Products and ratios of characteristic polynomials of random Hermitian matrices

Transcription

1 JOURAL OF ATHEATICAL PHYSICS VOLUE 44, UBER 8 AUGUST 2003 Products and ratios of characteristic polynomials of random Hermitian matrices Jinho Baik a) Department of athematics, Princeton University, Princeton, ew Jersey and Department of athematics, University of ichigan, Ann Arbor, ichigan Percy Deift b) Courant Institute of athematical Sciences, ew York University, ew York, ew York and School of athematics, Institute for Advanced Study, Princeton, ew Jersey Eugene Strahov c) Department of athematical Sciences, Brunel University, Uxbridge, UB8 3PH, United ingdom Received 7 April 2003; accepted 9 April 2003 We present new and streamlined proofs of various formulas for products and ratios of characteristic polynomials of random Hermitian matrices that have appeared recently in the literature American Institute of Physics. DOI: / I. ITRODUCTIO In random matrix theory, unitary ensembles of matrices H play a central role. 15 Such ensembles are described by a measure d with finite moments R x k d(x), k0,1,2,..., and the associated distribution function for the eigenvalues x i x i (H) of matrices H in the ensembles has the form dp, x 1 x 2 dx, 1.1 where d(x) i1 d(x i ), (x) i j1 (x i x j ) is the Vandermonde determinant for the x i s, and (x) 2 d(x) is the normalization constant. The special case d(x) e x2 dx is known as the Gaussian unitary ensemble GUE. For symmetric functions f (x) f (x 1,...,x ) of the x i s, f 1 f xx 2 dx 1.2 denotes the average of f with respect to dp,. Recently there has been considerable interest in the averages of products and ratios of the characteristic polynomials D,H i1 (x i (H)) of random matrices with respect to various ensembles. Such averages are used, in particular, in making predictions about the moments of the Riemann-zeta function see Refs circular ensembles and 3 unitary ensembles. any other uses are described, for example, in Refs. 1, 12, and 17. By 1.2, for unitary ensembles, such averages have the form a Electronic mail: jbaik@math.princeton.edu b Electronic mail: deift@cims.nyu.edu c Electronic mail: eugene.strahov@brunel.ac.uk /2003/44(8)/3657/14/$ American Institute of Physics

2 3658 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov j1 D j,h 1 D j j1,h j1 j1 j x i i1 j x i j1 x 2 dx. In this article we consider certain explicit determinantal formulas for 1.3 see 2.6, 2.24, 2.36, 3.3, and 3.12 below. Formula 2.6 is due to Brezin and Hikami 3 see also Ref. 16, and when all the j s are equal, see Ref. 10, whereas 2.24, 2.36, 3.3, and 3.12 are due to Fyodorov and Strahov. 17,11 References 17 and 11 also contain a discussion of the history of these formulas. The formulas 3.3 and 3.12 are particularly useful in proving universality results for the ratios 1.3 in the Dyson limit as see Ref. 17. For a discussion of other universality results, particularly the work of Brezin Hikami and Fyodorov in special cases, we again refer the reader to Ref. 17. The asymptotic analysis in Ref. 17 is based on the reformulation of the orthogonal polynomial problem as a Riemann Hilbert problem by Fokas, Its, and itaev. 9 The Riemann Hilbert problem is then analyzed asymptotically using the noncommutative steepest-descent method introduced by Deift and Zhou, 5 and further developed with Venakides in Ref. 6 to allow for fully nonlinear oscillations, and in Refs. 7 and 8. Our goal in this article is to give new, streamlined proofs of , using only the properties of orthogonal polynomials and a minimum of combinatorics. Along the way we will also need an integral version of the classical Binet Cauchy-formula due to C. Andréief dating back to 1883 see Lemma 2.1 below. Let j (z)x j denote the jth monic orthogonal polynomial with respect to the measure d, R j x k xdxc j c k jk ; j,k0, 1.4 where the norming constants c j s are positive. The key observation in our approach is that for 1 and 0 in1.3, D,H 1.5 see Ref. 18. In our words, the orthogonal polynomial () with respect to d is also precisely the average polynomial i1 (x i ) with respect to dp,. Formula 1.5 appears already in the work of Heine in the 1880s see Ref. 18. Set d,m t j1 m j1 j t j t 1.3 dt,,m0, 1.6 d 0,0 (t)d(t), and let,m j (t) denote the jth monic orthogonal polynomial with respect to d,m. With this notation we see immediately from 1.3 and 1.5 that Q j1 D j,h/q 1, j1 D j,h is proportional to ( ). Using a classical determinantal formula of Christoffel see Ref. 18 for,0 () and a more recent formula of Uvarov 19 for 0,m (), we are then led see Sec. II to 2.6, 2.24, and 2.36 in a rather straightforward way. Formula 3.3 appears to have a different character from 2.6, 2.24, and 2.36, and relies on Lemma 2.1 mentioned above, which computes the integral of the product of two determinants: formula 3.12 follows see Sec. III by combining 3.3 with 2.6 and In Ref. 17 the authors present a variety of additional formulas for Q j1 D j,h/q j1 D j,h for cases of and not covered by : we leave it to the interested reader to verify that the method of this article can also be used to derive these formulas in a straightforward manner. Remark 1.1: As is well known see, e.g., Ref. 18, each measure d gives rise to a tridiagonal operator

