Asymptotic correlations with corrections for the circular Jacobi β -ensemble
aa r X i v : . [ m a t h - ph ] A ug ASYMPTOTIC CORRELATIONS WITH CORRECTIONS FOR THECIRCULAR JACOBI β -ENSEMBLE PETER J. FORRESTER, SHI-HAO LI, AND ALLAN K. TRINH
Abstract.
Previous works have considered the leading correction term to the scaled limit ofvarious correlation functions and distributions for classical random matrix ensembles and their β generalisations at the hard and soft edge. It has been found that the functional form of thiscorrection is given by a derivative operation applied to the leading term. In the present workwe compute the leading correction term of the correlation kernel at the spectrum singularity forthe circular Jacobi ensemble with Dyson indices β = 1 , and 4, and also to the spectral densityin the corresponding β -ensemble with β even. The former requires an analysis involving theRouth-Romanovski polynomials, while the latter is based on multidimensional integral formulasfor generalised hypergeometric series based on Jack polynomials. In all cases this correction termis found to be related to the leading term by a derivative operation. Introduction
The establishment of exact limit laws for the spectral statistics of large random matrices fromclassical ensembles is a fundamental part of the theory of universality, whereby it is establishedthat the limit laws depend only on low order moments of the matrix entries [11]. Thus the limitingdistributions exhibited by the classical ensembles are shared by a much wider class of random ma-trices, and most importantly have the distinguishing property that they can be computed exactly.The first result of this type was obtained by Wigner [36, 37]. In these works it was established thatfor random real symmetric matrices, upper triangular entries independent, with finite variance nor-malised to / for convenience and bounded moments, the global spectral density ¯ ρ ( λ ) — obtainedby scaling the eigenvalues by dividing by √ N — has the functional form ρ W , ( x ) := π (1 − x ) / ,supported on | x | < , nowadays referred to as the Wigner semi-circle law. This implies that for asuitable class of test functions φ ( λ ) , lim N →∞ Z ∞−∞ φ ( λ )¯ ρ ( λ ) dλ = Z − φ ( λ ) ρ W , ( λ ) dλ. (1.1)The classical ensembles exhibiting the Wigner semi-circle law are the Gaussian orthogonal andunitary ensembles, with matrices G constructed out of standard real and complex Gaussian randommatrices X according to G = 2 − / ( X + X † ) — here the normalisation − / ensures the varianceof the upper triangular entries equals / . The special mathematical structures associated withthe classical ensembles permit the determination of the rate of convergence to the Wigner law. Forexample, with φ = φ x , where φ x ( λ ) = 1 for λ ≤ x , φ x ( λ ) = 0 otherwise, it has been proved byGötze and Tikhomirov [24] for the GUE, and by Kholopov et al. [27] in the case of the GOE, that Mathematics Subject Classification.
Key words and phrases. circular Jacobi β -ensemble; confluent hypergeometric kernel; Routh-Romanovski poly-nomials; asymptotic expansion. there is a constant C independent of N such that sup x (cid:12)(cid:12)(cid:12) Z ∞−∞ φ x ( λ )(¯ ρ ( λ ) − ρ W , ( λ )) dλ (cid:12)(cid:12)(cid:12) ≤ CN , (1.2)valid for all N . Most significantly, this O (1 /N ) rate is expected to hold for the wider class of randommatrices stated above (the Wigner class) in relation to (1.1). In keeping with the convergence rateexhibited in (1.2), for φ ( λ ) in (1.1) smooth, large N expansions of the form Z ∞−∞ φ ( λ )¯ ρ ( λ ) dλ = Z − φ ( λ ) ρ W , ( λ ) dλ + 1 N Z ∞−∞ φ ( λ ) ρ W , ( λ ) dλ + · · · (1.3)are well known in the context of the loop equation analysis of the Gaussian ensembles [39]. Moreover(1.3) can be extended to the Wigner class, with ρ W , ( λ ) now dependent on the fourth moment, butno higher moments; see the introduction to [19] for further discussion and references. The pointto note then is that the rate of convergence in this setting holds true generally, while the densityfunction at this order weakly depends on the details of the distribution of the entries.An effect very similar, but even more structured, was observed by Edelman, Guionnet and Péché[10] in relation to hard scaling for certain ensembles of positive definite matrices X † X , where theentries of X – itself rectangular of size n × N – are independent with zero mean. Since the spectraldensity is strictly zero for negative values, yet the eigenvalue density is nonzero for x > , theorigin is referred to as a hard edge. Hard edge scaling refers to a rescaling of the eigenvalues sothat the spacings between eigenvalues in the neighbourhood of the origin is of order unity. Thelimiting state depends on whether the entries of the matrix are real or complex (typically labelledby the Dyson index β = 1 and β = 4 ), and the parameter a = n − N which specifies the number ofextra rows relative to columns in X . With E N,β (0; J ; a ) denoting the probability that the domain J ⊂ R + contains no eigenvalues, it was conjectured in [10] that E N, (0; (0 , s/ N ); a ) = E hard2 (0; (0 , s ); a ) + a N s dds E hard2 (0; (0 , s ); a ) + O (cid:16) N (cid:17) , (1.4)where E hard β (0; (0 , s ); a ) = lim N →∞ E N,β (0; (0 , s/ N ); a ) . Proofs were subsequently given in [3, 33]. With