Moments of C$β$E field partition function, $\mathsf{Sine}_β$ correlations and stochastic zeta

MOMENTS OF C β E FIELD P AR TITION FUNCTION, Sine β CORRELA TIONS AND STOCHASTIC ZET A THEODOROS ASSIOTIS AND JOSEPH NAJNUDEL Abstract W e prove a conjecture of Fyodorov and Keating on the supercritical moments of the partition function of the C β E ﬁeld or equivalently the supercritical moments of moments of the characteristic polynomial of the C β E ensemble for general β > 0 and general real moment exponents. Moreover , we give the ﬁrst expression for all correlation functions of the Sine β point process for all β > 0. The main object behind both results is the Hua-Pickr ell stochastic zeta function introduced by Li and V alk ´ o. C ontents 1 Introduction 1 2 Proofs of main results 9 3 Proof of the main bound 21 References 31 1 I ntroduction The purpose of this paper is to solve two seemingly di ﬀ erent problems in random ma- trix theory: prove a conjectur e of Fyodorov and Keating on moments of characteristic polynomials of the circular β ensemble in Theorem 1.7 and give a ﬁrst expression for all β > 0 of all correlation functions of the Sine β point process in Theor em 1.8. Although these may seem quite di ﬀ erent, the basic object behind both statements, the Hua-Pickr ell stochastic zeta function of Li and V alk ´ o, and also the proof technique, based on random orthogonal polynomials on the unit circle, ar e the same. W e begin the article by discussing the general background. 1.1 M oments of the p ar tition function of the C β E field Let T be the unit cir cle that we identify with [ − π, π ). Deﬁne the C β E N ensemble to be the probability measur e on T N given by: 1 Z N ,β Y 1 ≤ j < k ≤ N    e i θ j − e i θ k    β d θ 1 · · · d θ N , Z N ,β = (2 π ) N Γ  β N 2 + 1  Γ  β 2 + 1  . W e denote by E C β E N expectation with respect to it. For β = 2 this is the law of eigenvalues of a Haar distributed random unitary matrix and is known as the Circular Unitary Ensemble 1 C β E moments , Sine β correla tions and stochastic zet a (CUE), see [37]. The choices β = 1 , 4 also have a natural conjugation-invariant random matrix interpretation, see [37]. Deﬁne the characteristic polynomial X N ( z ), with (e i θ j ) N j = 1 C β E N -distributed, by X N ( z ) def = N Y j = 1  1 − z e − i θ j  and the C β E N ﬁeld L N ( θ ) def = log | X N (e i θ ) | on T . Deﬁnition 1.1. Let β > 0 , k , s ∈ R + . Deﬁne the moments of the partition function of L N or moments of moments of X N , where the points { e i θ j } are C β E N -distributed, by M ( β ) N ( k ; s ) def = E C β E N       1 2 π Z π − π e 2 s L N ( θ ) d θ ! k       = E C β E N       1 2 π Z π − π    X N (e i θ )    2 s d θ ! k       . These quantities wer e ﬁrst considered by Fyodorov and Keating [45] (ﬁrst in the case β = 2 of CUE, then for general β > 0 [43, 53]) who made various predictions about their asymptotic behaviour as N → ∞ depending on the values of k , s , β . Such predictions have then been developed to predictions in number theory , for generalised moments of the Riemann zeta function, by Bailey and Keating in [14], following the Keating-Snaith philosophy [52, 51]. • The GMC regime. In a certain parameter range these asymptotics are conceptually well-understood owing to a connection to log-correlated Gaussian ﬁelds and the Gaussian multiplicative chaos (GMC), see [15, 53]. It is known that, with conver gence in distribution (denoted by d − → ) taking place in a negative Sobolev space, see [48], L N ( • ) d − → β − 1 2 G ( • ) , (1) where G is the Gaussian free ﬁeld on T with covariance: E  G ( x ) G ( y )  = − log    e i x − e i y    . From G one can deﬁne, for γ < √ 2, a non-trivial random measure GMC γ on T given informally by the expression: GMC γ (d θ ) = “ e γ G ( θ ) E  e γ G ( θ )  d θ = e γ G ( θ ) − γ 2 2 E [ G 2 ( θ ) ] d θ ” , and this is called the GMC on T , see [70, 18]. Hence, from (1) it is natural to expect that for 2 s 2 < β one should have the following weak convergence of random measures:    X N (e i θ )    2 s E C β E N     X N (e i θ )    2 s  d θ d − → GMC 2 s β − 1 2 ( d θ ) . (2) This was ﬁrst proven in the case β = 2 of CUE using Riemann-Hilbert techniques [76, 62] and recently established for general β > 0 [56] using a probabilistic approach. The critical case β = 2 s 2 , for which a di ﬀ erent renormalisation is required, is of particular inter est 2 T . A ssiotis and J. N ajnudel because of its relation to the following conjecture of Fyodorov-Hiary-Keating [45, 44] on the maximum of the ﬁeld L N on T , as N → ∞ : max θ ∈ T L N ( θ ) − s 2 β  log N − 3 4 log log N  d − → G β , (3) where G β is an explicit random variable, closely r elated to critical GMC, see [1, 66, 24, 67] for progr ess. An analogous conjecture exists for the Riemann ζ -function on the critical line ℜ ( z ) = 1 2 , motivated by the above random matrix prediction, see [45, 44, 61, 2, 47, 3, 4]. Back to M ( β ) N ( k ; s ), with 2 s 2 < β so that (2) holds, known as the moment-subcritical regime 2 ks 2 < β , M ( β ) N ( k ; s ) grows like a multiplicative constant times N 2 β ks 2 , see [69, 53, 56]. In this case, the leading order coe ﬃ cient is basically given by the moments of the total mass of the GMC, see [69, 53, 56] and the discussion in Remark 2.4 below . These moments have now been computed rigorously , see [69, 25] and [42] for the original physics prediction of Fyodorov-Bouchaud. In the moment-critical regime β = 2 ks 2 , M ( β ) N ( k ; s ) is conjectured by Keating and W ong [53] to grow like N log N times an explicit leading or der coe ﬃ cient again r elated to GMC, see [53]. For β = 2, k ∈ N , Keating and W ong proved their conjecture in the same paper . • The supercritical regime. When 2 k s 2 > β , the connection to moments of the GMC breaks down and it is not as clear how and why M ( β ) N ( k ; s ) should behave asymptotically . Nevertheless, the following remarkable prediction was made in [45, 43, 53], see Conjecture 2.4 in [15] for the special case β = 2. Conjecture 1.2. Let β > 0 , k ∈ N , s ∈ R + . For 2 ks 2 > β , as N → ∞ , M ( β ) N ( k ; s ) ∼ c ( β ) ( k ; s ) N 2 k 2 s 2 β − 1 − k + 1 . (4) The conjecture also makes sense for real exponents k and we will establish that as well for a certain parameter range. The leading order coe ﬃ cient c ( β ) ( k ; s ) was unknown at the time Conjecture 1.2 was ﬁrst made. Since then there has been signiﬁcant work on this problem using analytic and combinatorial techniques. V arious partial results have been obtained with di ﬀ er ent expr essions given for c ( β ) ( k ; s ), ranging from repr esentations in terms of Painlev ´ e equations to volumes of certain polytopes, that we now survey . Firstly , for k = 1, M ( β ) N (1; s ) can be computed completely explicitly using the Selber g integral and the N → ∞ asymptotics can readily be established, see [52, 32]. For k , 1, work on Conjecture 1.2 mainly focused on the case β = 2 of CUE for which a number of tools are available: • β = 2 . The asymptotics for k = 2, s ∈ R + , β = 2 were ﬁrst established by Claeys and Krasovsky in [30] using Riemann-Hilbert problem techniques who also gave a rep- resentation of c (2) (2; s ) in terms of the Painlev ´ e V equation. This follows from computing the asymptotics of N × N T oeplitz determinants with symbols with two merging Fisher- Hartwig singularities, a problem with very long history , see [77, 30, 27, 33, 22]. T wo alternative proofs of the asymptotics for k = 2, s ∈ N , β = 2 were given in [50]. One proof makes use of a multiple contour integral representation coming from [31] and the other one is algebraic combinatorial making use of symmetric function theory , see [23]. The complex analytic and combinatorial proofs were then extended in [13] and [11] respec- tively to prove the asymptotics for general k ∈ N , s ∈ N , β = 2. The combinatorial proof gives an expression for c (2) ( k ; s ) in terms of the volume of a Gelfand-T setlin type polytope. This expression, for k = 2, can be used to give an alternative proof of the repr esentation 3 C β E moments , Sine β correla tions and stochastic zet a of c (2) (2; s ) in terms of the Painlev ´ e V equation [17, 16]. For applications of these methods to allied problems, see [7, 9, 15]. Finally , for k ∈ N and s ∈ R + so that k s 2 > 1, Fahs in [33] using Riemann-Hilbert problem techniques gave the following upper and lower bound: M (2) N ( k ; s ) = e O k , s (1) N 2 k 2 s 2 β − 1 − k + 1 . Obtaining some expr ession for the e O k , s (1) term above when s ∈ R + , k > 2, and further connecting it to integrable systems is a well-known folklore problem in the Riemann- Hilbert pr oblem community . It boils down to a delicate understanding of asymptotics of N × N T oeplitz determinants with multiple (growing in number with k ) merging Fisher- Hartwig singularities that goes beyond Fahs’ tour-de-for ce work [33]. • β , 2 . For general β , 2 very little was known. Partial r esults, for k , s ∈ N , wer e ﬁrst obtained in [5], see also more recently [39], using combinatorics. The asymptotics were established for k = 2, s ∈ N , 4 s 2 > β and for k ∈ N , k > 2, s ∈ N , β ≤ 2. The leading order coe ﬃ cient was given as an integral over the polytope mentioned above. The integrand can be written in terms of the Dixon-Anderson probability distribution, see [37]. It was until now unclear whether a natural probabilistic object existed which would replace GMC in the supercritical regime. The results above did not give any indication. Our ﬁrst main result, Theorem 1.7 below , establishes the asymptotics of M ( β ) N ( k ; s ) in the supercritical regime for general β > 0 and generic real moment exponents k , s , in particular proving Conjecture 1.2, and reveals what this object is. Our approach is di ﬀ erent from previous works and is pr obabilistic in nature. 1.2 Sine β correla tions The general β > 0, Sine β point process, arguably one of the most fundamental objects in random matrix theory , is the universal scaling limit of general β -ensembles: see [73, 55, 21]. It was ﬁrst constructed for all β > 0 by V alk ´ o and V irag [73] and around the same time by Killip and Stoiciu [55]. The special case Sine 2 is the well-known determinantal point process [19, 49, 37] with corr elations given by the famous sine kernel [37]. The other classical values β = 1 , 4 also possess special algebraic structur e, with corr elations now given in terms of Pfa ﬃ ans, see [37]. W e now give the pr ecise deﬁnition of Sine β , see [73, 74, 55], via its counting function. Equivalent characterisations exist, such as being the spectrum of a certain random Dirac operator , see for example [74]. Deﬁne ( w C ( t )) t ∈ ( −∞ , ∞ ) to be a two-sided complex Brownian motion. Then, consider the unique strong solution ( p ( β ) x ( t )) t ∈ ( −∞ , ∞ ) to the one-parameter , in λ ∈ R , coupled (as they all share w C ) family of stochastic di ﬀ erential equations (SDE), d p ( β ) λ ( t ) = λ β 4 e β 4 t d t + ℜ  e − i p ( β ) λ ( t ) − 1  d w C ( t )  , (5) with t ∈ ( −∞ , ∞ ), and with initial condition lim t →−∞ p ( β ) λ ( t ) = 0. Finally , let U be a uniform random variable on [0 , 2 π ) independent of w C . Deﬁnition 1.3. For β > 0 , the Sine β point process is characterised as Sine β d = n λ ∈ R : p ( β ) λ (0) = U mod 2 π o . Here and thr oughout the paper d = denotes equality in distribution. 