3 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices 3659 JJda1 b1 0 b 1 a 2 b 2 0 b 2 a 3, bi0, 1.7 with generalized eigenfunctions given by the orthonormal polynomials i.e., p j xc j 1 j x, j0,1,..., 1.8 b j1 p j1 xa j p j xb j p j1 xxp j x, j1, 1.9 where b 0 0. Conversely, modulo certain essential self-adjointness issues, d is the spectral measure for J in the cyclic subspace generated by J and the vector e 1 (1,0,0,...) T see, e.g., Ref. 4. It follows that the transformation of measures d d,m 1.10 leads to the transformation of operators Jd Jd,m For appropriate choices of 1,..., m and 1,...,, such transformations correspond to removing m points from the spectrum of J(d) and inserting points: in the spectral theory literature, such transformations are known as Darboux transformations. The formulas in this article clearly provide formulas for the generalized eigenfunctions p j,m (x) of the Darboux-transformed operator J(d,m ), as well as the matrix entries, a j,m and b j,m, in terms of the corresponding objects for J(d). Again we leave the details to the reader. Here the elementary formulas b 2 n d n1 Z n dz n2 d, a n2 Z n1 d 2 n d d log Z nd t Z dtt0 n1 d t, 1.12 where d t (x)e tx d(x), are useful. Technical Remark 1.2: Formulas clearly do not make sense for all values of the parameters. In all the calculations that follow, we will assume that d has compact support, support (d)q,q, say, and that the i s and j s are distinct real numbers greater than Q: under these assumptions, d,m (t) becomes, in particular, a bona fide measure, etc. By analytic continuation one sees that the formulas remain true for complex values of i and j, as long as they remain distinct. Furthermore, if the j s and j s are distinct, and Im( j )0 for all j, then we can let Q and so the formulas are true for measures d with unbounded support. Finally, we can, for example, let j k for some jk, which leads to formulas involving derivatives of the j s, etc. II. FORULAS OF CHRISTOFFEL UVAROV TYPE We use the notations d, j, d,m,,m j,... of Sec. I. In addition, in all the calculations that follow we assume that d, j, k satisfy the conditions described in Technical Remark 1.2 above: the natural analytical continuation of the formulas obtained to complex values of the parameters, and the limit Q, is left to the reader. The following result of Christoffel see Ref. 18 plays a basic role in what follows. Lemma 2.1: Consider the measure d,0 (t) j1 ( j t)d(t), where 1,2,.... Then the nth monic orthogonal polynomial,0 n (t) associated with the new measure d,0 (t) can be expressed as follows:

4 3660 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov Proof: Set 1,0 n t t 1 t n1 n1 n n n t n t 2.1 n 1 n1 1 n n1. n1 q,0 n tn1 2.2 n n n t n t. We note that q n,0 (t) satisfies the condition t j q n,0 (t)d(t)0 for all j0,...,n1. Also q n,0 ( j )0, j1,...,, and so q n,0 (t)/( 1 t) ( t) is a polynomial of degree at most n. ow observe that t j q,0 n t 1 t td,0 t0, 0 jn, 2.3 which means that q,0 n (t) divided by the product ( 1 t) ( t) is proportional to the nth monic orthogonal polynomial,0 n (t) associated with the new measure d,0,0 (t). ow q n (t) cannot vanish for any t 1 Q, 1 1,...,. Indeed, if q,0 n ( 1 )0, then there exist i i0, not all zero, such that p(t) i0 i ni (t) vanishes at i 1 i1. Thus p (t)p(t)/ 1 i1 ( i t) is a polynomial of order n, and as above, p (t) is orthogonal to t j, 0 jn, with respect to the measure d 1,0 (t). Thus p (t)0 and hence 0 0, which is a contradiction. Replacing by 1, we conclude that n 1 n n n1 0. Taking the limit t and noting that the coefficient of the highest degree of,0 n (t) should be equal to 1, we find the coefficient of proportionality and establish formula 2.1. Representation 2.1 for the monic orthogonal polynomials associated with the measure d,0 (t) immediately leads to the following result: Corollary 2.2: The product of monic orthogonal polynomials j0 j,0 n ( j1 ) defined with respect to the different measures d j,0 (t)( j t) ( 1 t)d(t) is given by the formula n j0 j,0 j1 1 n1 n1 2.5 n 1 n 1, where () 1i j1 ( i j ). We observe that Corollary 2.2 gives the identity for the average of products of random characteristic polynomials obtained first by Brezin and Hikami. 3 Theorem 2.3: Let D,H be the characteristic polynomial of the Hermitian matrix H. The following identity is valid:

5 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices 3661 L j1 D j,h 1 L L L1 L, where the average is defined by (1.2). Proof: To prove formula 2.6 we use the representation for the monic orthogonal polynomials in the case L1 given in 1.5, 1 i1 x i 2 xdx. 2.7 Let,0 be defined by,0 2 xd,0 x, 1,2,..., 2.8 where d,0 (x) i1 d,0 (x i ). With this notation, we have L j1 D j,h Z L,0 Z L,0 L1,0 Z L1,0 L2,0 1, Equation 2.7 implies that n 1,0 ( ) can be represented as the ratio,0 / 1,0, where 0,0 () (), and 0,0. Thus we obtain L j1 L1 D j,h j0 j,0 j The above equation together with Corollary 2.2 proves formula 2.6. Remark 2.4: otice see Eqs. 2.7 and 2.10 that the average of products of characteristic polynomials can be rewritten as a product of averages. amely, L j1 D j,h j1 L D j,h j1,0, 2.11 where j1,0 means the average defined by Eq. 1.2 but with respect to the new measure d j1,0 (x), and d(x)d 0,0 (x). The formula of Christoffel Eq. 2.1 enables us to construct the orthogonal polynomials associated with the measure d,0 (t) j1 ( j t)d(t) in terms of the orthogonal polynomials associated with the measure d(t). ow we derive a formula due to Uvarov 19 expressing the monic orthogonal polynomials 0,m n (t) associated with the measure d 0,m (t) m j1 ( j t) 1 d(t), again in terms of the monic orthogonal polynomials n (t) associated with the measure d(t). Lemma 2.5: Suppose 0mn. The monic orthogonal polynomials 0,m n (t) associated with the measure d 0,m (t) can be expressed as ratios of determinants.

6 3662 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov hnm1 hn1 h nm m h n m nm t n t 0,m n t 2.12 h nm 1 h n1 1 h nm m h n1 m. Here the h k ( j ) s are the Cauchy transformations of the monic orthogonal polynomials k (t). Proof: Set h k j 1 2i ktdt t j hn1 q 0,m n thnm h nm m h n m nm t n t. ow q n 0,m (t) is proportional to the nth monic orthogonal polynomial n 0,m (t) with respect to the measure d 0,m (t). Indeed, first observe that Also, for 0kn, q 0,m n t dt0, j1,...,m t j t k m 1 t 1 t pt m 2.16 for suitable constants and for some polynomial p(t) of degree nm. But for 0kn, m t k q 0,m n td 0,m t 1 q n 0,m t t dt ptq n 0,m tdt The terms in the sum are zero by 2.15 and the final integral is zero by the construction 2.14 of q n 0,m (t) and the fact that deg p(t)nm. Thus q n 0,m (t) is proportional to n 0,m (t). An argument similar to the proof in Lemma 2.1, that n 1 n n n1 0, shows that the denominator in 2.12 does not vanish. Letting t in 2.14, and matching leading terms, we prove Lemma 2.5. Remark 2.6: In Ref. 19, Uvarov obtains formulas for 0,m n (t) of type 2.12 also in the case mn. These formulas can be used to obtain analogs of 2.24 and 2.36 below in the case.