Σ a fixed positive definite matrix, Hachen et al. [25]gave a generalisation of (1.4) for the product X † Σ X , and found that the structured form of the /N correction persists but with a renormalised. Such a modification was also found in [10] forthe setting that the Gaussian entries X are in convolution with a further distribution giving risefourth moments differing from the Gaussian — that latter deviation shows as an additive shift to a . A number of recent works [15, 21, 31] have focussed attention on generalising (1.4), involvingboth the parameter β and the details of the ensemble giving rise to the hard edge. For the classical β ensembles with a hard edge – namely the Laguerre and Jacobi weights – it is found in all casesconsidered that the leading correction term is related to the limiting distribution by a derivativeoperation. Moreover, this effect is also exhibited at the soft edge of the Gaussian and Laguerreensembles [14, 20, 22, 26, 29], now with the leading order correction proportional to /N / . Thusthese recent studies have revealed that the leading order correction to the limiting hard and softedge distributions for a number of classical random matrix ensembles is structured, and that therate of convergence is not dependent on the details of the ensemble.In this work we contribute to the study of this effect by considering convergence to the limitfor a singularity type distinct from the hard and soft edge — namely the spectrum singularity [13, IRCULAR JACOBI β -ENSEMBLE 3 §3.9]. This singularity is exhibited at infinity by the classical Cauchy β -ensemble, itself specifiedby the joint eigenvalue probability density function (PDF) proportional to N Y j =1 ω β ( x ) Y ≤ j Let z ( X ) = i e iX/N − e iX/N . (1.9) For β = 1 , and , the Christoffel-Darboux kernel S β,N ( x, y ) associated with the PDF (1.8) admitsthe large N expansion S N,β ( z ( X ) , z ( Y )) dzdX = ˆ K ( p,q ;0) ∞ ,β ( X, Y ) + 1 N ˆ L ( p,q ;0)1 ,β ( X, Y ) + O (cid:18) N (cid:19) , (1.10) where ˆ K ( p,q ;0) ∞ ,β ( X, Y ) , ˆ L ( p,q ;0)1 ,β ( X, Y ) are specified in (3.15) , (3.16) for β = 2 , in (3.24) for β = 1 ,and in (3.29) for β = 4 . Moreover, ˆ L ( p,q ;0)1 ,β ( X, Y ) = p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) ˆ K ( p,q ;0) ∞ ,β ( X, Y ) . (1.11) PETER J. FORRESTER, SHI-HAO LI, AND ALLAN K. TRINH The structural form of the /N correction in (1.11) as a simple first order derivative operationof the limiting kernel in (1.11) is precisely as found for the analysis of the rate of convergence tothe hard and soft edge limits. It implies that the optimally tuned scaling z ( X ) i iX/ ( N + p ))1 − exp(2 iX/ ( N + p )) (1.12)gives a convergence at the rate O (1 /N ) . In the case of the circular ensembles p = q = 0 , it impliesthat the leading order correction term is O (1 /N ) without any need for a shift; see [2, 4, 17] foran application of this fact in the case β = 2 to the analysis of the statistical properties of the largeRiemann zeros.Analysis beyond the classical values β = 1 , and is also possible in restricted circumstances.In particular, exact evaluations of the spectral density of (1.8), ρ N,β ( θ ) , for β even are known inthe framework of the Selberg integral theory and its associated special functions, allowing for thederivation of the following large N expansion. Proposition 1.2. For β even and N large, we have N ρ N,β (cid:18) θN (cid:19) = ρ ∞ ,β ( θ ) + 1 N p ddθ [ θρ ∞ ,β ( θ )] + O (cid:18) N (cid:19) , (1.13) where ρ ∞ ,β ( θ ) = lim N →∞ N ρ N,β (cid:18) θN (cid:19) (1.14) is an explicit β -dimensional integral given by (3.33) below. As in Proposition 1.1, the term proportional to /N is a simple derivative operation of thelimiting density and therefore trivially eliminated by tuning the hard edge scaling to x/ ( N + p ) .Thus the leading non-trivial correction term is O (1 /N ) . The same prescription was also given in[21] at the hard edge for the Laguerre β -ensemble for even β and the associated parameter a > − .We introduce the finite N formulas which enable the derivation of Propositions 1.1 and 1.2 inSection 2. The actual asymptotics is carried out in Section 3, with some of the more technicalworking deferred to Appendix A. 2. Preliminaries Correlations for β = 2 . The PDF (1.5) with Dyson index β = 2 is referred to as theCauchy unitary ensemble (CyUE) and has the special feature of being an example of an orthogonalpolynomial determinantal point process. This means that its general k -point correlation functionis given by a k × k determinant with entries determined by a single kernel function S n, ( x, y ) involving orthogonal polynomials, denoted ˜ I ( c, ¯ c ) n ( x ) , associated with the weight (1.6). Explicitly,the k -point correlation function is R ( k ) N, ( x , . . . , x k ) = det[ S N, ( x m , x n )] km,n =1 , (2.1)where S N, ( x, y ) = ( ω ( x ) ω ( y )) / N − X k =0 h k ˜ I ( c, ¯ c ) k ( x ) ˜ I ( c, ¯ c ) k ( y )= ( ω ( x ) ω ( y )) / h N − ˜ I ( c, ¯ c ) N ( x ) ˜ I ( c, ¯ c ) N − ( y ) − ˜ I ( c, ¯ c ) N ( y ) ˜ I ( c, ¯ c ) N − ( x ) x − y . (2.2) IRCULAR JACOBI β -ENSEMBLE 5 The second equality in (2.2) is the result of the Christoffel-Darboux summation; see e..g. [13,Prop. 5.1.3].The orthogonal polynomials ˜ I ( c, ¯ c ) n ( x ) in (2.2) are required to be monic and have the orthgonality Z ∞−∞ ω ( x ) ˜ I ( c, ¯ c ) n ( x ) ˜ I ( c, ¯ c ) m ( x ) dx = h n δ n,m (2.3)for some normalisation h n > . The polynomials with this property are known as the Routh-Romanovski polynomials (see e.g. the review [34]). They have the explicit hypergeometric form ˜ I ( c, ¯ c ) n ( x ) = ( − i ) n Γ( c + ¯ c + n + 1)Γ( n + c + 1)Γ( c + ¯ c + 2 n + 1)Γ( c + 1) F (cid:18) − n, n + 1 + c + ¯ c ; c + 1; 1 − ix (cid:19) , (2.4)and furthermore the normalisation has the gamma function form h n = 2 n +2+ c +¯ c π Γ( n + 1)Γ( − c − ¯ c − n )Γ( − c − ¯ c − n − − c − ¯ c − n )Γ( − c − n )Γ( − ¯ c − n ) . (2.5)It is furthermore true that (2.4) can be viewed as certain monic Jacobi polynomials i − n P ( c, ¯ c ) n ( ix ) of degree n .Bijectively related to the CyUE by the fractional linear transformation (1.7) is the eigenvaluePDF of the circular Jacobi unitary ensemble (cJUE) given by (1.8) with β = 2 . Its correspondingkernel function can therefore be attained from (2.2) under the same transformations. Consequently,local behavior near the spectrum singularity will involve the Routh-Romanovski polynomials (2.4)mapped onto the unit circle. As detailed in [18] in the case q = 0 , and to be extended to includethe cases q = 0 in Section 3.1 below, its asymptotics are evaluated from the confluent limit formula lim n →∞ F ( − n, b ; c ; t/n ) = F ( b ; c ; − t ) . (2.6)This limit formula follows immediately from the series forms F ( a, b ; c ; z ) = ∞ X α =0 ( a ) α ( b ) α ( c ) α z α α ! , F ( a ; c ; z ) = ∞ X α =0 ( a ) α ( c ) α z α α ! , (2.7)where ( n ) α = Γ( n + α ) / Γ( n ) denotes the Pochhammer symbol.From (2.2) and with z ( X ) given in (1.9), lim N →∞ S N, ( z ( X ) , z ( Y )) dz ( X ) dX = K ( p,q ;0) ∞ , ( X, Y ) (2.8)where K ( p,q ;0) ∞ , ( X, Y ) ∝ e − i ( X + Y ) ( XY ) ( p +1) X ( X − Y ) (cid:16) X ˜ C ( p,q, ( X ) ˜ C ( p,q, ( Y ) − ( X ↔ Y ) (cid:17) (2.9)with ˜ C ( p,q,k )0 ( X ) = F ( p + k − iq ; 2 p + 2 k ; 2 iX ); (2.10)hence the name confluent hypergeometric kernel used in [8]. The explicit proportionality is givenin (3.15). In the case q = 0 , from the formula F ( p ; 2 p ; 2 iX ) = Γ( p + 1 / (cid:16) x (cid:17) − p +1 / e iX J p − / ( X ) (2.11) ( X/Y ) K ( p, ∞ , ( X, Y ) simplifies to [32] ( XY ) / J p +1 / ( X ) J p − / ( Y ) − J p − / ( X ) J p +1 / ( Y )2( X − Y ) . (2.12) PETER J. FORRESTER, SHI-HAO LI, AND ALLAN K. TRINH In the special case when p, q = 0 , there is the absence of the singularity in (1.8), which is then recog-nised as the eigenvalue PDF for the circular unitary ensemble (CUE) of random Haar distributedunitary matrices; see e.g. [9]. Using the Bessel function formula J / ( X ) = (cid:18) πX (cid:19) / sin X, J − / ( X ) = (cid:18) πX (cid:19) / cos X (2.13)in (2.12), one recovers the sine kernel sin π ( X − Y ) π ( X − Y ) , well known for describing bulk statistics in the CUE.2.2. Correlations for β = 1 , . From the viewpoint of classical random matrix theory the Cauchyensemble (1.5) for the values β = 1 , is associated with orthogonal and symplectic symmetries inthe corresponding matrix models. The point processes of the eigenvalues are Pfaffian; see e.g. [13,Ch. 6]. For such ensembles with classical weights such as the generalised Cauchy ensemble, the k -point correlation function admits a Pfaffian with entries involving summations of skew orthogonalpolynomials. For the respective β values , , R ( k ) N,β ( x , · · · , x k ) = Pf " S N,β ( x i , x j ) I N,β ( x i , x j ) D N,β ( x i , x j ) S N,β ( x j , x i ) i,j =1 , ··· ,k . (2.14)The functions S N,β ( x, y ) in (2.14) are related to the kernel (2.2) via the relations [1] S N, ( x, y ) = s ω ( x ) ω ( y ) ˜ ω ( y )˜ ω ( x ) S N − , ( x, y )+ 12 γ N − ˜ ω ( y ) ˜ I ( c, ¯ c ) N − ( y ) Z ∞−∞ sgn ( x − t ) ˜ I ( c, ¯ c ) N − ( t )˜ ω ( t ) dt for N even , (2.15)and S N, ( x, y ) = 12 s ω ( y )˜ ω ( x )˜ ω ( y ) ω ( x ) S N, ( x, y ) − γ N − ω ( y ) p ˜ ω ( y ) ˜ I ( c, ¯ c )2 N ( y ) Z ∞ x ˜ I ( c, ¯ c )2 N − ( t ) ω ( t )(˜ ω ( t )) / dt. (2.16)Other functions in (2.14) are I ,N ( x, y ) = − Z yx S ,N ( x, z ) dz − sgn ( x − y ) , I ,N ( x, y ) = − Z yx S ,N ( x, z ) dz,D β,N ( x, y ) = ∂∂x S β,N ( x, y ) . In the above formulas, γ j = ( p − − j ) /h j with h j given by (2.5), and the weights in (2.15), (2.16)are obtained by identifying the Pearson pair ( f, g ) = (1 + x , − N + p ) x + 2 q ) of ω given in (1.6).According to [1, eq. (2.17)] we therefore have ˜ ω ( x ) = p ω ( x ) /f ( x ) = (1 − ix ) ( c − / (1 + ix ) (¯ c − / , ˜ ω ( x ) = ω ( x ) f ( x ) = (1 − ix ) c +1 (1 + ix ) ¯ c +1 . (2.17)By equaling the weights (2.17) and (1.6) requires that the parameter c in (2.15), (2.16) must be Re c = − N − p and Im c = 2 q IRCULAR JACOBI β -ENSEMBLE 7 for β = 1 and Re c = − N − p and Im c = q for β = 4 . We remark that similar Pffafian formulas for the case β = 1 and N odd can be stated(see [1, eq. (2.11)]).Explicit asymptotic forms under the scaling (1.9) in the neighbourhood of the spectrum singu-larity case q = 0 of the functions (2.15), (2.16) have the known limit [18] lim N →∞ S N,β ( z ( X ) , z ( Y )) dz ( X ) dX = S ∞ ,β ( X, Y ) where S ∞ , ( X, Y ) = S ∞ , ( X, Y ) | p → p +1 + π Γ( p/ p/ / J p +1 / ( Y )(2 Y ) / × − / Γ( p/ / p/ Z X s − / J p +3 / ( s ) ds ! (2.18)and S ∞ , ( X, Y ) = S ∞ , (2 X, Y ) | p → p − πp J p − / (2 Y ) Y / Z X s − / J p +1 / (2 s ) ds. (2.19)2.3. The density for even β . For even β , as with other classical β -ensembles, the spectral densityof the circular Jacobi ensemble can be expressed in terms of generalised hypergeometric functionsbased on Jack polynomials with β variables [13, Ch. 13]. These exact results are known in theframework of Selberg integral theory. They involve the generalised hypergeometric function p F ( α ) q ( a , . . . , a p ; b , . . . , b q ; x , . . . , x m ) = X k =( k ,...,k m ) | k | ! [ a ] ( α ) k · · · [ a p ] ( α ) k [ b ] ( α ) k · · · [ b q ] ( α ) k C ( α ) k ( x , . . . , x m ) where, with k a partition of no more than m parts, | k | = m X j =1 k j , [ a ] ( α ) k = m Y j =1 Γ( a − ( j − /α + k j )Γ( a − ( j − /α +) and C ( α ) k ( x , . . . , x m ) denotes the renormalised Jack polynomials.Upon normalisation the PDF of cJ β E (1.8) can be written P N,β ( θ , θ , . . . , θ N ) := 1(2 π ) N M N ( pβ/ iq, pβ/ − iq, β/ × N Y j =1 e − q ( θ j + π ) | − e iθ j | βp Y ≤ j For β even, it can be shown that the spectral density ρ N,β ( θ ) = N Z π dθ · · · Z π dθ N P N,β ( θ, θ , . . . , θ N ) , (2.22)can be expressed as [13, Proposition 13.1.2] ρ N +1 ,β ( θ )= ( N + 1) M N [( p − β/ iq, ( p + 1) β/ − iq, β/ M N +1 ( pβ/ iq, pβ/ − iq, β/ e − qθ + iNβθ/ iNβπ/ (cid:12)(cid:12) − e iθ (cid:12)(cid:12) pβ × F ( β/ ( − N, p + 1 − iq/β ; − N − p − iq ) /β + 2; ( e − iθ ) β ) . (2.23)Crucially, the hypergeometric function appearing in (2.23) has the β -dimensional integral rep-resentation given by [13, Proposition 13.1.4] F ( β/ ( − N, − ˜ b, β − /β + ˜ a + 1; ( e − iθ ) β ) =1(2 π ) β M β ( N + ˜ a, ˜ b, /β ) Z π − π dθ · · · Z π − π dθ β β Y j =1 e iθ j (˜ a − ˜ b ) / | e iθ j | ˜ a +˜ b (1 + (1 − e − iθ ) e iθ j ) N × Y ≤ j The correlation kernel for β = 2 . In this section Proposition 1.1 will be demonstratedfor case β = 2 . This is accomplished by computing the large N expansion of the correspondingcorrelation kernel up to the first correction term. We begin with the scaled correlation kernel (2.2)on the unit circle given by S N, (cid:18) i e iX/N − e iX/N , i e iY/N − e iY/N (cid:19) , (3.1)with our specific interest being the scaled large N form in the neighbourhood of the spectrum sin-gularity θ = 0 . As the Routh-Romanovski polynomials (2.4) play a crucial role in the computationof (2.2), it is imperative to first understand its large N asymptotics. It follows from (2.4), theexplicit value of ω ( x ) from (1.6), and the hypergeometric function transformation formula [40,Eq. (15.8.7)] F ( a, b ; c ; z ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b ; a + b + 1 − c ; 1 − z ) , a ∈ Z ≤ , (3.2)that for k = 0 , , · · · , N and p + k > , ˜ I ( c, ¯ c ) N − k (cid:18) i e iX/N − e iX/N (cid:19) = (cid:18) i − e iX/N (cid:19) N − k F (cid:16) − N + k, p + k − iq ; 2 p + 2 k ; 1 − e iX/N (cid:17) . (3.3)Introduce now the confluent function ˜ A ( p + k,q ) ( j ; X ) = ( p + k − iq ) j (2 p + 2 k ) j F ( p + k + j − iq ; 2 p + 2 k + j ; 2 iX ) , p + k > , j ≥ . (3.4) IRCULAR JACOBI β -ENSEMBLE 9 For subsequent use, note that it follows from properties of the confluent hypergeometric functionin (3.4) that ddX ˜ A ( p + k,q ) ( j ; X ) = 2 i ˜ A ( p + k,q ) ( j + 1; X ) (3.5)and iX ( ˜ A ( p + k,q ) (1; X ) − ˜ A ( p + k,q ) (2; X )) = 2( p + k ) ˜ A ( p + k,q ) (1; X ) − ( p + k − iq ) ˜ A ( p + k,q ) (0; X ) . (3.6)The equation (3.5) follows directly from the derivative of (A.1) and (3.6) is the result of thecontiguous relations for Kummer’s confluent functions (See for example, [40, Sec. 13.3]). Ofimmediate relevance is the fact that applying the formula (2.6) to (3.4) gives the limit lim N →∞ (cid:18) − e iX/N i (cid:19) N − k ˜ I ( c, ¯ c ) N − k (cid:18) i e iX/N − e iX/N (cid:19) = ˜ C ( p,q,k )0 ( X ) , (3.7)where ˜ C ( p,q,k )0 ( X ) = ˜ A ( p + k,q ) (0; X ) is given in (2.10).Just knowing the limit (3.7) is not sufficient to acquire the /N expansion to the (3.1). Anextension to the limits (2.6) and (3.7) to the first correction is also required. This can be achievedby studying factorial terms appearing from the series representation of (3.4). In particular, forlarge N we have ( − N + k ) α = ( − α N α (cid:18) − α (2 k + α − N + α ( α − N (cid:0) α + (12 k − α + (12 k − k + 2) (cid:1) + O (cid:18) N (cid:19) (cid:19) (3.8)valid for fixed α ∈ N and k ∈ C . This follows as a special case of the asymptotic formula [35] Γ( z + a )Γ( z + b ) = z a − b (cid:18) z ( a − b )( a + b − z (cid:18) a − b (cid:19) (3( a + b − − a + b − 1) + O (cid:18) z (cid:19) (cid:19) (3.9)when | z | → ∞ and | arg z | < π , and has been used in the context of studying correction terms tolimiting kernels in random matrix theory in the recent work [15]. Plugging the expansion (3.8) in(3.3), we get (see Appendix A.1 for the details) (cid:18) − e iX/N i (cid:19) N − k ω (cid:18) − e iX/N i (cid:19) ˜ I ( c, ¯ c ) N − k (cid:18) i e iX/N − e iX/N (cid:19) = ˜ C ( p,q,k )0 ( X ) + 1 N ˜ C ( p,q,k )1 ( X ) + O (cid:18) N (cid:19) , (3.10)where ˜ C ( p,q,k )0 is given in (2.10) and ˜ C ( p,q,k )1 ( X ) = 12 (2 iX ) (cid:16) ˜ A ( p + k,q ) (1; X ) − ˜ A ( p + k,q ) (2; X ) (cid:17) − k (2 iX ) ˜ A ( p + k,q ) (1; X ) + ( ik + q ) XC ( p,q,k )0 ( X ) . (3.11) The term appearing in the numerator from the Christoffel-Darboux summation in (2.2) promptsthe introduction