4 T . A ssiotis and J. N ajnudel Moving on to correlations, for a simple point process Ξ on R we denote by Ξ ( A ) the number of points of Ξ in A , where A is a Borel subset of R . The m -th correlation function of Ξ , if it exists, is the symmetric function ρ ( m ) : R m → R + that satisﬁes: for A 1 , . . . , A m disjoint Borel sets, E         m Y j = 1 Ξ ( A j )         = Z A 1 · · · Z A m ρ ( m ) ( x 1 , . . . , x m )d x 1 · · · d x m . The fact that corr elation functions of all orders for Sine β exist follows from moment bounds from [73], see the discussion in Section 2.1 of [68]. In particular , the following deﬁnition makes sense. Deﬁnition 1.4. For β > 0 and m ∈ N we deﬁne ρ ( m ) β to be the m-th correlation function of Sine β . The problem of giving an expression for ρ ( m ) β and obtaining some information out of it is a long-standing one. There has been very signiﬁcant body of work by Forrester in the case of even integer β = 2 n . Forrester using generalised hyper geometric functions and Jack polynomial theory obtains a multiple integral expression for the corr elations and also r elates them to higher order di ﬀ er ential equations, see [34, 35, 36, 37, 41, 40, 38]. Moreover , a certain expression for the correlations for rational values of β was obtained in the unpublished manuscript by Okounkov [63]. It was only very recently however , that Qu and V alk ´ o [68] obtained the ﬁrst expression for the pair corr elation ρ (2) β (0 , x ) of Sine β for general β > 0. This expression is given in terms of expectations of generalisations of the di ﬀ usions from (5) above. W e recall their formula, and compare to the one we obtain, in (14). Using it they can remarkably compute precise large x asymoptotics of ρ (2) β (0 , x ). Mor eover , for β = 2 n they can get a certain power series repr esentation for ρ (2) β (0 , x ). Our second main result Theorem 1.8 is an expression for all order correlation functions of Sine β for all β > 0. In some sense this gives implicitly a positive answer to Pr oblem 7 of [68] which asks for an expression of higher order correlation functions of Sine β in terms of di ﬀ usions like (5). W e also expect that the expression we obtain for the correlations will be useful in problems of inﬁnite-dimensional dynamics r elated to random matrices, see [64, 65, 71, 72]. 1.3 M ain resul ts In order to state our main results we need to introduce certain quantities from [74, 60]. For δ > 0 consider the unique strong solution ( p ( β,δ ) λ ( t )) t ∈ ( −∞ , ∞ ) to the generalisation of the coupled SDE (5) (which corresponds to δ = 0), indexed by λ ∈ R , d p ( β,δ ) λ ( t ) = λ β 4 e β 4 t d t + ℜ  e − i p ( β,δ ) λ ( t ) − 1  ( d w C ( t ) − i δ d t )  , (6) with t ∈ ( −∞ , ∞ ) and initial condition lim t →−∞ p ( β,δ ) λ ( t ) = 0. Let Θ δ be a random variable on [0 , 2 π ) with probability density: 1 2 π Γ (1 + δ ) 2 Γ (1 + 2 δ )    1 − e i θ    2 δ independent of the two-sided complex Brownian motion w C . Observe that Θ 0 d = U . 5 C β E moments , Sine β correla tions and stochastic zet a Deﬁnition 1.5. Deﬁne the Hua-Pickrell point process H P β,δ with parameters β, δ > 0 : H P β,δ d = n λ ∈ R : p ( β,δ ) λ (0) = Θ δ mod 2 π o . (7) Note that, H P β, 0 d = Sine β . H P β,δ can in fact be considered for ℜ ( δ ) > − 1 / 2 but we will not need this mor e general parameter range in this paper . For β = 2, it is well-known that H P 2 ,δ is determinantal with an explicit corr elation kernel generalising the sine kernel, see [20, 78]. Like Sine β , H P β,δ has multiple characterisations, including as the spectrum of a more general random Dirac operator , see [74, 60]. W e now deﬁne the Hua-Pickrell stochastic zeta function ξ β,δ of Li and V alk ´ o [60]. W e note that Li-V alk ´ o gave a di ﬀ erent equivalent repr esentation of it as a power series, see [60], and to get the principal value product in Deﬁnition 1.6 additional ar guments are required, see Proposition 2.8. The special case, with δ = 0, corresponding to Sine β was studied earlier by V alk ´ o and V irag [75] while the determinantal case of Sine 2 was introduced and studied by Chhaibi-Najnudel-Nikeghbali in [26]. Deﬁnition 1.6. Let β > 0 , δ > 0 . Deﬁne the following random entire function ξ β,δ ( z ) = lim R →∞ Y x ∈H P β,δ | x | < R  1 − z x  . (8) The fact that almost sur ely the principal value pr oduct in (8) conver ges uniformly on compact sets in z ∈ C will be a consequence of the proof of Proposition 2.8. W e will also need the special function Y β ( z ) = β 2 log G 1 + 2 z β ! −  z − 1 2  log Γ 1 + 2 z β ! + Z ∞ 0 1 2 x − 1 x 2 + 1 x (e x − 1) ! e − xz − 1 e x β/ 2 − 1 d x + z 2 β + z 2 , where G is the Barnes G-function. For special values of β , including β ∈ 2 N , Y β ( z ) has a much simpler expression, see Lemma 7.1 of [32]. W rite, for a ∈ R , R ≥ a = [ a , ∞ ). The following is the ﬁrst main result of the paper . Theorem 1.7. Let k ∈ N , s ∈ R + , β > 0 such that 2 ks 2 > β . Then, as N → ∞ , 1 N 2 k 2 s 2 β − 1 − k + 1 M ( β ) N ( k ; s ) − → F ( β ) k ; s E       Z ∞ −∞    ξ β, ks ( x )    2 s d x ! k − 1       , (9) where F ( β ) k ; s is given explicitly F ( β ) k ; s = (2 π ) 1 − k e Y β ( 1 + 2 ks − β/ 2 ) − 2 Y β ( 1 + ks − β/ 2 ) + Y β (1 − β/ 2) . (10) The same result holds for k ∈ R ≥ 1 whenever both 2 s 2 (2 k − ⌈ k ⌉ ) > β and, for k > 1 , 4 ⌈ k − 1 ⌉ ( k − ⌈ k − 1 ⌉ ) s 2 > β . The result is optimal for k ∈ N and completely proves Conjecture 1.2. The restriction for k ∈ R ≥ 1 \ N is technical and due to the fact that in the pr oof we compare to the next larger integer moment which we can contr ol explicitly . This also explains the fact that 6 T . A ssiotis and J. N ajnudel both restrictions are worse when k is just above an integer while they get closer to optimal as k approaches an integer from below . W e note that before our work Conjecture 1.2 had not been proven for any non-integer value of the exponent k , even when β = 2. By comparing to previous r esults in the literature [30, 50, 11, 13, 15, 33, 5] one r eadily obtains expressions for the integer moments of integrals of | ξ 2 , ks | 2 s in terms of Painlev ´ e equations and volumes of polytopes. It would be interesting and non-trivial to obtain them directly fr om ξ β, ks . Going further , it would be very interesting to use expression (9) along with the various descriptions of ξ β,δ to obtain new explicit information about the limit. W e mention in passing that joint moments of the T aylor coe ﬃ cients of ξ 2 ,δ , for suitable δ , also describe the limiting joint moments of characteristic polynomials of random unitary matrices along with their derivatives of any order , see [12, 8, 10]. Finally , we expect that an analogous strategy to the one we employ here for Theorem 1.7, along with additional technical innovations, could be used to tackle the problem of asymptotics of supercritical moments of moments for the orthogonal and unitary symplectic groups and for the Gaussian unitary ensemble (and its β -ensemble version) for which much less is known, see [46, 29, 28]. W e now move to our next main result on the corr elation functions ρ ( m ) β of Sine β . Theorem 1.8. For all β > 0 and m ∈ N , we have: ρ ( m ) β ( x 1 , . . . , x m ) = C ( m ) β Y 1 ≤ i < j ≤ m    x i − x j    β E         m Y j = 2     ξ β, m β/ 2  x j − x 1      β         , (11) where the constant C ( m ) β is explicitly given by: C ( m ) β = 2 m  β 2 − 1  π m β β m / 2 e Y β ( 1 + β ( m − 1 2 ) ) − 2 Y β  1 + β 2 ( m − 1)  + Y β  1 − β 2  . (12) The correlation functions ρ ( m ) β are uniformly bounded on R m . Mor eover , the function ( x 1 , . . . , x m ) 7→ E         m Y j = 2     ξ β, m β/ 2  x j − x 1      β         (13) is continuous and strictly positive for all ( x 1 , . . . , x m ) ∈ R m . A number of comments are in order . Firstly , using (11) it should be possible to prove continuity of β 7→ ρ ( m ) β ( x 1 , . . . , x m ). However , this requir es additional technical e ﬀ orts and we will not attempt to do this here. Moreover , it is easy to see from (11) that the even integer β = 2 n case is special. One can expand the β = 2 n powers of the entir e function z 7→ ξ β, m β/ 2 ( z ) and (formally) exchange the inﬁnite series and expectation. The computation then boils down to joint moments of the T aylor coe ﬃ cients of ξ β, m β/ 2 . These can in principle be computed using the Brownian motion repr esentation fr om [60] or , at least for β = 2, using exchangeability theory formulas or connections to Painlev ´ e equations as developed in [10]. Performing this computation systematically is an interesting but formidable task. W e also note that certain properties of the correlation functions, namely that they are uniformly bounded and behave like Q i < j | x i − x j | β , up to a well-behaved (continuous 7 C β E moments , Sine β correla tions and stochastic zet a and non-vanishing) multiplicative factor G β ; m deﬁned by (13), when variables are close, have signiﬁcant consequences in studying inﬁnite-dimensional log-interacting di ﬀ usions using Dirichlet form theory , see [64, 65]. The exact form of G β ; m , which for generic β , is unlikely to have a completely explicit evaluation is not important for such considerations. Finally , by comparing with the formula of Qu and V alk ´ o from [68] for the pair corre- lation, which is the only other known expression for general β > 0 (only for m = 2), we obtain the following intriguing equality , for x > 0, E     ξ β,β ( x )    β  = 1 C (2) β x β         1 4 π 2 + 1 2 π 2 ∞ X k = 1 Q k − 1 j = 0  − β 2 + j  Q k − 1 j = 0  1 + β 2 + j  E h cos  k p ( β,β/ 2) x (0) i         . (14) Observe that, the left hand side of (14) involves the H P β,β point process while the right hand side involves the family (in the variable x ∈ R ) of di ﬀ usions ( p ( β,β/ 2) x ( t )) t ∈ ( −∞ , ∞ ) and thus H P β,β/ 2 . It would be inter esting to relate the two expr essions directly . Formula (11) is very-well adapted to looking at the asymptotics when the variables ( x 1 , . . . , x m ) merge. For example, the following corollary , which is immediate from Theo- rem 1.8, provides an answer to leading order for Pr oblem 5 of [68] (in fact we consider the generalisation to all m -point correlation functions). It would be interesting to understand the lower order terms in | x i − x j | , at least for m = 2. Corollary 1.9. For β > 0 , m ∈ N , x ∗ ∈ R lim ( x 1 ,..., x m ) → ( x ∗ ,..., x ∗ ) ρ ( m ) β ( x 1 , . . . , x m ) Q 1 ≤ i < j ≤ m    x i − x j    β = C ( m ) β . (15) Finally , it is interesting to note that the answers to the two problems addressed in Theorem 1.7 and Theorem 1.8 are in fact directly connected in one special case. Namely , by comparing the two formulae for k = 2 , s = β/ 2 and m = 2 respectively , when β > 1 we have lim N →∞ N 1 − 2 β M ( β ) N (2; β/ 2) = πβ β 2 β − 1 Z ∞ −∞ 1 | x | β ρ (2) β (0 , x )d x . 