7 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices 3663 Remark 2.7: As noted in Refs. 11 and 17, the Cauchy transformations h k () of the k s occur explicitly, together with the k s, in the solution of the Fokas Its itaev Riemann Hilbert problem for orthogonal polynomials. 9 Lemma 2.5 implies the following analog of the Christoffel formula for the Cauchy transforms of monic orthogonal polynomials. Corollary 2.8: Let h k 0,m () be the Cauchy transform of the monic polynomial k 0,m (t). with respect to the measure d 0,m (t), 0,m t h 0,m k 1 k 2i t d 0,m t Let also 0mn. Then h n 0,m () has a representation similar to that for the monic orthogonal polynomials n,0 (t) [Eq. (2.1), h 0,m 1 m n m 1 hnm1 hn1 h nm m h n m h nm h n 2.20 h nm 1 h n1 1 h nm m h n1 m. Proof: The above representation follows from formula 2.12 and from the fact that m1 1 t m1 t 1 j1 1 t j k j 1 j k Indeed we find from formula 2.12 that h 0,m n () is the ratio of the determinants. The elements of the last row of the determinant in the numerator are the integrals 1 nk tdt 2i tt m t 1, 0km. Using identity 2.21 and noting that the only term 1 1 t m of the sum 2.21 contributes to the determinant, 2.20 follows. Equation 2.20 immediately implies the following analogy of 2.5 for the h 0,m k s. Corollary 2.9: Let 0mn. Then the product of the Cauchy transforms of monic orthogonal polynomials with respect to the measures d 0,j (t), 0 jm, can be written as a determinant, m j0 0,j h nm j j1 1mm1/2 hnm1 hn1 h nm m1 h n m ow we derive the identity for the average of the product of inverse random characteristic polynomials. Theorem 2.10: Suppose 1 and let n 2i/c n 2, where c n is the normality constant defined by Eq. (1.4). Then we have the following formula:

8 3664 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov j1 D 1 j,h 1 1/2 1 j j h1 h h h 1. Proof: When 1, we use the identity 2.21 together with 2.7 and the relation see, e.g., Ref. 18 n1 2in Z n1 Z n 2.25 to obtain D 1,H 1 h We rewrite the average in Eq as follows: j1 D 1 j,h Z 0, Z 1 0,1 1 Z 0,0 0,1 0,2 2 0,0, 2.27 where 0, 2 xd 0, x, ,0 and d 0,0 (x)d(x). The following relation can be observed from Eqs and 2.25: 0,m 0,m1 m. Z 2ih 0,m Inserting this relation in 2.27 we find j1 D 1 j,h j1 0, j h j j j Our result 2.24 immediately follows from the above equation and formula We now repeat the above considerations for the case d,m t 1t t 1 t m t dt The first result is a Christoffel-type formula for the measure 2.31, which is due to Uvarov. 19 Lemma 2.11: Suppose 0mn. Then the monic orthogonal polynomials n,m (t) s with respect to the measure d,m (t) have the following representation:

9 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices hn1 h m h n m 1 n 1 n 1,m n t 2.32 t t 1 hnm1 hnm1 hn1 h nm m h n m nm 1 n 1 nm n. nm t n t 3665 Proof: As in the previous cases we define q n,m (t) to be the determinant in the numerator of Observe that q n,m 1 q n,m and that q,m n tdt 1 t q,m n tdt 0. m t 2.34 The next steps are the same as in the proofs of Lemma 2.1 and Lemma 2.5. Corollary 2.12: h1 h11 h h j1 D j,h 0, 1 h 1 h 1 h h Proof: Identity 2.35 follows from Eqs and 2.32 once we note that Eq can be rewritten in a similar manner as Eq Finally we generalize Theorems 2.3 and 2.10 and obtain a formula for the average of ratios of characteristic polynomials. Theorem 2.13: Suppose 0. Then the average of ratios of characteristic polynomials of Hermitian matrices H is given by the following formula:

10 3666 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov j1 D j,h D j j1,h 1 1 1/2 j j Proof: Let 0,0, Z 0,0 n Z n. Then we have j1 D j,h j1 D j,h Z, 0,0 h1 h11 h h Z, 0, 0, Z, 0, i.e., j1 D j,h D j j1,h j1 D j,h 0, j1 D 1 j,h We use Corollary 2.12 and Theorem 2.10 to obtain formula Remark 2.14: Observe that formulas 2.6 and 2.24 do not follow immediately as special cases of 2.36: some further algebraic manipulation is required. Similarly, the process of adding and removing zeros is clearly reciprocal. ore precisely, given 1,...,, we can construct the polynomials 0, n (t;d 0, ) associated with the measure d 0, (t)( i1 ( i t) 1 )dt by 2.12: We can then construct,0 n (t;d( 0, ),0 ) with i i, inserting 0, n (t;d 0, ) for n (t) on the right-hand side of 2.1. We should find that,0 n (t;d( 0, ),0 ) n (t;d). However, again, this relation is not immediately clear, and requires further algebraic manipulation. III. FORULAS OF TWO-POIT FUCTIO TYPE The following integral version of the Binet Cauchy formula is due to Andréief, 2 and plays a basic role in our calculations. Lemma 3.1: Let (X,d) be a measure space and suppose f i, g j L 2 (X,d) for 1i, jk. Then X X det f i x j 1i, jk detg i x j 1i, jk dx 1 dx k k! det X f i xg j xdx Proof: Set c ij X f i (x)g j (x)d(x). Then X X det f i x j 1i, jk detg i x j 1i, jk dx 1 dx k i, jk,sk sgnsgnc 11 c kk sgn sgnc 11 c kk sgn 2 sgnc 11 c kk k! detc ij 1i, jk 3.2

11 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices 3667 as desired. In 3.2 we used sgn()(sgn )(sgn ) and the fact that c (1)(1) c (k)(k) c 1(1) c k(k) for all. Theorem 3.2: Let 1. Then the following identity is valid: j1 C, D j,hd j,h detw I, i, j 1i, j, 3.3 where W I, x,y x 1 y y 1 y xy 3.4 and 1 c 2 C, c where c is again the norming constant for given in (1.4). Proof: Let p j (x)c j 1 j (x), j0, denote the orthonormal polynomials with respect to d. From 1.2 we obtain j1 D j,hd j,h 1 x,x,dx. 3.6 Adding columns, we see that the Vandermonde determinant (x,) has the form 0x1 1x1 1x1 0 x 1 x 1 x and similarly for (x,). Here j (t) 0,0 j (t). The determinant (x,) can be evaluated by a Lagrange expansion of the form i1 i1,...,i 0i 1 i 2 i k 1 1 i 1 i1 i j1 x 1 j x 1 j1 x j x, 3.8 where i1,...,i 1 is an appropriate signature and ( j 1,...,j ):0 j 1 j 2 j 1 is the complement of i 1,...,i in 0,1,...,1. ultiplying 3.8 by a similar expansion for (x,), and inserting in 3.6, we obtain a sum of terms of the form j1 x 1 j x 1 j 1 j1 x j x x 1 j x j1 x j dx, x

12 3668 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov which is equal by Lemma 3.1 to! det( ji (x) j k (x)d(x)) 1i,k! det( ji j k c 2 jk ) 1i,k. From this we see that j1 D j,hd j,h! k1 0i 1 i k 1 i1 1 i 1 c 2 jk! q 1 2 c q x,x,! q 1 2 c q x,x, det i1 i 0i 1 i k 1 0i1 2 i1,...,i i1 1 i 1 i1 i detp i j k 1 j,k detp i j k 1 j,k p i j p i k, j,k where the last line follows by applying Lemma 3.1 to the discrete measure d 1 i0 i. But, by the Christoffel Darboux formula, p i j p i k j 1 k k 1 j, i1 j k which then implies 3.3 as! 1 0 c 2 see, e.g., Ref. 18. Theorem 3.3: Suppose 1. Then the following identity is valid: j1 D i,h, D j,h 1 1/ detw H, i, j 1i, j, 3.12 where W H, x,y h 1 h and again h k ()(1/2i) k (t)d(t)/(t) is the Cauchy transform of k (t) and 1 2 2i/C 1. Observe first that by linearity