of the functions J ( p,q,k )0 ( X, Y ) = X ˜ C ( p,q,k +1)0 ( X ) ˜ C ( p,q,k )0 ( Y ) − ( X ↔ Y ) , (3.12) J ( p,q,k )1 ( X, Y ) = X (cid:16) ˜ C ( p,q,k +1)0 ( X ) ˜ C ( p,q,k )1 ( Y ) + ˜ C ( p,q,k +1)1 ( X ) ˜ C ( p,q,k )0 ( Y ) (cid:17) − ( X ↔ Y ) (3.13)and Q ( p,q,k )1 = p (2 p + 2 k + 1) . (3.14)Direct substitution of (3.10) into (2.2) (again, for details see Appendix A.1) results in the expansionseen in (1.10) for the case β = 2 with ˆ K ( p,q ; k ) ∞ , ( X, Y ) = 2 p + k ) | Γ( p + k + 1 − iq ) | π Γ(2 p + 2 k + 2)Γ(2 p + 2 k + 1) e − i ( X + Y ) − qπ ( XY ) ( p + k +1) X ( X − Y ) J ( p,q,k )0 ( X, Y ) (3.15)and ˆ L ( p,q ; k )1 , ( X, Y ) =2 p + k ) | Γ( p + k + 1 − iq ) | π Γ(2 p + 2 k + 2)Γ(2 p + 2 k + 1) e − i ( X + Y ) − qπ ( XY ) ( p +1) X ( X − Y ) ( J ( p,q,k )1 ( X, Y ) + Q ( p,q,k )1 J ( p,q,k )0 ( X, Y )) . (3.16)Moreover, upon simplification (3.16) is seen to be a simple directive operation of (3.15), as givenby (1.11), which is stated as the following proposition. Proposition 3.1. In the β = 2 case, we have ˆ L ( p,q ;0)1 , ( X, Y ) = p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) ˆ K ( p,q ;0) ∞ , ( X, Y ) . (3.17) Proof. To verify this equation, we must show J ( p,q, ( X, Y ) + Q ( p,q, J ( p,q, ( X, Y )= p (cid:18) − iX − iY + 2 p + X ∂∂X + Y ∂∂Y (cid:19) J ( p,q, ( X, Y ) . (3.18)Now, from the contiguous formula (3.6) for the confluent hypergeometric function, one can find ˜ C ( p,q,k )1 ( X ) = p · iX ˜ A ( p + k,q ) (1; X ) − p · iX ˜ A ( p + k,q ) (0; X ) . Thus, the left hand side of (3.18) is given by p (cid:16) iXY ˜ A ( p +1 ,q ) (0; X ) ˜ A ( p,q ) (1; Y ) + 2 iX ˜ A ( p +1 ,q ) (1; X ) ˜ A ( p,q ) (0 , Y ) (cid:17) − p (cid:0) iXY + iX − (2 p + 1) X (cid:1) ˜ A ( p +1 ,q ) (0; X ) ˜ A ( p,q ) (0; Y ) − ( X ↔ Y ) . (3.19)Whereas the right hand side (3.18) coincides with (3.19) with use of the derivative formula (3.5).Therefore the large N expansion of (3.1) is S N, ( z ( X ) , z ( Y )) dzdX = ˆ K ( p,q ;0) ∞ , ( X, Y )+ 1 N p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) ˆ K ( p,q ;0) ∞ , ( X, Y ) + O (cid:18) N (cid:19) . (3.20) (cid:3) IRCULAR JACOBI β -ENSEMBLE 11 Now we prepare to proceed to different symmetries with regards to the random matrix ensemble.For the cases β = 1 , , computation of the k -point correlation function at the hard regime for thecircular Jacobi ensemble largely depends on the large N behaviour of kernel S N,β (cid:18) i e iX/N − e iX/N , i e iY/N − e iY/N (cid:19) which can be explicitly derived from (2.15) and (2.16). As a prerequisite, knowledge of the large N expansion of S N, is given in (3.20) and the remaining terms involving some integrals in (2.15),(2.16) will be deferred to the Appendices A.2 and A.3.3.2. The correlation kernel for β = 1 . The following variables and functions are relevant tothe case β = 1 . Denote η = 2 √ π Γ( p + 2)Γ( p + 5 / | Γ(( p + 3) / iq ) | , η = − ( p + 1) e − qπ p +2 | Γ( p + 2 − iq ) | π Γ(2 p + 4)Γ(2 p + 5) (3.21)and introduce the integral operator J o [ f ( s )] ( X ) = Z X e − is − qπ s p +1 f ( s ) ds. (3.22)For N even and z ( X ) be defined as in (1.9), direct large N expansion by following formula (2.15)gives (for details, please refer to Appendix A.2) S N, ( z ( X ) , z ( X )) dzdX = ˆ K ( p, q ;0) ∞ , ( X, Y ) + 1 N ˆ L ( p, q ;0)1 , ( X, Y ) + O (cid:18) N (cid:19) (3.23)where ˆ K ( p, q ;0) ∞ , ( X, Y ) = YX ˆ K ( p, q ;1) ∞ , ( X, Y ) + η X e − iY Y p +2 ˜ C ( p, q, ( Y ) (cid:16) J o [ ˜ C ( p, q, ( s )] − η (cid:17) ˆ L ( p, q ;0)1 , ( X, Y ) = YX ˆ L ( p, q ;1)1 , ( X, Y ) + η X e − iY Y p +2 (cid:18) ˜ C ( p, q, ( Y ) (cid:18) J o [ ˜ C ( p, q, ( s )] − p ( p + 2) η (cid:19) + ˜ C ( p, q, ( Y ) (cid:16) J o [ ˜ C ( p, q, ( s )] − η (cid:17) (cid:19) (3.24)with ˆ K ( p, q ;1) ∞ , , ˆ L ( p, q ;1)1 , specified by (3.15), (3.16). A direct computation gives the following propo-sition, as a special case of our main result in Proposition 1.1. Proposition 3.2. For β = 1 , we have ˆ L ( p, q ;0)1 , ( X, Y ) = p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) ˆ K ( p, q ;0) ∞ , ( X, Y ) . (3.25) Proof. The proof is divided into three parts according to the number of terms in the kernels. Thefirst part is easily verified similar to the proof of Proposition 3.1. The second term is to verify ˜ C ( p, q, ( Y ) J o [ ˜ C ( p, q, ( s )] + ˜ C (1)1 ( Y ) J o [ ˜ C ( p, q, ( s )] + p (2 p + 3) ˜ C ( p, q, ( Y ) J [ ˜ C ( p, q, ( s )]= p (cid:18) p + 1 − iY + Y ddY (cid:19) ( ˜ C ( p, q, ( Y ) J o [ ˜ C ( p, q, ( s )]) + p ˜ C ( p, q, ( Y ) X ddX J o [ ˜ C ( p, q, ( s )] . By making use of the confluent hypergeometric function, one has ddY ˜ C ( p, q, ( Y ) = 2 i ˜ A ( p +1 , q ) (1; Y ) , ˜ C ( p, q, ( Y ) = 2 ipY ˜ A ( p +1 , q ) (1; Y ) − ipY ˜ A ( p +1 , q ) (0; Y ) . (3.26) Substituting them into the above equation, the task becomes to verify that J o [ ˜ C ( p, q, ( s )] + p ( p + 2) J o [ ˜ C ( p, q, ( s )] = pX ddX J o [ ˜ C ( p, q, ( s )] = pe − iX − qπ X p +2 ˜ C ( p, q, ( X ) . Since when X = 0 , both sides of the equation are equal to zero, we can show the equation bytaking the derivative on both sides again and one gets ˜ C ( p, q, ( X ) = − ipX ˜ C ( p, q, ( X ) + pX ddX ˜ C ( p, q, ( X ) , which in fact could be verified by recognising that ˜ C ( p, q, = 2 ipX ˜ A ( p +2;2 q ) (1; X ) − ipX ˜ A ( p +2;2 q ) (0; X ) , ddX ˜ C ( p, q, ( X ) = 2 i ˜ A ( p +2;2 q ) (1; X ) . An equivalent form of the last term is to demonstrate ˜ C ( p, q, ( Y ) + p ( p + 1) ˜ C ( p, q, ( Y ) = p (cid:18) p + 1 − iY + Y ddY (cid:19) ˜ C ( p, q, ( Y ) , and it is established by using (3.26). (cid:3) Analogous expansion to (3.23) for the case N odd and β = 1 too can be made explicit from thekernel (2.15) using the formula [13, Eq. 6.112] S odd N, ( x, y ) = S N, ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) N → N − + ω ( y ) ˜ I ( c, ¯ c ) N − ( y )2˜ s N − − γ N − ˜ s N − ω ( y )˜ s N − Z ∞−∞ ω ( t ) sgn ( x − t ) (cid:16) ˜ I ( c, ¯ c ) N − ( t ) ˜ I ( c, ¯ c ) N − ( y ) − ˜ I ( c, ¯ c ) N − ( t ) ˜ I ( c, ¯ c ) N − ( y ) (cid:17) , (3.27)where ˜ s k := 12 Z ∞−∞ ω ( x ) ˜ I ( c, ¯ c ) k ( x ) dx. This yields the same large N expansion (3.23) with identical leading and correction terms.3.3. The correlation kernel for β = 4 . Here we define J s, = J s h ˜ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) i , J s, = J s h ˜ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) + ˜ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) i + 2 p (4 p + 1) J s, , where the integral operator J s is given by J s [ f ( s )]( X, Y ) = p p | Γ(2 p + 1 − iq ) | π Γ(4 p + 1)Γ(4 p + 2) e − qπ − iY Y p +1 Z X e − is s p f ( s ) ds. For large N , we have S N, ( z ( X ) , z ( X )) dzdX = ˆ K (2 p,q ;0) ∞ , ( X, Y ) + 1 N ˆ L (2 p,q ;0)1 , ( X, Y ) + O (cid:18) N (cid:19) , (3.28)where ˆ K (2 p,q ;0) ∞ , ( X, Y ) = Y X ˆ K (2 p,q ;0) ∞ , (2 X, Y ) + 1 X J s, , ˆ L (2 p,q ;0)1 , ( X, Y ) = Y X ˆ L (2 p,q ;0)1 , (2 X, Y ) + 1 X J s, . (3.29)The term O (1 /N ) term in (3.28), ˆ L (2 p,q ;0)2 , ( X, Y ) say, is made explicit in (A.17) below.The correction terms (3.24), (3.29) exhibit the structure (1.11), which means we can state thefollowing proposition. IRCULAR JACOBI β -ENSEMBLE 13 Proposition 3.3. For β = 4 case, we have ˆ L (2 p,q ;0)1 , ( X, Y ) = 2 p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) ˆ K (2 p,q ;0) ∞ , ( X, Y ) . (3.30) Proof. This will be demonstrated explicitly for the case β = 4 by a direct derivative computationof ˆ K (2 p,q ;0) ∞ , ( X, Y ) . Immediate from (1.11) with β = 2 , we have Y X ˆ L (2 p,q ;0)1 , (2 X, Y ) = 2 p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) (cid:18) Y X ˆ K (2 p,q ;0) ∞ , (2 X, Y ) (cid:19) . Therefore it is sufficient to show X J s, = 2 p (cid:18) X ∂∂X + Y ∂∂Y + 1 (cid:19) X J s, , (3.31)which is equivalent to ¯ C (0)1 ( Y ) ¯ J s h ¯ C (1)0 ( s ) i + ¯ C (0)0 ( Y ) ¯ J s h ¯ C (1)1 ( s ) i + 2 p (4 p + 1) ¯ C (0)0 ( Y ) ¯ J s h ¯ C (1)0 ( s ) i = 2 p (cid:18) (2 p − iY ) ¯ C (0)0 ( Y ) ¯ J s h ¯ C (1)0 ( s ) i + 2 pY ddY ¯ C (0)0 ( Y ) ¯ J s h ¯ C (1)0 ( s ) i(cid:19) + 2 pe − iX X p +1 ¯ C (0)0 ( Y ) ¯ C (1)0 ( X ) with ¯ J s [ f ( s )] = R X e − is s p f ( s ) ds and ¯ C ( k ) j ( X ) = ˜ C (2 p,q,k ) j (2 X ) for j = 0 , , . By substitutingthe formulas (3.5), (3.6), ddY ¯ C (0)0 ( Y ) = ddY ˜ A (2 p,q ) (0; 2 Y ) = 4 i ˜ A (2 p,q ) (1; 2 Y ) , ¯ C (0)1 ( Y ) = 8 ipY ˜ A (2 p,q ) (1; 2 Y ) − ipY ˜ A (2 p,q ) (0; 2 Y ) (3.32)into the above equality, it can shown that the difference of the derivatives of the LHS and RHS of(3.32) vanish. Additionally, since both the LHS and RHS of (3.32) vanish at X = 0 , (3.31) cannow be concluded. (cid:3) The density for β ∈ N . In (2.23) with the substitution (2.24), the θ and N dependenceare factorised in the integrand. Thus at the spectrum singularity, the large N asymptotics of the β -dimensional integral (2.24) can be obtained by Taylor expansion without the need of a saddlepoint analyses. Details for the large N expansions of the normalisation terms, identifications ofthe leading large N terms and their simplifications are given in Appendix A.4. This working leadsto Proposition 1.2, and moreover shows (see also [28]) ρ ∞ ( θ ) = C ( p,q ) β e iβθ/ θ pβ F ( β/ ( p + 1 − iq/β ; 2 p + 2; ( − iθ ) β ) , (3.33)where the generalised hypergeometric function has the β -dimensional integral representation givenby F ( β/ ( p + 1 − iq/β ; 2 p + 2; ( − iθ ) β ) = Z π − π dθ · · · Z π − π dθ β β Y j =1 e iθ j (˜ a − ˜ b ) / | e iθ j | ˜ a +˜ b e iθe iθj × Y ≤ j This work is part of a research program supported by the Australian Research Council (ARC)through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). AKTwould like to thank Mario Kieburg for many helpful discussions in preparation for this manuscript. Appendix A A.1. Case: β = 2 . The asymptotic expansion of (3.10) requires the expansion of each factorappearing on the right hand side. Note that (3.4) has the power series in the form (2 iX ) j ˜ A ( p + k,q ) ( j ; X ) = ∞ X α =0 ( p + k − iq ) α (2 p + 2 k ) α (2 iX ) α α ! ( α · · · ( α − j + 1)) . (A.1)From a simple Taylor expansion of the term (cid:16) − e iX/N i (cid:17) N − k and substituting the formula (3.9)into the series representation of (3.3) gives (cid:18) − e iX/N i (cid:19) N − k ˜ I ( c, ¯ c ) N − k (cid:18) − cot XN (cid:19) = ˜ A ( p + k,q ) (0; X )+ 1 N 12 (2 iX ) (cid:16) ˜ A ( p + k,q ) (1; X ) − ˜ A ( p + k,q ) (2; X ) (cid:17) − k (2 iX ) ˜ A ( p + k,q ) (1; X ) + O (cid:18) N (cid:19) . (A.2)The expansion of ω begins by noting ω ( x ) := (1 − ix ) c (1 + ix ) ¯ c = (1 + x ) − p exp(2 q arctan x ) , c = − p + iq, x ∈ R . (A.3)Use of the