1.4 O n the proof Our starting point to prove Theorems 1.7 and 1.8 is to write in Propositions 2.3 and 2.5 respectively both the moments of moments and the correlation functions of C β E N in terms of two di ﬀ erent types of moments of the normalised characteristic polynomial q β,δ N of the circular Jacobi beta ensemble CJ N ,β,δ , see Deﬁnitions 2.1 and 2.2. The expression we obtain for M ( β ) N ( k ; s ) in Proposition 2.3 is non-obvious and very judiciously chosen. It is worth comparing it with the natural r ewriting of the M ( β ) N ( k ; s ) formula to deal with the GMC regime, see Remark 2.4. It is then known from the work of Li and V alk ´ o [60], see Proposition 2.7, and [26, 75, 6, 57, 58] for related models, that under a certain scaling, q β,δ N converges in distribution, in the topology on entire functions induced by uniform convergence on compact sets in C , to a random entire function ˜ ξ β,δ which we subsequently show in Proposition 2.8 that is equal in distribution to ξ β,δ from Deﬁnition 1.6. T o then prove the desired convergence of moments and establish Theorems 1.7 and 1.8 we r equire very precise moment estimates which ar e uniform in both N and the variable 8 T . A ssiotis and J. N ajnudel of the polynomial q β,δ N . All of these estimates will be derived from the same main bound of Theorem 2.11 which is the main technical innovation of our paper . In equivalent form, see in particular Proposition 3.5, this uniform bound says the following about joint moments of ( | X N (e i θ j ) | 2 s j , j = 1 , . . . , ℓ ), E C β E N         ℓ Y j = 1     X N  e i θ j      2 s j         ≤ C β ; ℓ ;( s j ) ℓ j = 1 N 2 β P ℓ j = 1 s 2 j Y 1 ≤ j < m ≤ ℓ min          N , 1 2     sin  θ j − θ m 2               4 β s j s m for a constant C β ; ℓ ;( s j ) ℓ j = 1 which is independent of both N and the ( θ j ) ℓ j = 1 . A testament to the fact that this bound is sharp is that it captures perfectly the integrability requir ed to get the optimal range, for k ∈ N , in Theorem 1.7, see in particular Proposition 2.17 and Lemma 2.18. Moreover , this result specialises for β = 2 to the main result of Fahs [33], when applied to the CUE characteristic polynomial, which is proven using a very delicate asymptotic analysis of T oeplitz determinants with symbols with multiple merging Fisher- Hartwig singularities. For β , 2 such T oeplitz determinant structure is absent. Instead we make use of random orthogonal polynomials on the unit circle ﬁrst introduced by Killip and Nenciu in [54] in relation to C β E N . Parts of the argument are guided, rather implicitly , by some intuition coming from branching structur es used in the study of conjecture (3) on the maximum of L N ( · ), see for example [24]. W e believe this bound on joint moments is of independent interest and will have other applications in the futur e. 2 P roofs of main resul ts In this section we prove all our results except the main bound in Theor em 2.11 whose rather lengthy proof is deferr ed to Section 3. Deﬁnition 2.1. Deﬁne, for β, δ > 0 , the circular Jacobi beta ensemble CJ N ,β,δ to be the pr obability measure on T N given by 1 Z N ,β,δ Y 1 ≤ j < k ≤ N    e i θ j − e i θ k    β N Y j = 1    1 − e i θ j    2 δ d θ 1 · · · d θ N , where Z N ,β,δ is an explicit normalisation constant. Observe that, when δ = 0, CJ N ,β, 0 is simply C β E N . CJ N ,β,δ can in fact be deﬁned more generally for ℜ ( δ ) > − 1 / 2 but we will not need this here. The following notation will be very convenient. Deﬁnition 2.2. For β, δ > 0 , N ∈ N , deﬁne the random polynomial q β,δ N by q β,δ N ( z ) def = N Y j = 1 z − e i θ j 1 − e i θ j (16) where the random points (e i θ j ) N j = 1 are assumed to be CJ N ,β,δ -distributed. The following two exact ﬁnite N results will be our starting points for Theorem 1.7 and Theorem 1.8 r espectively . 9 C β E moments , Sine β correla tions and stochastic zet a Proposition 2.3. For β > 0 , k , s ∈ R + , we have M ( β ) N ( k ; s ) = E C β E N h | X N (1) | 2 ks i E       1 2 π Z π − π     q β, ks N (e i θ )     2 s d θ ! k − 1       . Proof. W e write, using rotation invariance of C β E N M ( β ) N ( k ; s ) = 1 2 π Z π − π E C β E N          X N (e i ϕ )    2 s 1 2 π Z π − π    X N (e i θ )    2 s d θ ! k − 1       d ϕ = E C β E N          | X N (1) | 2 ks         1 2 π Z π − π    X N (e i θ )    2 s | X N (1) | 2 s d θ         k − 1          . Then, a simple change of measure fr om C β E N to CJ N ,β, ks gives the result. □ Remark 2.4. It is worth comparing the formula for M ( β ) N ( k ; s ) obtained in Proposition 2.3 to the more obvious expr ession M ( β ) N ( k ; s ) =  E C β E N h | X N (1) | 2 s i k E C β E N                        1 2 π Z π − π    X N (e i θ )    2 s E C β E N     X N (e i θ )    2 s  d θ            k             . From (2) one expects and indeed it can be proven for part of the subcritical regime, at least when β = 2 , see Appendix A of [53], that the second expectation above converges to, 1 (2 π ) k E  GMC 2 s β − 1 2 ( T ) k  = Γ  1 − 2 ks 2 β  Γ  1 − 2 s β  k , with the explicit evaluation by virtue of the Fyodorov-Bouchaud formula [42, 69, 25]. Note that this formula does not make sense in the regime we ar e concerned with, 2 ks 2 > β . Proposition 2.5. Let ρ ( m ) N ,β be the m-th correlation function of C β E N . Then, for N > m, ρ ( m ) N ,β ( x 1 , . . . , x m ) = N ! ( N − m )! Z N − m ,β Z N ,β Y 1 ≤ j < k ≤ m    e i x j − e i x k    β × E C β E N − m h | X N − m (1) | m β i E         m Y j = 2     q β, m β/ 2 N − m  e i( x j − x 1 )      β         . Proof. By deﬁnition, we have ρ ( m ) N ,β ( x 1 , . . . , x m ) = N ! ( N − m )! 1 Z N ,β Z [ − π,π ) N − m Y 1 ≤ j < k ≤ m    e i x j − e i x k    β m Y j = 1 N Y k = m + 1    e i x j − e i θ k    β Y m + 1 ≤ j < k ≤ N    e i θ j − e i θ k    β d θ m + 1 · · · d θ N . 10 T . A ssiotis and J. N ajnudel Hence, we can write, ρ ( m ) N ,β ( x 1 , . . . , x m ) = N ! ( N − m )! Z N − m ,β Z N ,β Y 1 ≤ j < k ≤ m    e i x j − e ix k    β E C β E N − m         m Y j = 1    X N − m (e i x j )    β         . Using rotational invariance of C β E N , the expectation in the last display is equal to E C β E N − m         | X N − m (1) | β m Y j = 2     X N − m  e i( x j − x 1 )      β         and the result follows by a change of measur e to CJ N − m ,β, m β/ 2 . □ W e need some pr eliminary results fr om the literature. Firstly , the following lemma is well-known, see [32]. Lemma 2.6. For β > 0 , r ∈ R + , we have, as N → ∞ , N − 2 r 2 β E C β E N h | X N (1) | 2 r i − → e Y β ( 1 + 2 r − β/ 2 ) − 2 Y β ( 1 + r − β/ 2 ) + Y β (1 − β/ 2) . W e also r ecall the following r esult due to Li-V alk ´ o [60]. It generalises pr evious results of V alk ´ o-V irag [75] to δ > 0. For the special case β = 2 , δ = 0 see even earlier work [26]. Proposition 2.7. Let β, δ > 0 . Then, there exists a coupling of the ( CJ N ,β,δ ) ∞ N = 1 and H P β,δ such that almost surely , as N → ∞ , q β,δ N (e i z / N )e − i z / 2 − → ˜ ξ β,δ ( z ) , uniformly on compacts in z ∈ C and the entire function’ s ˜ ξ β,δ ∞ zero set is exactly given by H P β,δ . Li-V alk ´ o give an interesting characterisation of ˜ ξ β,δ ∞ in terms of the solution of an entire function-valued SDE and also a repr esentation of the T aylor coe ﬃ cients of ˜ ξ β,δ ∞ in terms of iterated integrals of Brownian motions, see [60]. T o connect to the principal value product fr om Deﬁnition 1.6 we prove the following. Proposition 2.8. For β, δ > 0 , we have ξ β,δ d = ˜ ξ β,δ . (17) Moreover , convergence in the pr oduct representation (8) of ξ β,δ is uniform on compact sets in C . Proof. Firstly , we claim that almost sur ely ˜ ξ β,δ ∞ belongs to the Cartwright class of entir e functions CC deﬁned as CC = ( f entire : ∃ ∆ > 1 so that | f ( z ) | ≤ ∆ 1 + | z | , for all z ∈ C , and Z ∞ −∞ log + | f ( x ) | 1 + x 2 d x < ∞ ) . This is proven in Proposition 2.10 below . Then, the argument follows the proof of Proposition 34 of [75] which treats the case δ = 0. Namely , by Theorem 11 of Section V .4.4 of [59] for f ∈ CC , we have f ( z ) = cz m e i bz lim R →∞ Y | λ k | < R  1 − z λ k  11 C β E moments , Sine β correla tions and stochastic zet a where b , c ∈ R , m ∈ N ∪ { 0 } and ( λ k ) k ∈ Z are the non-zero roots of f . W e now apply this to ˜ ξ β,δ . It is easy to see that, since ˜ ξ β,δ (0) = 1, and ˜ ξ β,δ maps the real line to the real line, see [60], and has real roots, we must have m , b = 0 and c = 1, which gives the proof for pointwise equality . Now , by construction of ˜ ξ β,δ as the secular function of the random Dirac operator associated to the H P β,δ point process, see [60], we have ˜ ξ β,δ ( z ) = e − γ z Y x ∈H P β,δ  1 − z x  e z / x , the convergence being uniform on compact sets z ∈ C , for some random variable γ ( γ has a meaning in terms of random Dirac operators, see [75, 60], but we will not need this). W e now determine γ . Observe that we have, uniformly on compact sets in z ∈ C , ˜ ξ β,δ ( z ) = lim R →∞ exp              z              − γ + X x ∈H P β,δ | x | < R 1 / x                           Y x ∈H P β,δ | x | < R  1 − z x  . By taking z ∈ C with z , 0 and ˜ ξ β,δ ( z ) , 0, we obtain (using the pointwise equality and the fact that the roots of ˜ ξ β,δ are given by the point pr ocess H P β,δ ) that almost surely γ = lim R →∞ X x ∈H P β,δ | x | < R 1 x . Putting everything together completes the proof. □ Remark 2.9. Observe that, the proof of Proposition 2.8 above in fact shows almost sure equality between ˜ ξ β,δ and ξ β,δ in the coupling a ﬀ orded by Proposition 2.7. Proposition 2.10. For any β, δ > 0 , almost surely ˜ ξ β,δ ∈ CC . W e will prove this pr oposition later in this section. W e now arrive to the following sharp bound which is the main technical contribution of our paper . It will be pr oven in Section 3 using random orthogonal polynomials on the unit circle. Theorem 2.11. Let β, δ > 0 , ℓ ∈ N and ( x 1 , . . . , x ℓ ) ∈ R ℓ . Suppose r 1 , . . . , r ℓ ∈ R + are such that P ℓ j = 1 r j ≤ 2 δ . Then, E      q β,δ N (e i x 1 / N )     r 1 · · ·     q β,δ N (e i x ℓ / N )     r ℓ  ≤ C ℓ Y j = 1 1 1 + | x j | (2 δ − P ℓ m = 1 r m ) r j /β Y 1 ≤ i < j ≤ ℓ 1 1 + | x i − x j | r i r j /β , (18) where the constant C > 0 depends only on β , δ , ℓ and ( r m ) 1 ≤ m ≤ ℓ , but not on N and ( x m ) 1 ≤ m ≤ ℓ . The following corollary is immediate fr om Theorem 2.11 using Pr oposition 2.7 and Fatou’s lemma. 