13 J. ath. Phys., Vol. 44, o. 8, August 2003 Polynomials of random Hermitian matrices 3669 h1 hl11 h h L1 1 L1 1 d 2i 1 L1 L j1 j j L11 L L1 1 1 L1 L. Inserting 2.36 on the left-hand side, and using 2.5 to reexpress the integrand on the right-hand side, we obtain the following result, which is of independent interest. The result expresses averages of ratios of characteristic polynomials in terms of averages of products of such polynomials. Proposition 3.4: Let 1. Then But L j1 D i,h D j j1,h 1 1 1/2 j j j1 L d 2i j1 j j, D j,h D j j1,h Proof of Theorem 3.2: For L, by3.15 and 3.3, 1 1/2 1 j j1 j j1 d D i,h 2i j1 j j D j,h C, j1 i1 1 d j 2i j j i1 i j j 1 k 1 j k j k 1 2i d j 1 1 j j j 1 2i d j 1 j j j i2 i j ik i2 i j ik i j detw I, i, j 1i, j. j 1 k 1 j k j 1 k 1 j k

14 3670 J. ath. Phys., Vol. 44, o. 8, August 2003 Baik, Deift, and Strahov as d( j ) j 1 ( j ) d( j ) j ( j )0 for 021. Continuing in this way, the integral reduces to i1 ( i j )W H, ( i, k ). Thus we find 1 j 1 1/2 j D i,h j1, detw I, i, k 1i,k D j j1,h 3.18 and 3.12 follows. ACOWLEDGETS The authors would like to thank Jeff Geronimo for useful conversations and for pointing out the paper of U. B. Uvarov. The authors would also like to thank ick Witte for many useful remarks. The work of the first author was supported in part by SF Grant o. DS The work of the second author was supported in part by SF Grant o. DS and by the Institute for Advanced Study in Princeton. The work of the third author was supported in part by EPSRC Grant o. GR/13838/01 Random atrices close to Unitary or Hermitian and by the Brunel University Vice-Chancellor grant. 1 Andreev, A. V. and Simons, V. D., Correlators of spectral determinants in quantum chaos, Phys. Rev. Lett. 75, Andréief, C., ote sur une relation les intégrales définies des produits des fonctions, ém. Soc. Sci. Bordeaux 2, Brezin, E. and Hikami, S., Characteristic polynomials of random matrices, Commun. ath. Phys. 214, Deift, P., Orthogonal Polynomials and Random atrices: a Riemann-Hilbert Approach, Vol.3ofCourant Lecture otes in athematics AS, Providence, Deift, P. and Zhou, X., A steepest descent method for oscillatory Riemman-Hilbert problems. Asymptotics for the dv equation, Ann. ath. 137, Deift, P., Venakides, S., and Zhou, X., ew results in small dispersion dv by an extension of the steepest descent method for Riemann-Hilbert problems, Int. ath. Res. otices 6, Deift, P., riecherbauer, T., claughlin,., Venakides, S., and Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Commun. Pure Appl. ath. 52, Deift, P., riecherbauer, T., claughlin,., Venakides, S., and Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Commun. Pure Appl. ath. 52, Fokas, A., Its, A., and itaev, V., Discrete Painlevé equations and their appearance in quantum gravity, Commun. ath. Phys. 142, Forrester, P. and Witte,., Applications of the tau-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Commun. ath. Phys. 219, Fyodorov, Y. V. and Strahov, E., An exact formula for general spectral correlation functions of random matrices, J. Phys. A 36, Hughes, C., eating, J., and O Connell,., Random matrix theory and the derivative of the Riemann-zeta function, Proc. R. Soc. London, Ser. A 456, Hughes, C., eating, J., and O Connell,., On the characteristic polynomials of a random unitary matrix, Commun. ath. Phys. 220, eating, J. and Snaith,., Random matrix theory and (1/2it), Commun. ath. Phys. 214, ehta,., Random atrices, 2nd ed. Academic, San Diego, ehta,. and ormand, J-., oments of the characteristic polynomial in the three ensembles of random matrices, J. Phys. A 34, Strahov, E. and Fyodorov, Y. V., Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach, math-ph/ Szegö, G.,Orthogonal Polynomials, Vol. 23 of American athematical Society, Colloquium Publications, 4th ed. AS, Providence, RI, Uvarov, V. B., The connection between systems of polynomials orthogonal with respect to different distribution functions, USSR Comput. ath. ath. Phys. 9,