How universal is the Wigner distribution?

Ho w univ ersal is the Wigner distribution? ∗ Mthokozisi Masuku and Jo ˜ ao P . Ro drigues † Natio nal Instit ut e fo r Theor etical Physics Sc ho ol of Physics and Cen tr e for Theor etical P hysics Universit y of the Witw atersr and, Johannesburg Wits 2 050, South Africa June 29, 2018 Abstract W e consider Gaussian ensembles o f m N × N complex matrices. W e iden tify an enhanced sym metry in the system and the resu ltan t closed subsector, whic h is nat u rally associated with the radial sector of the t h eory . The densit y of radial eigen v alues is obtained in the la r ge N limit. It is of the Wigner f orm only for m = 1. F or m ≥ 2, the n ew form of the d ensit y is obtained. ∗ WITS-CTP-07 7 † Email: joao .ro drigues@ wits.ac.za 1 1 In tro duction The Wigner semicircle distribution [1 ] f ( x ) = 2 π R 2 √ R 2 − x 2 , − R < x < R ; f ( x ) = 0 , | x | > R , describes the densit y of eigen v alues of a gaussian ense mble of single la rge hermitean, symmetric or quaternionic matrices [2], and finds applications in the descripiton of systems in man y areas o f Phy sics, fr om Nuclear Ph ysics to Condensed Matter Phy sics. The Gaussian ensem ble of a single complex matrix, or equiv alen tly , of t w o hermitean matrices, is also described by a Wigner t yp e distribution of eigen v alues of a “radial” v ariable [3] [4], the definition of whic h will b e made precise in the follo wing. It is of great in t erest and imp ortance to in ves tig ate if this con tinues to b e a general prop erty of gaussian ensem bles of more matrices, and in general, to study the prop erties of systems with a finite num b er of matrices, part icularly in their la rge N limit [5]. The reason for this imp ortance includes, for instance, the fact that, as it has b een established already some time ago [6], [7], [8], QC D can b e reduced to a finite n umber of matrices with quenc hed momen ta. Of more r ecent in terest, the matrix description o f D branes [9] has for instance lead to the prop osal that the large N limit of the quantum mec hanics of the m ulti matrix description of D0 branes pro vides a definition of M theory [1 0]. In the con t ext of the AdS/CFT dualit y [11], [12], [13 ], due to sup ersymmetry and conformal in v ariance, correlators of sup ergra vity and 1 / 2 BPS states reduce to calculation of free matrix mo del ov erlaps [14], [15] or consideration of related matrix hamiltonians [16]. F o r stringy states, in the con text o f the BMN limit [17] and N = 4 SYM, similar considerations apply [18], [1 9 ], [20]. A plane-wa v e matrix theory [21] is related to the N = 4 SYM dilatation op era t o r [22]. In this communic at ion w e will consider m complex N × N matrices Z A A = 1 , ..., m , or equiv alently , a n ev en n um b er 2 m of N × N hermitian matrices and a gaussian ensem ble of suc h matrices: 2 p [ Z † A , Z A ] = e − w 2 2 T r( P m A =1 Z † A Z A ) Z , Z = Z [ d Z A † d Z A ] e − w 2 2 T r( P m A =1 Z † A Z A ) . (1) W e will obtain the large N description of the system in terms of the densit y o f eigen v alues of the matrix m X A =1 Z † A Z A (2) This matrix has a v ery natural in terpretation as a matrix v alued ra dial co ordinate. W e will establish that for m = 1 the densit y of eigenv alues of the ra dial matrix is still describ ed by a Wigner distribution, but that this is no longer the case for m ≥ 2 and obtain the form of the new eigenv alue distribution. In order to obtain these densities, w e will first need to establish a new result, the measure for the probabilit y