12 T . A ssiotis and J. N ajnudel Corollary 2.12. Let β, δ > 0 . For r 1 , . . . , r ℓ ∈ R + such that P ℓ j = 1 r j ≤ 2 δ we have, E      ˜ ξ β,δ ( x 1 )     r 1 · · ·     ˜ ξ β,δ ( x ℓ )     r ℓ  ≤ C ℓ Y j = 1 1 1 + | x j | (2 δ − P ℓ m = 1 r m ) r j /β Y 1 ≤ i < j ≤ ℓ 1 1 + | x i − x j | r i r j /β , (19) where the constant C > 0 depends only on β , δ , ℓ and ( r m ) 1 ≤ m ≤ ℓ . W e now prove a number of intermediate r esults needed to prove Theorem 1.8. Lemma 2.13. For β > 0 , N , m ∈ N , with N ≥ m, we have the following asymptotics as N → ∞ N ! ( N − m )! Z N − m ,β Z N ,β E C β E N − m h | X N − m (1) | m β i ∼ 2 m  β 2 − 1  π m β β m / 2 e Y β ( 1 + β ( m − 1 2 ) ) − 2 Y β  1 + β 2 ( m − 1)  + Y β  1 − β 2  N β m 2 ( m − 1) + m . Proof. This is direct computation using Lemma 2.6 and standard asymptotics for Gamma functions. □ Proposition 2.14. Let β > 0 , m ∈ N , ( x 1 , . . . , x m ) ∈ R m . W e have as N → ∞ , E         m Y j = 2     q β, m β/ 2 N − m  e i 1 N ( x j − x 1 )      β         Y 1 ≤ j < k ≤ m     e i x j N − e i x k N     β ∼ N − ( m 2 ) β E         m Y j = 2     ˜ ξ β, m β/ 2 ( x j − x 1 )     β         Y 1 ≤ j < k ≤ m    x j − x k    β . Proof. The asymptotics of the V andermonde-type term are obvious. Thus, we only need to show that for any ( x 1 , . . . , x m ) ∈ R m , as N → ∞ , E         m Y j = 2     q β, m β/ 2 N − m  e i 1 N ( x j − x 1 )      β         − → E         m Y j = 2     ˜ ξ β, m β/ 2 ( x j − x 1 )     β         . For this we only need some uniform integrability , since by virtue of Pr oposition 2.7 we have almost sure convergence (in the coupling of Proposition 2.7). It moreover su ﬃ ces to have for some r > β , sup N ≥ m + 1 E         m Y j = 2     q β, m β/ 2 N − m  e i 1 N ( x j − x 1 )      r         < ∞ . Picking r so that β < r ≤ m m − 1 β , this follows by virtue of the bound from Theorem 2.11. □ Proposition 2.15. Let β > 0 , m ∈ N . For any continuous function f m with compact support on R m , we have Z R m ρ ( m ) β ( x 1 , . . . , x m ) f m ( x 1 , . . . , x m )d x 1 · · · d x m = lim N →∞ 1 N m Z R m ρ ( m ) N ,β  x 1 N , . . . , x m N  f m ( x 1 , . . . , x m )d x 1 · · · d x m . 13 C β E moments , Sine β correla tions and stochastic zet a Proof. Let us write Ξ N ,β = { θ 1 , . . . , θ N } for the random point process with ( θ 1 , . . . , θ N ) being C β E N -distributed. It is known that, see [73, 55], that as random point processes, as N → ∞ , N Ξ N ,β d − → Sine β . (20) W e only need to show that the correlation functions ( λ ( m ) N ,β ) ∞ m = 1 of N Ξ N ,β also converge to the corresponding ones for Sine β . First, by a simple change of variables we readily get ( λ ( m ) N ,β ) ∞ m = 1 = ( N − m ρ ( m ) N ,β ( x 1 / N , . . . , x m / N )) ∞ m = 1 . Now , for all m ∈ N , making use of the formula from Proposition 2.5, the bound from Theorem 2.11, along with the working in Lemma 2.13 and Proposition 2.14, we obtain for some constant C m ,β : sup N ≥ m + 1 sup ( x 1 ,..., x m ) ∈ R m λ ( m ) N ,β ( x 1 , . . . , x m ) ≤ C m ,β . (21) Hence, for all ℓ ∈ N with A being an arbitrary compact Borel set we have, for universal constants c ℓ, m > 0, E   ( N Ξ N ,β ) ( A )  ℓ  = ℓ X m = 1 c ℓ, m Z A m λ ( m ) N ,β ( x 1 , . . . , x m )d x 1 · · · d x m ≤ C A ,ℓ,β , (22) for some constant C A ,ℓ,β > 0 independent of N . From (20), we get, as N → ∞ , X y 1 , y 2 ··· , y m ∈ N Ξ N ,β f m ( y 1 , . . . , y m ) d − → X y 1 , y 2 ··· , y m ∈ Sine β f m ( y 1 , . . . , y m ) . (23) Since by deﬁnition of correlation function [19, 49], we have E         X y 1 , y 2 ··· , y m ∈ N Ξ N ,β f m ( y 1 , . . . , y m )         = 1 N m Z R m ρ ( m ) N ,β  x 1 N , . . . , x m N  f m ( x 1 , . . . , x m )d x 1 · · · d x m , E         X y 1 , y 2 ··· , y m ∈ Sine β f m ( y 1 , . . . , y m )         = Z R m ρ ( m ) β ( x 1 , . . . , x m ) f m ( x 1 , . . . , x m )d x 1 · · · d x m , the desired result follows using (23) by virtue of dominated convergence and uniform integrability , which is a consequence of the uniform bound (22) since f m is continuous with compact support. □ W e next prove a number of intermediate r esults used to establish Theorem 1.7 Proposition 2.16. Let β, δ , s > 0 and suppose 4 s ( δ − s ) > β . Then, as N → ∞ , Z π N − π N     q β,δ N (e i x / N )     2 s d x d − → Z ∞ −∞     ˜ ξ β,δ ( x )     2 s d x . (24) Proof. It is enough to prove that, with the coupling given in Proposition 2.7, Y N tends to zero in pr obability , where Y N def =       Z π N − π N     q β,δ N (e i x / N )     2 s d x − Z ∞ −∞     ˜ ξ β,δ ( x )     2 s d x       . 14 T . A ssiotis and J. N ajnudel For 0 < R ≤ π N , we have Y N ≤ Y N , R + Z N , R , where Y N , R def =       Z R − R     q β,δ N (e i x / N )     2 s d x − Z R − R     ˜ ξ β,δ ( x )     2 s d x       , Z N , R def = Z R \ [ − R , R ]     q β,δ N (e i x / N )     2 s d x + Z R \ [ − R , R ]     ˜ ξ β,δ ( x )     2 s d x . Hence, for all R , ϵ > 0, lim sup N →∞ P ( Y N ≥ ϵ ) ≤ lim sup N →∞ P ( Y N , R ≥ ϵ/ 2) + lim sup N →∞ P ( Z N , R ≥ ϵ/ 2) . By Proposition 2.7, Y N , R almost surely converges to zer o when N → ∞ , so the ﬁrst upper limit in the right-hand side is equal to zero. By virtue of the bound from Theorem 2.11 and Corollary 2.12 we get, since δ > s by assumption, lim sup N →∞ E  Z N , R  ≤ C Z R \ [ − R , R ] d x 1 + | x | 4 s ( δ − s ) /β , for C > 0 depending only on β, δ and s . Hence, lim sup N →∞ P ( Y N ≥ ϵ ) ≤ lim sup N →∞ P ( Z N , R ≥ ϵ/ 2) ≤ 2 C ϵ Z R \ [ − R , R ] d x 1 + | x | 4 s ( δ − s ) /β for any R > 0. Since 4 s ( δ − s ) /β > 1 by assumption, the right-hand side is ﬁnite and tends to zero when R → ∞ , which implies lim sup N →∞ P ( Y N ≥ ϵ ) = 0 , concluding the proof. □ Proposition 2.17. Let β, δ , s > 0 and m ∈ N be such that 2 s (2 δ − ( m + 1) s ) > β and 4 m ( δ − ms ) s > β . Then, we have, as N → ∞ , E       Z π N − π N     q β,δ N (e i x / N )     2 s d x ! m       − → E " Z ∞ −∞     ˜ ξ β,δ ( x )     2 s d x ! m # . (25) Proof. Since m ∈ N , we can expand the power and use Fubini-T onelli theorem to bring the expectation inside the multiple integral. Deﬁne the functions: F β,δ N , m ( x 1 , . . . , x m ) = E      q β,δ N (e i x 1 / N )     2 s · · ·     q β,δ N (e i x m / N )     2 s  , F β,δ ∞ , m ( x 1 , . . . , x m ) = E      ˜ ξ β,δ ( x 1 )     2 s · · ·     ˜ ξ β,δ ( x m )     2 s  . W e make two claims from which the conclusion immediately follows by the dominated convergence theor em. The ﬁrst claim is that for all ( x 1 , . . . , x m ) ∈ R m , as N → ∞ , F β,δ N , m ( x 1 , . . . , x m ) − → F β,δ ∞ , m ( x 1 , . . . , x m ) . (26) 15 C β E moments , Sine β correla tions and stochastic zet a The second claim is that for all N ∈ N and ( x 1 , . . . , x m ) ∈ R m , F β,δ N , m ( x 1 , . . . , x m ) ≤ C D m ( x 1 , . . . , x m ) , (27) for some constant C independent of N ∈ N and ( x 1 , . . . , x m ) ∈ R m , where D m is given by: D m ( x 1 , . . . , x m ) = m Y j = 1 1 1 + | x j | 4( δ − ms ) s /β Y 1 ≤ j < k ≤ m 1 1 + | x j − x k | 4 s 2 /β , and satisﬁes Z R m D m ( x 1 , . . . , x m )d x 1 · · · d x m < ∞ . (28) T o show (26), by virtue of the almost sure convergence a ﬀ orded by Pr oposition 2.7 (in the coupling therein) we only need some uniform integrability . In particular , it su ﬃ ces to have that for some r > s , sup N ≥ 1 E      q β,δ N (e i x 1 / N )     2 r · · ·     q β,δ N (e i x m / N )     2 r  < ∞ . Picking r such that s < r ≤ δ m (since ms < δ ) this follows from Theorem 2.11. For the second claim, the bound (27) is immediate from Theorem 2.11 while the integrability condition (28) follows from Lemma 2.18 by taking a = 4( δ − ms ) s /β and b = 4 s 2 /β and observing that 2 s (2 δ − ( m + 1) s ) > β and and 4 m ( δ − ms ) s > β are exactly equivalent to the conditions of Lemma 2.18. This completes the pr oof. □ Lemma 2.18. Let m ∈ N , a , b ∈ R + such that a + ( m − 1) b / 2 > 1 and ma > 1 . Then, Z R m m Y j = 1 1 1 + | x j | a Y 1 ≤ i < j ≤ m 1 1 + | x i − x j | b d x 1 · · · d x m < ∞ . Proof. The result is obvious for m = 1 or b = 0. Let us assume m ≥ 2 and b > 0. Since | x i − x j | ≥    | x i | − | x j |    , the integral is, by symmetry , bounded by 2 m m ! Z 0 ≤ x 1 ≤ x 2 ≤···≤ x m < ∞ m Y j = 1 1 (1 + x j ) a Y 1 ≤ i < j ≤ m 1 (1 + x j − x i ) b d x 1 · · · d x m . It is then enough to check the ﬁniteness of this last integral. By making the change of variables y 1 = x 1 , y 2 = x 2 − x 1 , . . . , y m = x m − x m − 1 , it is equal to Z R m + m Y j = 1 1  1 + P j ℓ = 1 y ℓ  a Y 2 ≤ i ≤ j ≤ m 1  1 + P j ℓ = i y ℓ  b d y 1 · · · d y m . Now , for 1 ≤ i ≤ j ≤ m , we have m        1 + j X ℓ = i y ℓ        ≥ j X ℓ = i (1 + y ℓ ) . Hence, it is enough to show Z R m + m Y j = 1 1  P j ℓ = 1 (1 + y ℓ )  a Y 2 ≤ i ≤ j ≤ m 1  P j ℓ = i (1 + y ℓ )  b d y 1 · · · d y m < ∞ , 16 T . A ssiotis and J. N ajnudel or equivalently Z R m ≥ 1 m Y j = 1 1  P j ℓ = 1 y ℓ  a Y 2 ≤ i ≤ j ≤ m 1  P j ℓ = i y ℓ  b d y 1 · · · d y m < ∞ , which is in turn implied by Z R m ≥ 1 m Y j = 1 1  max { y j , y 1 }  a Y 2 ≤ i ≤ j ≤ m 1  max { y i , y j }  b d y 1 · · · d y m < ∞ . W rite P erm [ i ; j ] for the set of permutations of { i , . . . , j } . Let us decompose R m ≥ 1 as follows R m ≥ 1 = [ 1 ≤ i ≤ m ,σ ∈ P erm [2; m ] H i ; σ , H i ; σ = n ( y 1 , y 2 , . . . , y m ) ∈ R m : 1 ≤ y σ (2) ≤ y σ (3) ≤ · · · ≤ y σ ( i ) ≤ y 1 ≤ y σ ( i + 1) ≤ · · · ≤ y σ ( m ) o , with the obvious convention for i = 1 , m , namely y 1 ≤ y j and y 1 ≥ y j respectively for all j = 2 , . . . , m . Now , observe that on each H i ; σ we have: m Y j = 1 max { y j , y 1 } = y i 1 m Y j = i + 1 y σ ( j ) , Y 2 ≤ i ≤ j ≤ m max { y i , y j } = m Y j = 2 y j − 1 σ ( j ) . W e can then bound the last integral by X 1 ≤ i ≤ m ,σ ∈ P erm [2; m ] Z H i ; σ y − ia 1 i Y j = 2 y − ( j − 1) b σ ( j ) m Y j = i + 1 y − a − ( j − 1) b σ ( j ) d y 1 · · · d y m . Observe that, for ﬁxed 1 ≤ i ≤ m , the integrals over H i ; σ are the same for all σ ∈ P er m [2; m ] and hence it su ﬃ ces to check that each of the following m integrals, where 1 ≤ i ≤ m , is indeed ﬁnite: Z 1 ≤ x 1 ≤···≤ x i − 1 ≤ t ≤ x i ≤···≤ x m − 1 < ∞ t − ia i − 1 Y j = 1 x − jb j m − 1 Y j = i x − a − jb j d x 1 · · · d x m − 1 d t < ∞ . By performing the integrations sequentially in x m − 1 , x m − 2 , . . . , x i , t , x i − 1 , . . . , x 1 we require that each exponent on the x j ’s or t is more negative than − 1. A simple counting argument then gives the following constraints, for each 1 ≤ i ≤ m (note that for i = m , constraint (29) is absent while for i = 1, constraint (30) is absent): − ( m − j − 1) + ( m − j ) a + b m ( m − 1) 2 − j ( j − 1) 2 ! − 1 > 0 , j = i , . . . , m − 1 , (29) − ( m − i ) + ma + b m ( m − 1) 2 − i ( i − 1) 2 ! − 1 > 0 , − ( m − j ) + ma + b m ( m − 1) 2 − j ( j − 1) 2 ! − 1 > 0 , j = 1 , . . . , i − 1 . (30) Observe that, for b > 0, the functions in the variable j deﬁned by the left hand sides of (29) and (30) ar e concave and hence it su ﬃ ces to only check the inequalities for j = i , m − 1 and 17 C β E moments , Sine β correla tions and stochastic zet a j = 1 , i − 1 respectively . Thus, overall we only need to check that the following constraints are satisﬁed: a + ( m − 1) b − 1 > 0 , (31) − ( m − i − 1) + ( m − i ) a + b m ( m − 1) 2 − i ( i − 1) 2 ! − 1 > 0 , i = 1 , . . . , m − 1 , (32) − ( m − i ) + ma + b m ( m − 1) 2 − i ( i − 1) 2 ! − 1 > 0 , i = 1 , . . . , m , (33) − ( m − i + 1) + ma + b m ( m − 1) 2 − ( i − 1)( i − 2) 2 ! − 1 > 0 , i = 2 , . . . , m , (34) − ( m − 1) + ma + b m ( m − 1) 2 − 1 > 0 . (35) First, observe that (31) is implied by (35) which is in turn equivalent to a + ( m − 1) b / 2 > 1. Then, observe again that, when b > 0, the functions in the variable i deﬁned by the left- hand sides of (32), (33) and (34) are concave and so it su ﬃ ces to only check the constraints for i = 1 , m − 1, i = 1 , m and i = 2 , m respectively . The r esulting inequalities are given by , − ( m − 2) + ( m − 1) a + bm ( m − 1) / 2 − 1 > 0 , a + b ( m − 1) − 1 > 0 , − ( m − 1) + ma + bm ( m − 1) / 2 − 1 > 0 , ma − 1 > 0 , − ( m − 1) + ma + bm ( m − 1) / 2 − 1 > 0 , − 1 + ma + b ( m − 1) − 1 > 0 . These inequalities can then be easily checked from a + ( m − 1) b / 2 > 1 and ma > 1 (also recalling m ≥ 2) which completes the proof. □ Proposition 2.19. Let β, s > 0 and k ∈ R ≥ 1 such that 2 s 2 (2 k − ⌈ k ⌉ ) > β and moreover for k > 1 , 4 ⌈ k − 1 ⌉ ( k − ⌈ k − 1 ⌉ ) s 2 > β . Then, as N → ∞ , E        Z π N − π N     q β, ks N (e i x / N )     2 s d x ! k − 1        − → E       Z ∞ −∞     ˜ ξ β, ks ( x )     2 s d x ! k − 1       . Proof. The case k = 1 is trivial. For k ∈ N , with k ≥ 2, the statement follows by taking δ = ks and m = k − 1 in Proposition 2.17. Let us assume then that k ∈ R ≥ 1 \ N . By taking δ = ks in Proposition 2.16, which is allowed since for k > 1, 4( k s − s ) s ≥ 2 s 2 (2 k − ⌈ k ⌉ ) > β , we get, as N → ∞ , Z π N − π N     q β, ks N (e i x / N )     2 s d x d − → Z ∞ −∞     ˜ ξ β, ks ( x )     2 s d x . Thus, to get the requir ed statement we only need some uniform integrability which will be implied by conver gence of the next larger integer moment. Let us take δ = ks and m = ⌈ k − 1 ⌉ in Proposition 2.17. This is valid since the two conditions 2 s 2 (2 k − ⌈ k ⌉ ) > β and 4 ⌈ k − 1 ⌉ ( k − ⌈ k − 1 ⌉ ) s 2 > β exactly imply the conditions 2 s (2 δ − ( m + 1) s ) > β and 4 m ( δ − ms ) s > β requir ed in Proposition 2.17. This concludes the pr oof. □ W e now return to the pr oof of Proposition 2.10. Proof of Pr oposition 2.10. The integral condition on the r eal line follows immediately from Corollary 2.12, since for 0 < s ≤ 2 δ , E  log +     ˜ ξ β,δ ( x )      ≤ c s E      ˜ ξ β,δ ( x )     s  ≤ Cc s 1 + | x | (2 δ − s ) s /β ≤ Cc s ∀ x ∈ R , 18 T . A ssiotis and J. N ajnudel for c s > 0 depending only on s . T o get the exponential bound for all z ∈ C , suppose we could show that for some c , ˜ c > 0, we have for all k ∈ N , P max | z |≤ k     ˜ ξ β,δ ( z )     ≥ 2 c k ! ≤ 2 − ˜ c k . (36) Then, by the Bor el-Cantelli lemma, the conclusion would follow . Let s ∈ (0 , 2 δ ]. W e have P max | z |≤ k     ˜ ξ β,δ ( z )     ≥ 2 c k ! = P max | z |≤ k     ˜ ξ β,δ ( z )     s ≥ 2 c sk ! ≤ E  max | z |≤ k     ˜ ξ β,δ ( z )     s  2 c sk . Observe that, since ˜ ξ β,δ is an entire function, for any s > 0, | ˜ ξ β,δ | s is subharmonic (this is a consequence of two well-known facts: (a) for holomorphic z 7→ h ( z ), z 7→ log | h ( z ) | is subharmonic and (b) the composition of a subharmonic function g with an increasing convex function ϕ , ϕ ◦ g is again subharmonic; thus taking ϕ ( x ) = exp( sx ), for any s > 0, gives the claim). W e now require two facts: for a subharmonic function f on C , we have the maximum principle max | z |≤ k f ( z ) = max | z | = k f ( z ) and moreover , if f is non-negative, we have by Poisson’s inequality , again for a generic constant C > 0 used throughout this proof, which may change fr om line to line, max | z | = k f ( z ) ≤ max | z | = k 1 2 π Z 2 π 0 f  ( k + 1)e i θ  ( k + 1) 2 − k 2    ( k + 1)e i θ − z    2 d θ ≤ C ( k + 1) Z | z | = k + 1 f ( z ) | d z | . Hence, applying this to f ( · ) = | ˜ ξ β,δ ( · ) | s we obtain, P max | z |≤ k     ˜ ξ β,δ ( z )     ≥ 2 c k ! ≤ C ( k + 1) 2 c sk E " Z | z | = k + 1     ˜ ξ β,δ ( z )     s | d z | # ≤ C ( k + 1) 2 2 c sk sup | z | = k + 1 E      ˜ ξ β,δ ( z )     s  . For z = x + i y we have, by virtue of Proposition 2.7 and Fatou’s lemma, E      ˜ ξ β,δ ( x + i y )     s  ≤ e ys / 2 lim inf N →∞ E      q β,δ N (e − y / N e i x / N )     s  . (37) For r < 1, with P r ( θ ) = (2 π ) − 1 P ∞ n = −∞ r | n | e i n θ the normalised Poisson kernel, we have, E      q β,δ N ( r e i θ )     s  ≤ Z π − π P r ( θ − t ) E      q β,δ N (e i t )     s  d t ≤ C , (38) the ﬁnal inequality due to the bound in Theorem 2.11. Now , by virtue of rotational invariance of C β E N we have, for r > 1, the functional equation     q β,δ N ( r e i θ )     s d = r Ns     q β,δ N ( r − 1 e − i θ )     s , and so we get the following bound on the whole complex plane: E      q β,δ N ( r e i θ )     s  ≤        C , r ≤ 1 , θ ∈ T , Cr Ns , r > 1 , θ ∈ T . Thus, by plugging in (37) we obtain, E      ˜ ξ β,δ ( z )     s  ≤ C e s | z | / 2 , ∀ z ∈ C . 19 C β E moments , Sine β correla tions and stochastic zet a Putting everything together we get, P max | z |≤ k     ˜ ξ β,δ ( z )     ≥ 2 c k ! ≤ C ( k + 1) 2 2 sk (1 / (2 log 2) − c ) . Hence, picking c > 1 / (2 log 2) (and appropriately ˜ c > 0 small enough) gives (36) and completes the proof. □ Proof of Theor em 1.7. This follows immediately by computing the asymptotics of the for- mula in Pr oposition 2.3 by virtue of Lemma 2.6 and Pr oposition 2.19 (note that for k ∈ N , k ≥ 2, both conditions are implied by 2 ks 2 > β ) and ﬁnally recalling Proposition 2.8. □ Proof of Theor em 1.8. W e combine the expression given in Proposition 2.5 with Lemma 2.13 and Proposition 2.14, and we use the uniform bound (21) in order to apply dominated convergence. In this way , we obtain the value of the limit lim N →∞ 1 N m Z R m ρ ( m ) N ,β  x 1 N , . . . , x m N  f m ( x 1 , . . . , x m )d x 1 · · · d x m , for any continuous function f m of compact support on R m . Combining the explicit expres- sion of this limit with Proposition 2.8 and Pr oposition 2.15, we deduce Z R m ρ ( m ) β ( x 1 , . . . , x m ) f m ( x 1 , . . . , x m )d x 1 · · · d x m = C ( m ) β Z R m Y 1 ≤ i < j ≤ m    x i − x j    β E         m Y j = 2     ξ β, m β/ 2  x j − x 1      β         f m ( x 1 , . . . , x m )d x 1 · · · d x m . Since such f m form a determining class we obtain expr ession (11). Then, uniform bound- edness follows immediately by plugging into (11) the bound from Cor ollary 2.12. T o show continuity of (13) we only need some uniform integrability to get convergence of moments (since almost surely z 7→ ξ β,δ ( z ) is continuous, in fact entire). It su ﬃ ces to have, with some r > β , for all R > 0, sup ( x 1 ,..., x m − 1 ) ∈ [ − R , R ] m − 1 E         m − 1 Y j = 1    ξ β, m β/ 2 ( x j )    r         < ∞ . Picking r such that β < r ≤ m m − 1 β , this follows from Cor ollary 2.12 and Proposition 2.8. W e ﬁnally show that (13) is strictly positive for all ( x 1 , . . . , x m ) ∈ R m . Suppose other- wise. Then, by virtue of representation (8) of ξ β, m β/ 2 there must exist a deterministic y ∈ R such that with positive probability q , the point process H P β, m β/ 2 has a point at y . Hence, for all ϵ > 0, q ≤ E h H P β, m β/ 2  y − ϵ, y + ϵ  i = Z y + ϵ y − ϵ ρ (1) β, m β/ 2 ( x )d x ≤ 2 ϵ sup x ∈ [ y − 1 , y + 1] ρ (1) β, m β/ 2 ( x ) , where ρ (1) β, m β/ 2 is the ﬁrst correlation function of H P β, m β/ 2 which is known to be continuous (we only need that is bounded), see [68], and this gives a contradiction by taking ϵ small enough. □ 20 T . A ssiotis and J. N ajnudel 3 P roof of the main bound W e will need several preliminaries. W e consider the family ( Φ k , Φ ∗ k ) k ≥ 0 , of orthogonal poly- nomials on the unit cir cle T corresponding to the sequence ( α j ) j ≥ 0 of random V erblunsky coe ﬃ cients: these coe ﬃ cients being independent, invariant in distribution by multipli- cation by complex numbers of modulus one, | α j | 2 being distributed as a beta random variable with parameters 1 and ( β/ 2)( j + 1), namely with density for x ∈ [0 , 1] given by , β ( j + 1) 2 (1 − x ) β 2 ( j + 1) − 1 . W e have Φ 0 = Φ ∗ 0 = 1 and the Szeg ¨ o recursion, for k ≥ 0: Φ k + 1 ( z ) = z Φ k ( z ) − α k Φ ∗ k ( z ) , Φ ∗ k + 1 ( z ) = Φ ∗ k ( z ) − z α k Φ k ( z ) . Killip and Nenciu [54] have proven that for all N ≥ 1, θ 7→ X N (e i θ ) has the same distribu- tion as θ 7→ Φ ∗ N − 1 (e i θ ) − e i θ η Φ N − 1 (e i θ ) where η is an independent, uniform random variable on T . Moreover , for example by Lemma 2.3 of [24], we have for all k ≥ 0, Φ ∗ k (e i θ ) = k − 1 Y j = 0 (1 − α j e i Ψ j ( θ ) ) , where Ψ j ( θ ) = ( j + 1) θ − 2 j − 1 X r = 0 ℑ log  1 − α r e i Ψ r ( θ )  , taking the principal branch of the logarithm in the last expression. For j ≥ 0, we denote by F j the σ -algebra generated by the ( α r ) 0 ≤ r ≤ j − 1 . W e have the following bound on ﬂuctations of the variations of Ψ j : Proposition 3.1. For j , ℓ ≥ 0 , θ ∈ R , t > 0 , P  | Ψ j + ℓ ( θ ) − Ψ j ( θ ) − ℓ θ | ≥ t |F j  ≤ 2 exp         − t 2 β 8 log  1 + βℓ 1 + β j          . Proof. By rotational invariance and independence of the V erblunsky coe ﬃ cients, Ψ j + ℓ ( θ ) − Ψ j ( θ ) − ℓ θ = − 2 j + ℓ − 1 X r = j ℑ log(1 − α r ) in distribution, and is independent of F j . By applying Proposition 2.5 of [24], we get, for all λ ∈ R , E h e λ ( Ψ j + ℓ ( θ ) − Ψ j ( θ ) − ℓθ )    F j i ≤ exp         2 λ 2 j + ℓ − 1 X r = j 1 1 + β ( r + 1)         . 