)with the branch of λ ± α chosen with < arg λ < π . Here L ( λ ) = e − λ/ e − iπ ( α + β ) Γ(1+ α − β )Γ(1+2 α ) ϕ ( α + β, α, λ ) e iπ ( α − β ) Γ(2 α )Γ( α + β ) ϕ ( − α + β, − α, λ ) − e − iπ ( α − β ) Γ(1+ α + β )Γ(1+2 α ) ϕ (1 + α + β, α, λ ) e iπ ( α + β ) Γ(2 α )Γ( α − β ) ϕ (1 − α + β, − α, λ ) (D.9)is an entire function, with ϕ ( a, b ; z ) = 1 + ∞ (cid:88) n =1 a ( a + 1) · · · ( a + n − c ( c + 1) · · · ( c + n − z n n ! , c (cid:54) = 0 , − , − , ..., (D.10)and ˜ G is the constant matrix ˜ G = (cid:18) g (cid:19) , ˜ g = − sin π ( α + β )sin 2 πα . (D.11)If α is an integer, we have M (3) ( λ ) = ˜ L ( λ ) λ ασ (cid:18) m ( λ )0 1 (cid:19) , (D.12) m ( λ ) = ( − α +1 π sin π ( α + β ) ln( λe − iπ ) , (D.13)where ˜ L ( λ ) is analytic at , and the branch of the logarithm corresponds to the argument of λ between and π . 61 eferences [1] Arguin, L.P., Belius, D. and Bourgade, P., 2017. Maximum of the characteristic polynomial ofrandom unitary matrices. 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