density (1) in terms o f the eigen v alues ρ i = r 2 i , i = 1 , ..., N of (2): Y A [ d Z A † d Z A ] = C m Y i dρ i ρ m − 1 i Y i>j ρ m − 1 i ρ m − 1 j ( ρ i − ρ j ) 2 = D m Y i dr i r 2 m − 1 i Y i>j r 2 m − 2 i r 2 m − 2 j ( r 2 i − r 2 j ) 2 (3) = C m Y i dρ i ρ m − 1 i ∆ 2 RM ( ρ i ) = D m Y i dr i r 2 m − 1 i ∆ 2 RM ( r 2 i ) , where the a n tisymmetric pro duct ∆ RM ( ρ i ) ≡ Y i>j ρ m − 1 2 i ρ m − 1 2 j ( ρ i − ρ j ) generalizes the w ell known V a n der Monde determinan t ∆ = Q i>j ( ρ i − ρ j ), and C m and D m are num erical constants . This com unication is organized as f ollo ws: In Section 2 we restate the problem, identify the enlarged symmetries of the gaussiand ensem ble and iden tify a complete subset of op erato rs inv ariant under t his enlarg ed symme- try . In Section 3 , based o n the remark able fact t hat this subset o f in v ariant 3 op erators closes under Sc h winger D yson equations, the Jacobian o f the trans- formation to these in v ariant states and to the eigen v alues of (2) is obtained. In Section 4, the large N densit y of states on the p ositive real line is obtained for m ≥ 2. In this case, the effectiv e p oten tial con tains a new lo garithmic term which keep s the supp ort of the densit y of eigen v alues strictly p o sitiv e, i.e., 0 < ρ − ≤ ρ ≤ ρ + . In Section 5 , the densit y is appropriately extended to the whole real line, allow ing for a common description of b oth the m = 1 and m ≥ 2 cases. F or m = 1, a (restricted) Wigner distribution emerges, whereas for m ≥ 2 the r equired tw o cut solution is sho wn to agree with tha t of the previous section, when suitably restricted to the p ositiv e real line. In Section 6, these densities are related to the densit y of zeros of certain p olynomials. Section 7 is left for a summary and brief discussion. 2 Gaussian ensem ble and symmetries As stat ed in the In tro duction, we consider a g aussian ensem ble of m complex N × N matrices ( Z A ) ij , A = 1 , ..., m. The gaussian p oten tial S g = w 2 2 T r( X A Z † A Z A ) is in v ariant under the U ( N ) m +1 symmetry Z A → V A Z A V † , A = 1 , ..., m, (4) with V A , A = 1 , ..., m and V unitary matrices. The p oten tia l dep ends o nly on the eigen v alues of the p ositiv e definite, hermitean matrix X A Z † A Z A , (5) whic h are denoted ρ i = r 2 i , i = 1 , ..., N , ρ i ≥ 0 . In the conte xt o f a phy sical fra mework where the m complex matrices are naturally a sso ciated with 2 m hermitean mat rix v alued co ordinat es, as 4 appropriate, f o r instance, in t he description of t he dynamics of branes, these eigen v alues hav e a na tural interpretation as the eigen v alues of a matrix v alued radial co ordinat e. The purp ose of this comm unication is to obtain the large N distribution of these eigenv alues for the gaussian partition function Z = Z Y A Y ij d Z A † ij d Z A ij e − S g W e will do so b y “integrating out” the “angular matrix v alued degrees of freedom or equiv alen tly , by obtaining the jacobian J ( ρ i ) of the change of v ariables to the “radial” eigen v alues: Z = Z Y i dρ i J ( ρ i ) e − S g ( ρ i ) In order to do so, w e will consider correlators of op erators t hat are in- v arian t under the symmetry (4), i.e., in the subsector of the theory with this enlarged symmetry . Suc h in v ariant op erato r s can b e built as the trace (sin- gle traces) of p o wers of the matrix (5), and hence they dep end again on the eigen v alues of this matrix o nly . A generating function for suc h op earators is giv en b y Φ k = T r e ik P A Z † A Z A = X i e ik ρ i = X i e ik r 2 