21 C β E moments , Sine β correla tions and stochastic zet a Adding these estimates for λ and − λ gives, for all λ ≥ 0, E h e λ | Ψ j + ℓ ( θ ) − Ψ j ( θ ) − ℓθ ) |    F j i ≤ 2 exp         2 λ 2 j + ℓ − 1 X r = j 1 1 + β ( r + 1)         ≤ 2 exp 2 λ 2 Z j + ℓ − 1 j − 1 d t 1 + β ( t + 1) ! = 2 exp 2 λ 2 β − 1 log 1 + β ( j + ℓ ) 1 + β j !! . Using the Cherno ﬀ bound, we get P  | Ψ j + ℓ ( θ ) − Ψ j ( θ ) − ℓ θ | ≥ t |F j  ≤ 2 exp − λ t + 2 λ 2 β − 1 log 1 + βℓ 1 + β j !! . Finally , taking λ = β t 4 log  1 + βℓ 1 + β j  gives the desired r esult. □ Now , we obtain a bound for the conditional on F j , with j ≥ ℓ , moment of the product | Φ ∗ j + ℓ (e i θ 1 ) | 2 s 1 · · · | Φ ∗ j + ℓ (e i θ p ) | 2 s p . This will be used as seed for iteration in Pr oposition 3.3. Proposition 3.2. For p ≥ 2 , let ( θ m ) 1 ≤ m ≤ p ∈ R p , ( s m ) 1 ≤ m ≤ p ∈ R p + , and j , ℓ ∈ N ∪ { 0 } so that j ≥ ℓ . Then, E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp        2 ℓ β ( j + 1) p X m = 1 s 2 m + K j + 1 min       ℓ, min 1 ≤ m 1 , m 2 ≤ p || θ m 1 − θ m 2 || ! − 1       . . . · · · + K ℓ j + 1 ! 3 / 2         1 + s log 1 + j 1 + ℓ !                 , where K > 0 depends only on β, p and on the sequence ( s m ) 1 ≤ m ≤ p , and || a || denotes the distance between a and the set 2 π Z . Proof. W e have, for ℓ ≥ 1, E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j + ℓ − 1        = p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m E        p Y m = 1    1 − α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        . If we take the principal branch of the powers and use Newton’s expansion, we get, for any ρ ∈ (0 , 1), p Y m = 1 (1 − ρ α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m ) ) s m = ∞ X k = 0 C k ρ k α k j + ℓ − 1 22 T . A ssiotis and J. N ajnudel where C k = X ( r m ) 1 ≤ m ≤ p ∈{ 0 , 1 ,..., k } p , r 1 + ··· + r p = k p Y m = 1 ( − e i Ψ j + ℓ − 1 ( θ m ) ) r m s m r m ! . W e have, for all r ≥ s m ≥ 0,       s m r + 1 !       =       s m r !       r − s m r + 1 ≤       s m r !       , which shows that  s m r  is bounded for ﬁxed s m ≥ 0. W e deduce that each term in the sum giving C k has a modulus bounded by a quantity D > 0 depending only on the sequence ( s m ) 1 ≤ m ≤ p , and that | C k | ≤ D ( k + 1) p . Expanding the squared modulus gives: p Y m = 1    1 − ρ α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m = ∞ X k 1 , k 2 = 0 C k 1 C k 2 ρ k 1 + k 2 α k 1 j + ℓ − 1 α k 2 j + ℓ − 1 , the double sum being absolutely convergent, uniformly in α j + ℓ − 1 on the unit disc. From rotational invariance of the distribution of α j + ℓ − 1 and independence with F j + ℓ − 1 , we get E        p Y m = 1    1 − ρ α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        = ∞ X k = 0 | C k | 2 ρ 2 k E [ | α j + ℓ − 1 | 2 k ] . By dominated convergence for ρ tending to 1 from below , E        p Y m = 1    1 − α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        = ∞ X k = 0 | C k | 2 E [ | α j + ℓ − 1 | 2 k ] , and then E        p Y m = 1    1 − α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        = 1 + 1 1 + ( β/ 2)( j + ℓ )        p X m = 1 s m e i Ψ j + ℓ − 1 ( θ m )        2 + O        D 2 ∞ X k = 2 ( k + 1) 2 p E [ | α j + ℓ − 1 | 2 k ]        since the expectation of | α j + ℓ − 1 | 2 , which is beta-distributed with parameters equal to 1 and ( β/ 2)( j + ℓ ), is equal to 1 / (1 + ( β/ 2)( j + ℓ )). Mor eover , ∞ X k = 2 ( k + 1) 2 p E [ | α j + ℓ − 1 | 2 k ] = ( β/ 2)( j + ℓ ) Z 1 0        ∞ X k = 2 x k ( k + 1) 2 p        (1 − x ) ( β/ 2)( j + ℓ ) − 1 d x where (1 − x ) − 2 p − 1 = ∞ X k = 0 x k (2 p + 1)(2 p + 2) . . . (2 p + k ) k ! = ∞ X k = 0 x k ( k + 1)( k + 2) . . . ( k + 2 p ) (2 p )! , 23 C β E moments , Sine β correla tions and stochastic zet a x 2 (1 − x ) − 2 p − 1 = ∞ X k = 2 x k ( k − 1) k ( k + 1) . . . ( k + 2 p − 2) (2 p )! ≥ (1 / 3)(2 / 3) (2 p )! ∞ X k = 2 ( k + 1) 2 p x k and then ∞ X k = 2 ( k + 1) 2 p E [ | α j + ℓ − 1 | 2 k ] ≤ (9 β/ 4)(2 p )!( j + ℓ ) Z 1 0 x 2 (1 − x ) ( β/ 2)( j + ℓ ) − 2 p − 2 d x . If j + ℓ ≥ (2 /β )(2 p + 2), the last integral is ﬁnite, and we get ∞ X k = 2 ( k + 1) 2 p E [ | α j + ℓ − 1 | 2 k ] ≤ (9 β/ 4)(2 p )!( j + ℓ ) Γ (3) Γ (( β/ 2)( j + ℓ ) − 2 p − 1) Γ (( β/ 2)( j + ℓ ) − 2 p + 2) ≤ (9 β/ 2)(2 p )!( j + ℓ ) (( β/ 2)( j + ℓ ) − 2 p − 1)(( β/ 2)( j + ℓ ) − 2 p )(( β/ 2)( j + ℓ ) − 2 p + 1) , which, for j + ℓ ≥ (2 /β )(4 p + 2), is bounded, since ( β/ 2)( j + ℓ ) − 2 p − 1 ≥ ( β/ 4)( j + ℓ ) in this case, by (9 β/ 2)(2 p )!( j + ℓ ) [( β/ 4)( j + ℓ )] 3 = 288(2 p )! β 2 ( j + ℓ ) 2 . W e deduce, for j + ℓ ≥ (2 /β )(4 p + 2), E        p Y m = 1    1 − α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        ≤ 1 + 2 β ( j + ℓ )        p X m = 1 s m e i Ψ j + ℓ − 1 ( θ m )        2 + K ( j + ℓ ) 2 , where K > 0 depends only on β , p and the sequence ( s m ) 1 ≤ m ≤ p . The r estriction j + ℓ ≥ (2 /β )(4 p + 2) can trivially be removed by suitably adjusting the value of K . W e deduce, after adjusting K again, E        p Y m = 1    1 − α j + ℓ − 1 e i Ψ j + ℓ − 1 ( θ m )    2 s m      F j + ℓ − 1        ≤ 1 + 2 β ( j + ℓ )        p X m = 1 s m e i( Ψ j ( θ m ) + ( ℓ − 1) θ m )        2 + K j + ℓ  t + 1 ∃ m ∈{ 1 ,..., p } , | Ψ j + ℓ − 1 ( θ m ) − Ψ j ( θ m ) − ( ℓ − 1) θ m |≥ t  + K ( j + ℓ ) 2 for any t > 0. Multiplying this bound by the product of | Φ ∗ j + ℓ − 1 (e i θ m ) | 2 s m and taking the conditional expectation given F j , we deduce E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤         1 + 2 β ( j + ℓ )        p X m = 1 s m e i( Ψ j ( θ m ) + ( ℓ − 1) θ m )        2 + Kt j + ℓ + K ( j + ℓ ) 2         E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m      F j        + K j + ℓ E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m 1 ∃ m ∈{ 1 ,..., p } , | Ψ j + ℓ − 1 ( θ m ) − Ψ j ( θ m ) − ( ℓ − 1) θ m |≥ t      F j        . 24 T . A ssiotis and J. N ajnudel T aking t = 1 and bounding the indicator function by 1, we get in particular , for a possibly di ﬀ erent value of K chosen to be larger than 1, E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ 1 + K j + ℓ ! E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m      F j        ≤ 1 + 1 j + ℓ ! K E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m      F j        , which by induction on ℓ , gives E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ j + ℓ + 1 j + 1 ! K p Y m = 1     Φ ∗ j (e i θ m )     2 s m for all ℓ ≥ 0. By the Cauchy-Schwarz inequality and an application of the inequality just above to 2 s 1 , . . . , 2 s p instead of s 1 , . . . , s p , we deduce, for ℓ ≥ 1, E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m 1 ∃ m ∈{ 1 ,..., p } , | Ψ j + ℓ − 1 ( θ m ) − Ψ j ( θ m ) − ( ℓ − 1) θ m |≥ t      F j        ≤        E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     4 s m      F j               1 / 2 × P  ∃ m ∈ { 1 , . . . , p } , | Ψ j + ℓ − 1 ( θ m ) − Ψ j ( θ m ) − ( ℓ − 1) θ m | ≥ t |F j  1 / 2 ≤ j + ℓ + 1 j + 1 ! K p Y m = 1     Φ ∗ j (e i θ m )     2 s m p 2 p exp         − t 2 β 16 log  1 + β ( ℓ − 1) 1 + β j          1 ℓ ≥ 2 ≤ j + ℓ + 1 j + 1 ! K p Y m = 1 | Φ ∗ j (e i θ m ) | 2 s m p 2 p exp − t 2 β j 16 ℓ ! , for some K > 0 depending only on β , p and ( s m ) 1 ≤ m ≤ p , where the second inequality is obtained from a union bound on m and Proposition 3.1, and the last inequality is due to the fact that for ℓ ≥ 2, log 1 + β ( ℓ − 1) 1 + β j ! ≤ log 1 + βℓ β j ! = log(1 + ℓ/ j ) ≤ ℓ/ j . Now , if we assume ℓ ≤ j , ( j + ℓ + 1) / ( j + 1) is smaller than 2, and then its K -th power is bounded by a quantity depending only on β , p and ( s m ) 1 ≤ m ≤ p . W e then deduce, after adjusting K , E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤         1 + 2 β ( j + ℓ )        p X m = 1 s m e i( Ψ j ( θ m ) + ( ℓ − 1) θ m )        2 + Kt j + ℓ + K ( j + ℓ ) 2         E        p Y m = 1     Φ ∗ j + ℓ − 1 (e i θ m )     2 s m      F j        + K j + ℓ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp − t 2 β j 16 ℓ ! . 25 C β E moments , Sine β correla tions and stochastic zet a By induction on ℓ ∈ { 0 , 1 , . . . , j } , we deduce E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ ℓ Y r = 1         1 + 2 β ( j + r )        p X m = 1 s m e i( Ψ j ( θ m ) + ( r − 1) θ m )        2 + K ( t + e − t 2 β j / (16 r ) ) j + r + K ( j + r ) 2         p Y m = 1     Φ ∗ j (e i θ m )     2 s m ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m × exp         2 β ( j + 1) ℓ X r = 1        p X m = 1 s m e i( Ψ j ( θ m ) + ( r − 1) θ m )        2 + K ℓ ( t + e − t 2 β j / (16 ℓ ) ) j + 1 + K ℓ ( j + 1) 2         . T aking, for ℓ ≥ 1, and then j ≥ 1, t = s 8 log((1 + j ) / (1 + ℓ )) β j /ℓ , we get t + e − t 2 β j / (16 ℓ ) = s 8 log((1 + j ) / (1 + ℓ )) β j /ℓ + e − log((1 + j ) / (1 + ℓ )) / 2 , which provides a bound of the form E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m × exp         2 β ( j + 1) ℓ X r = 1        p X m = 1 s m e i( Ψ j ( θ m ) + ( r − 1) θ m )        2 + K ℓ (1 + p log((1 + j ) / (1 + ℓ )) ( j + 1) p (1 + j ) / (1 + ℓ )         , the term K ℓ/ ( j + 1) 2 being absorbed by the previous term after adjusting K . This bound obviously occurs also for ℓ = 0. If we expand the squared modulus and sum in r , we get a double sum, for 1 ≤ m 1 , m 2 ≤ p , of geometric series. Separating the geometric series corresponding to the terms where m 1 = m 2 and the series for m 1 , m 2 gives a bound of the following form: E        p Y m = 1     Φ ∗ j + ℓ (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp        2 ℓ β ( j + 1) p X m = 1 s 2 m + K j + 1 min       ℓ, min 1 ≤ m 1 , m 2 ≤ p || θ m 1 − θ m 2 || ! − 1       . . . · · · + K ℓ (1 + p log((1 + j ) / (1 + ℓ )) ( j + 1) p (1 + j ) / (1 + ℓ )       . This completes the proof of the pr oposition. □ By iterating Proposition 3.2 we ar e able to deduce the following. 26 T . A ssiotis and J. N ajnudel Proposition 3.3. For p ≥ 2 , let ( θ m ) 1 ≤ m ≤ p ∈ R p , ( s m ) 1 ≤ m ≤ p ∈ R p + . Let µ def = min 1 ≤ m 1 , m 2 ≤ p || θ m 1 − θ m 2 || . Then, for all j ∈ N so that j ≥ µ − 1 , E        p Y m = 1     Φ ∗ 2 j (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp        2 log 2 β p X m = 1 s 2 m + K log(1 + j µ ) j µ ! 1 / 3        where K > 0 depends only on β, p and on the sequence ( s m ) 1 ≤ m ≤ p . Proof. For q ∈ N such that 1 ≤ q ≤ j , we consider j 0 , . . . , j q ∈ N such that j = j 0 ≤ j 1 ≤ · · · ≤ j q = 2 j , and | j r + 1 − j r − j / q | ≤ 1 for 0 ≤ r ≤ q − 1. For 0 ≤ r ≤ q − 1, applying Proposition 3.2 to j = j r , ℓ = j r + 1 − j r ≤ 2 j − j ≤ j r , and conditioning on F j gives an inequality of the form E        p Y m = 1     Φ ∗ j r + 1 (e i θ m )     2 s m      F j        ≤ E        p Y m = 1     Φ ∗ j r (e i θ m )     2 s m      F j        exp        2( j r + 1 − j r ) β j r p X m = 1 s 2 m + K µ − 1 j + Kq − 3 / 2 p log(1 + q )        . Combining these inequalities for all values of r between 0 and q − 1 gives E        p Y m = 1     Φ ∗ 2 j (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m × exp         q − 1 X r = 0 2( j r + 1 − j r ) β j r p X m = 1 s 2 m + Kq j µ + K s log(1 + q ) q         . W e have, for all x ∈ [0 , 1], 0 ≤ x − log(1 + x ) = x 2 2 − x 3 3 + x 4 4 − · · · ≤ x 2 2 and then 0 ≤ j r + 1 − j r j r − log( j r + 1 / j r ) ≤ 1 2 j r + 1 − j r j r ! 