trigonometric formula arctan (cid:18) i e iX/N − e iX/N (cid:19) = i i cot XN − i cot XN , cot XN = i e iX/N + e − iX/N e iX/N − e − iX/N tells us that arctan (cid:18) i e iX/N − e iX/N (cid:19) = i (cid:16) − e − iX/N (cid:17) = i − 1) + XN . Thus we see from (A.3) that ω (cid:18) i e iX/N − e iX/N (cid:19) / = (cid:18) sin XN (cid:19) N + p e q ( XN − π ) . (A.4)Combining (A.2) with (A.4) yields (3.10).We now take account of the scaling by (1.9). The expansion of the normalisation (2.5) appearingin (2.2) can be achieved by using the formula (3.9). We have h N − k (cid:12)(cid:12)(cid:12)(cid:12) p = N + p, q = q = h ( p + k,q ) N p +2 k − × (cid:18) p (2 p + 2 k − N + ( p + k − p + 2 k − p − p − k )6 N + O (cid:18) N (cid:19) (cid:19) , (A.5) IRCULAR JACOBI β -ENSEMBLE 15 where h ( p + k,q ) = 2 p + k ) − | Γ( p + k − iq ) | π Γ(2 p + 2 k )Γ(2 p + 2 k − . Finally, sin X/N sin Y /N sin( X − Y ) /N = XYN ( X − Y ) (cid:18) − XY N + O (cid:18) N (cid:19)(cid:19) . (A.6)Therefore, by substituting (3.10), (A.5), (A.6) into (2.2), we get S N − k, (cid:18) i e iX/N − e iX/N , i e iY/N − e iY/N (cid:19) = e − i ( X + Y ) − qπ h ( p + k +1 ,q ) ( XY ) p + k +1 N ( X − Y ) (cid:18) ( J ( p,q,k )0 + 1 N (cid:16) J ( p,q,k )1 + Q ( p,q,k )1 J ( p,q,k )0 (cid:17) + 1 N (cid:16) J ( p,q,k )2 + Q ( p,q,k )1 J ( p,q,k )1 + Q ( p,q,k )2 J ( p,q,k )0 (cid:17) + O (cid:18) N (cid:19) (cid:19) . (A.7)Here J , J , Q are given in (3.12), (3.13), (3.14), Q ( p,q,k )2 = − XY p + k )(2 p + 2 k + 1)(6 p − p − k − (A.8)and J ( p,q,k )2 = X (cid:18) ˜ C ( p,q,k +1)2 ( X ) ˜ C ( p,q,k )0 ( Y )+ ˜ C ( p,q,k +1)1 ( X ) ˜ C ( p,q,k )1 ( Y ) + ˜ C ( p,q,k +1)0 ( X ) ˜ C ( p,q,k )2 ( Y ) (cid:19) − ( X ↔ Y ) (A.9)with ˜ C ( p,q,k )2 ( X ) = 18 (2 iX ) (cid:16) ˜ A ( p + k,q ) (2; X ) − A ( p + k,q ) (3; X ) + ˜ A ( p + k,q ) (4; X ) (cid:17) + 16 (2 iX ) (cid:16) ˜ A ( p + k,q ) (1; X ) − k + 1) ˜ A ( p + k,q ) (2; X ) + (3 k + 2) ˜ A ( p + k,q ) (3; X ) (cid:17) + 14 (2 iX ) (cid:16) − k ˜ A ( p + k,q ) (1; X ) + 2 k ( k + 1) ˜ A ( p + k,q ) (2; X ) (cid:17) + ˜ C ( p,q,k )1 ( X ) − ˜ C ( p,q,k )0 ( X ) + (cid:18) ( ik + q ) − k + p (cid:19) X ˜ C ( p,q,k )0 ( X ) . (A.10)Together with the Jacobian as a result of the change of variables (1.9), this working shows thelarge N expansion of (2.2) up to the first is given by (1.10), and furthermore shows that explicitform of the O (1 /N ) term is given by ˆ L ( p,q ;0)2 , ( X, Y )= f ( X, Y ) (cid:18) J ( p,q, ( X, Y ) + Q ( p,q, J ( p,q, ( X, Y ) + (cid:18) Q ( p,q, + 13 X (cid:19) J ( p,q, ( X, Y ) (cid:19) , (A.11)where f ( X, Y ) is equal to the prefactor of J ( p, q, in (3.15). The expressions for Q ( p,q, , J ( p,q, ( X, Y ) are given in (A.8), (A.9). A.2. Case: β = 1 . Recall that for β = 1 we take Re c = − N − p and Im c = 2 q . For the case N even, the integral term (2.15) evaluates as [13, Eq. 6.113] Z ∞−∞ sgn ( x − t ) ˜ I ( c, ¯ c ) N − ( t ) ω ( t ) dt = 2 Z x −∞ ˜ I ( c, ¯ c ) N − ( t ) ω ( t ) dt − (cid:18)Z ∞−∞ ω ( t ) dt (cid:19) N/ − Y j =0 γ j γ j +1 (A.12)where N/ − Y j =0 γ j γ j +1 = Γ (cid:0) p +52 (cid:1) Γ (cid:0) p +22 (cid:1) Γ (cid:0) p +32 (cid:1) √ π (cid:12)(cid:12) Γ (cid:0) p +3+2 iq (cid:1)(cid:12)(cid:12) Γ (cid:0) N − (cid:1) Γ (cid:16) N + p +12 (cid:17) (cid:12)(cid:12)(cid:12) Γ (cid:16) N + p +1+2 iq (cid:17)(cid:12)(cid:12)(cid:12) Γ (cid:16) N + p (cid:17) Γ (cid:16) N +2 p +32 (cid:17) (cid:12)(cid:12)(cid:12) Γ (cid:16) N + p +12 (cid:17)(cid:12)(cid:12)(cid:12) . (A.13)Recall γ j given below (2.16). The orthogonality property of the Routh-Romanovski polynomials(2.3) implies Z ∞−∞ ω ( t ) dt = 2 − p π Γ( p )Γ (cid:0) p +1+ iq (cid:1) Γ (cid:0) p +1 − iq (cid:1) (A.14) = √ π Γ (cid:16) N + p (cid:17) Γ (cid:16) N + p +12 (cid:17) Γ (cid:16) N + p +12 (cid:17) Γ (cid:16) N + p +1+2 iq (cid:17) Γ (cid:16) N + p +1 − iq (cid:17) . (A.15)The second line in (A.14) is the result of the duplication formula for the gamma function. Bycombining (A.13), (A.14) with (A.12) and making use of (3.9), we have Z ∞−∞ ω ( t ) dt = √ π Γ (cid:16) N + p (cid:17) Γ (cid:16) N + p +12 (cid:17) Γ (cid:16) N + p +12 (cid:17) Γ (cid:16) N + p +1+2 iq (cid:17) Γ (cid:16) N + p +1 − iq (cid:17) and hence Z ∞−∞ ˜ I ( c, ¯ c ) N − ( t ) ω ( t ) dt = Γ (cid:0) p +22 (cid:1) Γ (cid:0) p +52 (cid:1) Γ (cid:0) p +32 (cid:1) Γ (cid:0) p +3+2 iq (cid:1) Γ (cid:0) p +3 − iq (cid:1) Γ (cid:0) N − (cid:1) Γ (cid:16) N +2 p +32 (cid:17) = η N − p − (cid:18) − p ( p + 2) N + ( p + 2)( p + 3)(3 p + p + 1)6 N + O (cid:18) N (cid:19)(cid:19) . Moreover the expansion of the Routh-Romanovski polynomials as given by (A.2) gives ω (cid:18) − cot YN (cid:19) ˜ I ( c, ¯ c ) N − k (cid:18) − cot YN (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p = N + p, q =2 q = (cid:18) YN (cid:19) p + k +1 e − qπ − iY ( − k (cid:18) ˜ C ( p, q,k )0 ( Y ) + 1 N ˜ C ( p, q,k )1 ( Y ) + 1 N ˜ C ( p, q,k )2 ( Y ) + O (cid:18) N (cid:19)(cid:19) and therefore after a change of variables the first term on the left hand side of (A.12) becomes Z − cot( X/N ) −∞ ˜ I ( c, ¯ c ) N − ( t ) ω ( t ) dt = 1 N p +2 Z X e − qπ − is s p +1 × (cid:18) ˜ C ( p, q, ( s ) + 1 N ˜ C ( p, q, ( s ) + 1 N (cid:18) ˜ C ( p, q, ( s ) + s C (2)0 ( s ) (cid:19) + O (cid:18) N (cid:19)(cid:19) ds. Therefore the large N expansion of the kernel (2.15) under the scaling (1.9) results in (3.23). IRCULAR JACOBI β -ENSEMBLE 17 A.3. Case: β = 4 . The process for the asympotic expansion of the symplectic kernel (2.16) issimilar. With Re c = − N − p and Im c = q , from the known expansion (A.2) one can find Z ∞− cot( X/N ) ˜ I ( c, ¯ c )2 N − ( t ) ω ( t )( ω ( t )) / dt = − Z − cot( X/N ) −∞ ˜ I ( c, ¯ c )2 N − ( t ) ω ( t ) dt = e − qπ N p +1 Z X e − is s p (cid:18) ˜ C (2 p,q, (2 s ) + 1 N ˜ C (2 p,q, (2 s ) + 1 N ˜ C (2 p,q, (2 s ) + O (cid:18) N (cid:19)(cid:19) . (A.16)Together with the expansion γ N − = 2 ph N − = 2 ph (2 p +1 ,q ) N p +1 (cid:18) p (4 p + 1) N + p (4 p + 1)(24 p − p − N + O (cid:18) N (cid:19)(cid:19) which is the result of (3.9), (A.16) and (2.16), this implies (3.28). Moreover, the term proportionalto /N in (3.28) is explicitly given by ˆ L (2 p,q ;0)2 , ( X, Y ) = Y X ˆ L (2 p,q ;0)2 , (2 X, Y ) + X − Y Y X ˆ K (2 p,q ;0) ∞ , (2 X, Y ) + 1 X J s, , (A.17)where J s, = J s h ¯ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) + ˜ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) + ˜ C (2 p,q, (2 Y ) ˜ C (2 p,q, (2 s ) i + 2 p (4 p + 1) J s, + p (4 p + 1)(24 p − p − J s, . (A.18)A.4. Case: β even. The large N expansion of (2.23) comes in two parts – the first is the hyper-geometric function component and second is the normalisation terms expressed in terms of Morrisintegrals. Expanding the hypergeometic function. The feasibility of expanding the functional part of (2.23)requires expressing the N -dimensional integral to one with β dimensions via known duality trans-formations. For negative integer a, b , from the relation [13, Proposition 13.1.7] F (1 /α )1 ( a, b ; c ; ( t ) m ) = F (1 /α )1 ( a, b ; a + b + 1 + α ( m − − c ; (1 − t ) m ) F (1 /α )1 ( a, b ; a + b + 1 + α ( m − − c ; (1) m ) , (in the case m = 1 this reduces to (3.2)) the hypergeometric function in (2.23) becomes F ( β/ ( − N, p + 1 − iq/β ; 2 p + 2; (1 − e − iθ ) β ) F ( β/ ( − N, p + 1 − iq/β ; 2 p + 2; (1) β ) . (A.19)With the parameters ˜ a, ˜ b defined in (2.25), the numerator in (A.19) has the β -dimensional integralrepresentation given by I N +1 ( θ ) := 1(2 π ) β M β (˜ a, ˜ b, /β ) Z π − π dθ · · · Z π − π dθ β β Y j =1 e iθ j (˜ a − ˜ b ) / | e iθ j | ˜ a +˜ b (1 + (1 − e − iθ ) e iθ j ) N × Y ≤ j 11 + e iθ p ( θ ) . (A.23)The second equality in (A.23) is the result of integration by parts. Similarly, I − iθ β X j =1 e iθ j + θ β X j =1 e iθ j ( θ ) = Z π − π dθ · · · Z π − π dθ β β Y j =1 e iθ j (˜ a − ˜ b ) / | e iθ j | ˜ a +˜ b × β X p =1 ∂ ∂θ p β Y j =1 e iθe iθj Y ≤ j 11 + e iθ p ( θ ) + i ˜ aθ I " β X p =1 e iθ p ( θ )+ i β θ I β X j,k =1 j = k e iθ j (cid:18) e iθ j e iθ k − e iθ j − e − iθ j e − iθ k − e − iθ j (cid:19) ( θ ) . (A.24) IRCULAR JACOBI β -ENSEMBLE 19 Using the fact that the sum in the last term (A.24) is invariant when interchanging the indices θ j and θ k , it follows that I β X j,k =1 j = k e iθ j (cid:18) e iθ j e iθ k − e iθ j − e − iθ j e − iθ k − e − iθ j (cid:19) ( θ ) = ( β − I " β X p =1 e iθ p ( θ ) . Combining (A.23), (A.24) with (A.22) and observing iθI ′∞ ( θ ) = − θ I " β X p =1 e iθ p ( θ ) , we get I N +1 (cid:18) θN + 1 (cid:19) = I ∞ ( θ ) + 1 N [( q + iβ/ ipβ/ θI ∞ ( θ ) + pθI ′∞ ( θ )] + O (cid:18) N (cid:19) . (A.25) Expanding the Morris integral. As a corollary of the multiplication formula for the gamma functionwe have β − Y j =0 Γ(2 j/β + z ) = (2 π ) β/ − ( β/ − β (1+2 z ) / Γ( βz/ β (1 + z ) / . The Morris integral (2.21) specifying the normalisation (2.24) becomes M β ( N + ˜ a, ˜ b, /β ) = (2 π ) β/ − ( β/ β/ − β − Y j =0 Γ(1 + 2 /β )Γ(2( j + 1) /β + 1) × Γ( β/ − pβ/ iq )Γ( − pβ/ iq )Γ( N β/ p + β/ N β/ pβ + 1)Γ( β ( N − / βp/ iq + 1)Γ( N β/ βp/ iq + 1) . (A.26)It follows from this that the ratio of the Morris integrals appearing in (2.23) evaluates to Γ( N β/ pβ/ iq + 1)Γ( N β/ p − β/ iq + 1)Γ( pβ/ − iq + 1)Γ(1 + β/ βN/ pβ + 1)Γ( N β/ β/ p − β/ iq + 1) . (A.27)Hence with use of (3.9) we get M β ( q, ˜ b, /β ) M N [( p − β/ iq, ( p + 1) β/ − iq, β/ M N ( pβ/ iq, pβ/ − iq, β/ 2) = C ( p,q ) β M β (˜ a, ˜ b, /β ) Γ( N β/ pβ + β/ N β/ β/ C ( p,q ) β M β (˜ a, ˜ b, /β ) (cid:18) N β (cid:19) pβ (cid:20) N ( p β + pβ + p ) + O (cid:18) N (cid:19)(cid:21) , (A.28)where C ( p,q ) β is given by (3.35).The β -dimensional integral I ∞ ( θ ) given below (A.21) can be expressed in terms of a generalisedhypergeometric function [13, Chapter 13 Q4(i)] I ∞ ( θ ) = (2 π ) β M β (˜ a, ˜ b, /β ) F ( β/ ( p + 1 − iq/β ; 2 p + 2; ( − iθ ) β ) . and this together with (A.28) and (A.25) establishes Proposition 1.2. References [1] M. Adler, P. Forrester, T. Nagao and P. van Moerbeke. Classical skew orthogonal polynomials andrandom matrices . J. Stat. 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(2014), 083302.[40] Digital Library of Mathematical Functions, https://dlmf.nist.gov. School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Sta-tistical Frontiers, The University of Melbourne, Victoria 3010, Australia E-mail address : [email protected] School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Sta-tistical Frontiers, The University of Melbourne, Victoria 3010, Australia E-mail address : [email protected] School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Sta-tistical Frontiers, The University of Melbourne, Victoria 3010, Australia E-mail address ::