i , or its fourier transform, the densit y of eigen v alues: Φ( ρ ) = Z dk 2 π e − ik ρ Φ k = X i δ ( ρ − ρ i ) = X i δ ( ρ − r 2 i ) . It turns that these correlators close in this subsector, as it will b e sho wn in the f o llo wing by use of Sch winger Dyson equations 3 Jacobian Sc h winger Dyson equations can b e obtained fro m the iden tity: Z Y A Y ij d Z A † ij d Z A ij ∂ ∂ ( Z A ) j i ∂ Φ k ∂ ( Z A ) † ij F [Φ ] e − S g ! = 0 , (6) 5 where F [Φ] is an arbitrar y pro duct o f in v arian t op erato rs. This yields: < ∂ 2 Φ k ∂ ( Z A ) † ij ∂ ( Z A ) j i F [Φ ] > + < ∂ Φ k ∂ ( Z A ) † ij ∂ F [Φ] ∂ ( Z A ) j i > − < F [Φ] ∂ Φ k ∂ ( Z A ) † ij ∂ S g ∂ ( Z A ) j i > = 0 . (7) W e denoted in the ab ov e: < G [Φ] > ≡ Z Y A Y ij d Z A † ij d Z A ij G [Φ] e − S g = Z [ d Φ] J (Φ ) G [Φ] e − S g . F ollow ing [23], w e now consider the iden tity Z [ d Φ] Z dk ′ ∂ ∂ Φ ′ k " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # J (Φ ) F [Φ] e − S g ! = 0 . Then Z dk ′ < ∂ ∂ Φ ′ k " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # F [Φ] > + Z dk ′ < " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # ∂ ln J (Φ) ∂ Φ ′ k F [Φ] > (8) + < ∂ Φ k ∂ ( Z A ) † ij ∂ F [Φ] ∂ ( Z A ) j i > − < F [Φ] ∂ Φ k ∂ ( Z A ) † ij ∂ S g ∂ ( Z A ) j i > = 0 where in the la st tw o terms we used the c hain rule. Comparing (7) with (8), whic h are equiv alen t for arbitrary F [Φ], it follows that: Z dk ′ " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # ∂ ln J (Φ) ∂ Φ ′ k (9) + Z dk ′ ∂ ∂ Φ ′ k " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # = ∂ 2 Φ k ∂ ( Z A ) † ij ∂ ( Z A ) j i 6 F ourier transforming, and defining Ω ρρ ′ = Z dk 2 π Z dk ′ 2 π e − ik ρ e − ik ′ ρ ′ " ∂ Φ k ∂ ( Z A ) † ij ∂ Φ k ′ ∂ ( Z A ) j i # (10) w ρ = Z dk 2 π e − ik ρ ∂ 2 Φ k ∂ ( Z A ) † ij ∂ ( Z A ) j i , the differen tial equation for the jacobian t hen take s the form Z dρ ′ Ω ρρ ′ ∂ ln J (Φ) ∂ Φ( ρ ′ ) + Z dρ ′ ∂ Ω ρρ ′ ∂ Φ( ρ ′ ) = w ρ . (11) Ω ρρ ′ and w ρ ha ve b een obtained in [4]: Ω ρρ ′ = ∂ ρ ∂ ρ ′ [ ρ Φ( ρ ) δ ( ρ − ρ ′ )] (12) w ρ = − ∂ ρ  ρ Φ( ρ )  2 − Z dρ ′ Φ( ρ ′ ) ρ − ρ ′ + N ( m − 1) ρ  (13) As a result, the second term in (11) v anishes a nd the Jacobian satisfies: ∂ ρ ∂ ∂ Φ( ρ ) ln J = 2 − Z dρ ′ Φ( ρ ′ ) ρ − ρ ′ + N ( m − 1) ρ This equation w as previously o bt a ined in [4], using collectiv e field theory metho ds [24]. The solution is ln J = Z dρ Φ( ρ ) − Z dρ ′ Φ( ρ ) ln | ρ − ρ ′ | + N ( m − 1) Z dρ Φ( ρ ) ln ρ In terms of the eigen v alues, J = Y i ρ m − 1 i Y i 6 = j ρ m − 1 2 i ρ m − 1 2 j | ρ i − ρ j | = Y i ρ m − 1 i Y i>j ρ m − 1 i ρ m − 1 j ( ρ i − ρ j ) 2 Since, up to a constan t, R [ d Φ] ∼ R Q i dρ i , w e ha ve obtained the result that w e sough t to establish: 7 Z Y A Y ij d Z A † ij d Z A ij e − S = Z Y i dρ i J ( ρ i ) e − S ( ρ i ) = Z Y i dρ i J ( ρ i ) e − S ( ρ i ) = C m Z Y i dρ i ρ m − 1 i " Y i>j ρ m − 1 i ρ m − 1 j ( ρ i − ρ j ) 2 # e − S ( ρ i ) , for p otentials in v arian t under (4). A couple of commen ts are in order. When m = 1, the ab ov e result reduces to the result first obtained in [4], where an explicit parametrization of the 2 N degrees of freedom of t wo hermitean matrices was obtained, in terms of radial and angular matrix v alued co ordinates. There is a classic result [25] [2] parametrizing a single complex matrix in t erms of its complex eigenv alues and upp er diagonal matrix. This parametrization of degrees of freedom, useful for holomorphic pro j ections, is different from the one considered in this comm unication. F or m = 1, the existence of a closed hermitean subsector has also b een iden