2 ≤ 1 2 j / q + 1 j ! 2 ≤ 1 2 2 j / q j ! 2 ≤ 2 q 2 . W e deduce a bound of the form E        p Y m = 1     Φ ∗ 2 j (e i θ m )     2 s m      F j        27 C β E moments , Sine β correla tions and stochastic zet a ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp         q − 1 X r = 0 2 β log( j r + 1 / j r ) p X m = 1 s 2 m + K q + Kq j µ + K s log(1 + q ) q         . W e have j µ ≥ 1 by assumption. W e choose q = f ( j µ ) where f is a function from [1 , ∞ ) to Z such that f ( x ) = 1 for x ∈ [1 , π ], 1 ≤ f ( x ) ≤ x /π for x ≥ π and f ( x ) is equivalent to x 2 / 3 (log x ) 1 / 3 when x → ∞ . The inequalities satisﬁed by f guarantee that 1 ≤ q ≤ j . Indeed, since µ is always at most π , we have 1 ≤ f ( j µ ) ≤ j µ/π ≤ j if j µ ≥ π , and f ( j µ ) = 1 ≤ j if j µ ∈ [1 , π ]. For j µ ≥ 2 su ﬃ ciently large, this choice of q gives 1 q + q j µ + s log(1 + q ) q ≤ 1 ( j µ ) 2 / 3 (log( j µ )) 1 / 3 / 2 + 2( j µ ) 2 / 3 (log( j µ )) 1 / 3 j µ + s log(1 + 2( j µ ) 2 / 3 (log( j µ )) 1 / 3 ) ( j µ ) 2 / 3 (log( j µ )) 1 / 3 / 2 , which is dominated by ( j µ ) − 1 / 3 (log(1 + j µ )) 1 / 3 . This bound remains true for j µ ≥ 1 smaller than a universal constant, since q = f ( j µ ) ∈ [1 , max(1 , j µ/π )] takes ﬁnitely many values in this case. This completes the pr oof of Proposition 3.3, after observing that q − 1 X r = 0 log( j r + 1 / j r ) = log( j q / j 0 ) = log 2 . □ W e can now relatively easily deduce the following bound on conditional moments on F j for general Φ ∗ N instead of Φ ∗ 2 j . Proposition 3.4. For p ≥ 2 , let ( θ m ) 1 ≤ m ≤ p ∈ R p , ( s m ) 1 ≤ m ≤ p ∈ R p + . Let µ def = min 1 ≤ m 1 , m 2 ≤ p || θ m 1 − θ m 2 || . Then, for all N , j ∈ N so that N ≥ j ≥ µ − 1 , E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m      F j        ≤ K N j ! 2 β P p m = 1 s 2 m p Y m = 1     Φ ∗ j (e i θ m )     2 s m where K > 0 depends only on β, p and on the sequence ( s m ) 1 ≤ m ≤ p . Proof. If j ≥ µ − 1 , using Proposition 3.3, we deduce, by induction on r ≥ 0, E        p Y m = 1     Φ ∗ 2 r j (e i θ m )     2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m exp        2 r log 2 β p X m = 1 s 2 m + K r X b = 1 log(1 + 2 b j µ ) 2 b j µ ! 1 / 3        . Applying Proposition 3.2 to 2 r j instead of j and to ℓ = N − 2 r j , and then conditioning with respect to F j , we deduce, for 2 r j ≤ N ≤ 2 r + 1 j , an inequality of the form E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m      F j        ≤ p Y m = 1     Φ ∗ j (e i θ m )     2 s m × 28 T . A ssiotis and J. N ajnudel exp        K + 2 r log 2 β p X m = 1 s 2 m + K r X b = 1 log(1 + 2 b j µ ) 2 b j µ ! 1 / 3        , since the exponential factor in Proposition 3.2 is uniformly bounded by a quantity de- pending only on β, p and ( s m ) 1 ≤ m ≤ p . Now , we are done, since the sum in b is unifomly bounded for j µ ≥ 1, and exp        2 r log 2 β p X m = 1 s 2 m        = (2 r ) 2 β P p m = 1 s 2 m ≤ ( N / j ) 2 β P p m = 1 s 2 m . □ W e can now prove the following sharp bound on the joint moments of ( | Φ ∗ N (e i θ m ) | 2 s m ) p m = 1 by an inductive argument on the number of factors in the pr oduct Q p m = 1 | Φ ∗ N (e i θ m ) | 2 s m . Proposition 3.5. Let ( θ m ) 1 ≤ m ≤ p ∈ R p , ( s m ) 1 ≤ m ≤ p ∈ R p + . Then, for all N ∈ N , E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m        ≤ K N 2 β P p m = 1 s 2 m Y 1 ≤ m 1 < m 2 ≤ p min( N , | e i θ m 2 − e i θ m 1 | − 1 ) 4 β s m 1 s m 2 , where K > 0 depends only on β, p and on the sequence ( s m ) 1 ≤ m ≤ p . Proof. W e prove this result by induction on p ≥ 1. The case p = 1 is a direct consequence of the fact that    Φ ∗ N (e i θ 1 )    2 s 1 = N − 1 Y j = 0    1 − α j    2 s 1 in distribution, and from the moment estimate E [ | 1 − α j | 2 s 1 ] ≤ e 2 s 2 1 / (1 + β ( j + 1)) proven in Pr oposition 2.5 of [24], which implies E [ | Φ ∗ N (e i θ 1 ) | 2 s 1 ] ≤ e (2 s 2 1 /β ) P N − 1 j = 0 ( j + 1) − 1 ≤ e (2 s 2 1 /β )(1 + log N ) . For p ≥ 2, let us assume that the proposition holds for p − 1 points on the unit circle. By symmetry , we can assume that the closest pair of points among (e i θ m ) 1 ≤ m ≤ p is given by θ p − 1 and θ p . Let j = min( N , ⌈ || θ p − θ p − 1 || − 1 ⌉ ) . If || θ p − θ p + 1 || > 1 / N , we have j ≥ || θ p − θ p − 1 || − 1 and then we can apply Proposition 3.4, which implies E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m        ≤ K N j ! 2 β P p m = 1 s 2 m E        p Y m = 1     Φ ∗ j (e i θ m )     2 s m        . This inequality remains obviously true for || θ p − θ p − 1 || ≤ 1 / N , in which case j = N . W e deduce E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m        ≤ 29 C β E moments , Sine β correla tions and stochastic zet a K N j ! 2 β P p m = 1 s 2 m E        max      Φ ∗ j (e i θ p − 1 )     2( s p + s p − 1 ) ,     Φ ∗ j (e i θ p )     2( s p + s p − 1 )  p − 2 Y m = 1     Φ ∗ j (e i θ m )     2 s m        . Bounding the maximum by a sum and applying the induction hypothesis, we deduce an equality of the form E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m        ≤ K N j ! 2 β P p m = 1 s 2 m j 2 β  ( s p − 1 + s p ) 2 + P p − 2 m = 1 s 2 m  . . . · · · × X r ∈{ p − 1 , p } p − 2 Y m = 1 min( j , | e i θ m − e i θ r | − 1 ) 4 β s m ( s p − 1 + s p ) Y 1 ≤ m 1 < m 2 ≤ p − 2 min( j , | e i θ m 2 − e i θ m 1 | − 1 ) 4 β s m 1 s m 2 . Notice that her e, the product in m is equal to 1 for p = 2 and the product in ( m 1 , m 2 ) is equal to 1 for p ∈ { 2 , 3 } . Now , for 1 ≤ m ≤ p − 2, | e i θ m − e i θ p | ≤ | e i θ m − e i θ p − 1 | + | e i θ p − e i θ p − 1 | ≤ 2 | e i θ m − e i θ p − 1 | because e i θ p − 1 and e i θ p are the closest points among (e i θ m ) 1 ≤ m ≤ p . Similarly , | e i θ m − e i θ p − 1 | ≤ 2 | e i θ m − e i θ p | . Hence, the ratio between the two terms for r = p − 1 and r = p is bounded from above and below by a positive quantity depending only on β and ( s m ) 1 ≤ m ≤ p . In the estimate, we can then replace the sum of the two terms by any of their weighted geometric means after adjusting K . W e deduce E        p Y m = 1    Φ ∗ N (e i θ m )    2 s m        ≤ K N j ! 2 β P p m = 1 s 2 m j 2 β  ( s p − 1 + s p ) 2 + P p − 2 m = 1 s 2 m  . . . · · · × Y r ∈{ p − 1 , p } p − 2 Y m = 1 min( j , | e i θ m − e i θ r | − 1 ) 4 β s m s r Y 1 ≤ m 1 < m 2 ≤ p − 2 min( j , | e i θ m 2 − e i θ m 1 | − 1 ) 4 β s m 1 s m 2 = KN 2 β P p m = 1 s 2 m j 4 β s p − 1 s p Y 1 ≤ m 1 ≤ p − 2 , m 1 < m 2 ≤ p min( j , | e i θ m 2 − e i θ m 1 | − 1 ) 4 β s m 1 s m 2 ≤ K N 2 β P p m = 1 s 2 m min( N , ⌈ || θ p − θ p − 1 || − 1 ⌉ ) 4 β s p − 1 s p Y 1 ≤ m 1 ≤ p − 2 , m 1 < m 2 ≤ p min( N , | e i θ m 2 − e i θ m 1 | − 1 ) 4 β s m 1 s m 2 . This proves the desir ed result, since ⌈ || θ p − θ p − 1 || − 1 ⌉ ≤ 1 + || θ p − θ p − 1 || − 1 ≤ 1 + | e i θ p − e i θ p − 1 | − 1 ≤ 3 | e i θ p − e i θ p − 1 | − 1 . □ W e can ﬁnally prove the main bound fr om Theorem 2.11. Proof of Theor em 2.11. By a change of measur e, we can write, for 2 δ ≥ P ℓ j = 1 r j , so that in particular all exponents are non-negative, E      q β,δ N (e i x 1 / N )     r 1 · · ·     q β,δ N (e i x ℓ / N )     r ℓ  = E C β E N  | X N (1) | 2 δ − P ℓ j = 1 r j    X N (e i x 1 / N )    r 1 · · ·    X N (e i x ℓ / N )    r ℓ  E C β E N h | X N (1) | 2 δ i . 30 T . A ssiotis and J. N ajnudel Now , recall that the function θ 7→ X N (e i θ ) has the same distribution as θ 7→ Φ ∗ N − 1 (e i θ ) − e i θ η Φ N − 1 (e i θ ), wher e η is uniformly distributed on T , and also | Φ ∗ N − 1 (e i θ ) − e i θ η Φ N − 1 (e i θ ) | ≤ 2 | Φ ∗ (e i θ ) | . Then, plugging in the bound from Proposition 3.5, making use of the asymp- totics from Lemma 2.6 for the denominator and using the elementary inequality N − t j t k β min  N ,    e i x j / N − e i x k / N    − 1  t j t k β ≤ C 1 1 +    x j − x k    t j t k /β , for some constant C depending only on β > 0, t j , t k ≥ 0, we obtain the desired result. □ R eferences [1] A rguin , L.-P ., B elius , D., and B ourgade , P . Maximum of the characteristic polyno- mial of random unitary matrices. Comm. Math. Phys. 349 , 2 (2017), 703–751. [2] A rguin , L.-P ., B elius , D., B ourgade , P ., R adziwiłł , M., and S oundararajan , K. Maximum of the Riemann zeta function on a short interval of the critical line. Comm. Pure Appl. Math. 72 , 3 (2019), 500–535. [3] A rguin , L.-P ., B ourgade , P ., and R adziwiłł , M. The fyodorov-hiary-keating conjec- ture. i. arXiv preprint arXiv:2007.00988 (2020). [4] A rguin , L.-P ., B ourgade , P ., and R adziwiłł , M. The fyodorov-hiary-keating conjec- ture. ii. arXiv preprint arXiv:2307.00982 (2023). [5] A ssiotis , T . On the moments of the partition function of the C β E ﬁeld. J. Stat. Phys. 187 , 2 (2022), Paper No. 14, 22. [6] A ssiotis , T . Random entire functions from random polynomials with real zeros. Adv . Math. 410 (2022), Paper No. 108701, 28. [7] A ssiotis , T ., B ailey , E. C., and K ea ting , J. P . On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices. Ann. Inst. Henri Poincar ´ e D 9 , 3 (2022), 567–604. [8] A ssiotis , T ., B edert , B., G unes , M. A., and S oor , A. On a distinguished family of random variables and Painlev ´ e equations. Probab. Math. Phys. 2 , 3 (2021), 613–642. [9] A ssiotis , T ., E riksson , E., and N i , W . On the moments of moments of random matrices and Ehr hart polynomials. Adv . in Appl. Math. 149 (2023), Paper No. 102539, 30. [10] A ssiotis , T ., G unes , M. A., K ea ting , J. P ., and W ei , F . Exchangeable arrays and integrable systems for characteristic polynomials of random matrices. arXiv pr eprint arXiv:2407.19233 (2024). [11] A ssiotis , T ., and K ea ting , J. P . Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts. Random Matrices Theory Appl. 10 , 2 (2021), Paper No. 2150019, 16. [12] A ssiotis , T ., K ea ting , J. P ., and W arren , J. On the joint moments of the characteristic polynomials of random unitary matrices. Int. Math. Res. Not. IMRN , 18 (2022), 14564–14603. 31 C β E moments , Sine β correla tions and stochastic zet a [13] B ailey , E. C., and K ea ting , J. P . On the moments of the moments of the characteristic polynomials of random unitary matrices. Comm. Math. Phys. 371 , 2 (2019), 689–726. [14] B ailey , E. C., and K ea ting , J. P . On the moments of the moments of ζ (1 / 2 + it ). J. Number Theory 223 (2021), 79–100. [15] B ailey , E. C., and K ea ting , J. P . Maxima of log-correlated ﬁelds: some recent developments. J. Phys. A 55 , 5 (2022), Paper No. 053001, 76. [16] B asor , E., C hen , Y ., and E hrhardt , T . Painlev ´ e V and time-dependent Jacobi poly- nomials. J. Phys. A 43 , 1 (2010), 015204, 25. [17] B asor , E., G e , F ., and R ubinstein , M. O. Some multidimensional integrals in number theory and connections with the Painlev ´ e V equation. J. Math. Phys. 59 , 9 (2018), 091404, 14. [18] B erestycki , N. An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22 (2017), Paper No. 27, 12. [19] B orodin , A. Determinantal point processes. In The Oxford handbook of random matrix theory . Oxford Univ . Press, Oxfor d, 2011, pp. 231–249. [20] B orodin , A., and O lshanski , G. Inﬁnite random matrices and ergodic measures. Comm. Math. Phys. 223 , 1 (2001), 87–123. [21] B ourgade , P ., E rd ˝ os , L., and Y au , H.-T . Universality of general β -ensembles. Duke Math. J. 163 , 6 (2014), 1127–1190. [22] B ourgade , P ., and F alconet , H. Liouville quantum gravity from random matrix dynamics. arXiv preprint arXiv:2206.03029 (2022). [23] B ump , D., and G amburd , A. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 , 1 (2006), 227–274. [24] C hhaibi , R., M adaule , T ., and N ajnudel , J. On the maximum of the C β E ﬁeld. Duke Math. J. 167 , 12 (2018), 2243–2345. [25] C hhaibi , R., and N ajnudel , J. On the circle, gmc γ = lim ← − − c β e n for γ = q 2 β , ( γ ≤ 1). arxiv:1904.00578 (2019). [26] C hhaibi , R., N ajnudel , J., and N ikeghbali , A. The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios. Invent. Math. 207 , 1 (2017), 23–113. [27] C laeys , T ., and F ahs , B. Random matrices with merging singularities and the Painlev ´ e V equation. SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 031, 44. [28] C laeys , T ., F orkel , J., and K ea ting , J. P . Moments of moments of the characteristic polynomials of random orthogonal and symplectic matrices. Proc. A. 479 , 2270 (2023), Paper No. 20220652, 22. [29] C laeys , T ., G lesner , G., M inakov , A., and Y ang , M. Asymptotics for averages over classical orthogonal ensembles. Int. Math. Res. Not. IMRN , 10 (2022), 7922–7966. 32 T . A ssiotis and J. N ajnudel [30] C laeys , T ., and K rasovsky , I. T oeplitz determinants with merging singularities. Duke Math. J. 164 , 15 (2015), 2897–2987. [31] C onrey , J. B., F armer , D. W ., K ea ting , J. P ., R ubinstein , M. O., and S naith , N. C. Integral moments of L -functions. Proc. London Math. Soc. (3) 91 , 1 (2005), 33–104. [32] D al B orgo , M., H ovhannisy an , E., and R ouaul t , A. Mod-Gaussian convergence for random determinants. Ann. Henri Poincar ´ e 20 , 1 (2019), 259–298. [33] F ahs , B. Uniform asymptotics of Toeplitz determinants with Fisher-Hartwig singu- larities. Comm. Math. Phys. 383 , 2 (2021), 685–730. [34] F orrester , P . J. Selberg correlation integrals and the 1 / r 2 quantum many-body system. Nuclear Phys. B 388 , 3 (1992), 671–699. [35] F orrester , P . J. Addendum to: “Selberg corr elation integrals and the 1 / r 2 quan- tum many-body system” [Nuclear Phys. B 388 (1992), no. 3, 671–699; MR1201273 (94e:33030)]. Nuclear Phys. B 416 , 1 (1994), 377–385. [36] F orrester , P . J. Integration formulas and exact calculations in the Caloger o- Sutherland model. Modern Phys. Lett. B 9 , 6 (1995), 359–371. [37] F orrester , P . J. Log-gases and random matrices , vol. 34 of London Mathematical Society Monographs Series . Princeton University Press, Princeton, NJ, 2010. [38] F orrester , P . J. Di ﬀ erential identities for the structure function of some random matrix ensembles. J. Stat. Phys. 183 , 2 (2021), Paper No. 33, 28. [39] F orrester , P . J. Dualities in random matrix theory . arXiv preprint (2025). [40] F orrester , P . J., and R ahman , A. A. Relations between moments for the Jacobi and Cauchy random matrix ensembles. J. Math. Phys. 62 , 7 (2021), Paper No. 073302, 22. [41] F orrester , P . J., and R ains , E. M. A Fuchsian matrix di ﬀ erential equation for Selberg correlation integrals. Comm. Math. Phys. 309 , 3 (2012), 771–792. [42] F yodorov , Y . V ., and B ouchaud , J.-P . Freezing and extreme-value statistics in a random ener gy model with logarithmically correlated potential. J. Phys. A 41 , 37 (2008), 372001, 12. [43] F yodorov , Y . V ., G nutzmann , S., and K ea ting , J. P . Extreme values of CUE charac- teristic polynomials: a numerical study . J. Phys. A 51 , 46 (2018), 464001, 22. [44] F yodorov , Y . V ., H iary , G. A., and K ea ting , J. P . Fr eezing transition, characteristic polynomials of random matrices, and the riemann zeta function. Phys. Rev . Lett. 108 (Apr 2012), 170601. [45] F yodorov , Y . V ., and K ea ting , J. P . Freezing transitions and extreme values: random matrix theory , and disorder ed landscapes. Philos. T rans. R. Soc. Lond. Ser . A Math. Phys. Eng. Sci. 372 , 2007 (2014), 20120503, 32. [46] F yodorov , Y . V ., and S imm , N. J. On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. Nonlinearity 29 , 9 (2016), 2837–2855. 33 C β E moments , Sine β correla tions and stochastic zet a [47] H arper , A. J. On the partition function of the riemann zeta function, and the fyodorov–hiary–keating conjectur e. arXiv preprint arXiv:1906.05783 (2019). [48] H ughes , C. P ., K ea ting , J. P ., and O’C onnell , N. On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 , 2 (2001), 429–451. [49] J ohansson , K. Random matrices and determinantal processes. In Mathematical statistical physics . Elsevier B. V ., Amsterdam, 2006, pp. 1–55. [50] K ea ting , J. P ., R odgers , B., R oditty -G ershon , E., and R udnick , Z. Sums of divisor functions in F q [ t ] and matrix integrals. Math. Z. 288 , 1-2 (2018), 167–198. [51] K ea ting , J. P ., and S naith , N. C. Random matrix theory and L -functions at s = 1 / 2. Comm. Math. Phys. 214 , 1 (2000), 91–110. [52] K ea ting , J. P ., and S naith , N. C. Random matrix theory and ζ (1 / 2 + it ). Comm. Math. Phys. 214 , 1 (2000), 57–89. [53] K ea ting , J. P ., and W ong , M. D. On the critical-subcritical moments of moments of random characteristic polynomials: a GMC perspective. Comm. Math. Phys. 394 , 3 (2022), 1247–1301. [54] K illip , R., and N enciu , I. Matrix models for circular ensembles. International Math- ematics Research Notices 2004 , 50 (01 2004), 2665–2701. [55] K illip , R., and S toiciu , M. Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146 , 3 (2009), 361–399. [56] L ambert , G., and N ajnudel , J. Subcritical multiplicative chaos and the characteristic polynomial of the C β E. arXiv preprint arXiv:2407.19817 (2024). [57] L ambert , G., and P aquette , E. Str ong approximation of gaussian β -ensemble charac- teristic polynomials: the edge r egime and the stochastic airy function. arXiv pr eprint arXiv:2009.05003 (2020). [58] L ambert , G., and P aquette , E. Bulk asymptotics of the gaussian β -ensemble charac- teristic polynomial. arXiv preprint arXiv:2508.01458 (2025). [59] L evin , B. J. Distribution of zeros of entire functions , r evised ed., vol. 5 of T ranslations of Mathematical Monographs . American Mathematical Society , Providence, RI, 1980. T ranslated from the Russian by R. P . Boas, J. M. Danskin, F . M. Goodspeed, J. Korevaar , A. L. Shields and H. P . Thielman. [60] L i , Y ., and V alk ´ o , B. Operator level limit of the circular Jacobi β -ensemble. Random Matrices Theory Appl. 11 , 4 (2022), Paper No. 2250043, 41. [61] N ajnudel , J. On the extreme values of the Riemann zeta function on random intervals of the critical line. Probab. Theory Related Fields 172 , 1-2 (2018), 387–452. [62] N ikula , M., S aksman , E., and W ebb , C. Multiplicative chaos and the characteristic polynomial of the CUE: the L 1 -phase. T rans. Amer . Math. Soc. 373 , 6 (2020), 3905–3965. [63] O kounkov , A. On n-point correlations in the log-gas at rational temperature. arXiv preprint hep-th / 9702001 (1997). 34 T . A ssiotis and J. N ajnudel [64] O sada , H. Dirichlet form approach to inﬁnite-dimensional Wiener processes with singular interactions. Comm. Math. Phys. 176 , 1 (1996), 117–131. [65] O sada , H. Non-collision and collision pr operties of Dyson’s model in inﬁnite di- mension and other stochastic dynamics whose equilibrium states ar e determinantal random point ﬁelds. In Stochastic analysis on large scale interacting systems , vol. 39 of Adv . Stud. Pure Math. Math. Soc. Japan, T okyo, 2004, pp. 325–343. [66] P aquette , E., and Z eitouni , O. The maximum of the CUE ﬁeld. Int. Math. Res. Not. IMRN , 16 (2018), 5028–5119. [67] P aquette , E., and Z eitouni , O. The extr emal landscape for the C β E ensemble. arXiv e-prints (Sept. 2022), 118pp. [68] Q u , Y ., and V alk ´ o , B. On the pair correlation function of the sine β process. arXiv preprint arXiv:2509.15446 (2025). [69] R emy , G. The Fyodorov-Bouchaud formula and Liouville conformal ﬁeld theory . Duke Math. J. 169 , 1 (2020), 177–211. [70] R hodes , R., and V argas , V . Gaussian multiplicative chaos and applications: a review . Probab. Surv . 11 (2014), 315–392. [71] S uzuki , K. On the ergodicity of interacting particle systems under number rigidity . Probab. Theory Related Fields 188 , 1-2 (2024), 583–623. [72] S uzuki , K. Curvature bound of Dyson Brownian motion. Comm. Math. Phys. 406 , 7 (2025), Paper No. 154, 56. [73] V alk ´ o , B., and V ir ´ ag , B. Continuum limits of random matrices and the Br ownian carousel. Invent. Math. 177 , 3 (2009), 463–508. [74] V alk ´ o , B., and V ir ´ ag , B. The Sine β operator . Invent. Math. 209 , 1 (2017), 275–327. [75] V alk ´ o , B., and V ir ´ ag , B. The many faces of the stochastic zeta function. Geom. Funct. Anal. 32 , 5 (2022), 1160–1231. [76] W ebb , C. The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos—the L 2 -phase. Electron. J. Pr obab. 20 (2015), no. 104, 21. [77] W idom , H. T oeplitz determinants with singular generating functions. Amer . J. Math. 95 (1973), 333–383. [78] W itte , N. S., and F orrester , P . J. Gap probabilities in the ﬁnite and scaled Cauchy random matrix ensembles. Nonlinearity 13 , 6 (2000), 1965–1986. S chool of M a thema tics , U niversity of E dinburgh , J ames C lerk M axwell B uilding , P eter G uthrie T ait R d , E dinburgh EH9 3FD, U.K. theo.assiotis@ed.ac.uk S chool of M a thema tics , U niversity of B ristol , F ry B uilding , W oodland R oad , B ristol , BS8 1UG, U.K. joseph.najnudel@bristol.ac.uk 35

Moments of C$β$E field partition function, $\mathsf{Sine}_β$ correlations and stochastic zeta

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