tified in [26]. Gaussian ensem bles of r ectangula r M × N matrices hav e also b een dis- cussed in [27], [28] and [2 9]. They can b e related to the ensem bles discussed in this communic a tion when M = mN . In this con t ext, the approac h follow ed is equiv a lent to using the symmetries of the system to set m − 1 o f the N × N matrices to zero. This corresp onds to a ”g auge fixed” treatmen t, as opp o sed to the gauge inv ariant approach describ ed in this comm unication, for whic h the eigen v alues r i ha ve a natural identification as a radial co ordinate. 4 Large N densit y of the eigenv alues W r it ing Z = Z Y i dρ i J ( ρ i ) e − S g ( ρ i ) = Z Y i dρ i e − S ef f , w e hav e 8 S ef f = w 2 2 Z dρ Φ( ρ ) ρ (14) − Z dρ Φ( ρ ) − Z dρ ′ Φ( ρ ′ ) ln | ρ − ρ ′ | − N ( m − 1 ) Z dρ Φ( ρ ) ln ρ , Z dρ Φ( ρ ) = N . In o r der to exhibit explicitly the N dependence, we rescale ρ → N ρ and Φ → Φ, so that S ef f = N 2 [ w 2 2 Z dρ Φ( ρ ) ρ (15) − Z dρ Φ( ρ ) − Z dρ ′ Φ( ρ ′ ) ln | ρ − ρ ′ | − ( m − 1 ) Z dρ Φ( ρ ) ln ρ ] , Z dρ Φ( x ) = 1 The large N → ∞ configuration is then determined b y the stationar y condition ∂ ρ ∂ S ef f ∂ Φ( ρ ) = 0, o r − Z dρ ′ Φ( ρ ′ ) ρ − ρ ′ = w 2 4 − ( m − 1) 2 ρ (16) W e note a ma jor difference b etw een the case of one complex ma t rix ( m = 1) and the case of more than o ne complex matrix: for mo r e than one complex matrix, a n additional log a rithmic p oten tial is presen t, in addition to t he standard V an der Monde repulsion amongst the eigen v alues. As a result, the eigen v alues are “pushed aw a y” from ρ = 0. The m = 1 solutio n will b e describ ed in the next section, where the range of the densit y of eigen v alues is extended to the full real line. In this section, w e obta in the the densit y of eigen v alues fo r m ≥ 2. The solution to the in tegral equation (16), g eneralizing that asso ciated with P enner p oten tials [3 0], is o btained using standard metho ds [31], tog ether with a careful treatmen t of the ρ → 0 b ehav io ur, follo wing T an [32]. One in tro duces the function G ( z ) in the complex plane G ( z ) = Z ρ + ρ − dρ ′ Φ( ρ ′ ) z − ρ ′ 9 where Φ( ρ ) ha s supp or t only in the in terv al [ ρ − , ρ + ], ρ + > ρ − > 0. F or large | z | , G ( z ) ∼ 1 z , and as z approache s the supp o r t of Φ( ρ ), G ( ρ ± iη ) = − Z ρ + ρ − dρ ′ Φ( ρ ′ ) ρ − ρ ′ ∓ iπ Φ( ρ ) = w 2 4 − ( m − 1) 2 ρ ∓ iπ Φ( ρ ) . Therefore, this suggests the single cut anzatz G ( z ) = w 2 4 − ( m − 1) 2 z − w 2 4 z p ( z − ρ − )( z − ρ + ) One also requires [3 2] that G ( z ) has no p ole as z → 0. Thes e conditions fix: ρ ± = 2 w 2 ( m + 1) ± 4 w 2 √ m (17) It follo ws that the densit y of eigen v alues is giv en b y Φ( ρ ) = w 2 4 π ρ p ( ρ + − ρ )( ρ − ρ − ) , ρ − ≤ ρ ≤ ρ + = w 2 4 π r ( ρ + ρ − 1)(1 − ρ − ρ ) (18) = 1 π ρ r 1 − w 4 16 ( ρ − 2 w 2 ( m + 1)) 2 , no longer of the Wigner for m. 5 Symmetric so lutions In this section, w e extend the domain of definition of the densit y of eigen v al- ues, allow ing us to prov ide a unified description of the single complex matrix case ( m = 1) and that of more than tw o complex matr ices ( m ≥ 2). With ρ = r 2 , r > 0, define 2 r Φ( r 2 ) ≡ φ ( r ) ≡ φ ( − r ) In this wa y , for an a rbitrary function f ( r 2 ) Z + ∞ −∞ dr f ( r 2 ) φ ( r ) = 2 Z + ∞ 0 dρf ( ρ )Φ( ρ ) . (19) 10 Returning to (1 6), w e remark that ( ρ = r 2 ) − Z ∞ 0 dρ ′ Φ( ρ ′ ) ρ − ρ ′ = − Z ∞ 0 2 r ′ dr ′ Φ( r ′ 2 ) r 2 − r ′ 2 = 1 2 r − Z ∞ 0 dr ′ φ ( r ′ )( 1 r − r ′ + 1 r + r ′ ) = 1 2 r − Z ∞ −∞ dr ′ φ ( r ′ ) r − r ′ (20) As a result, (16) is equiv alen tly written as − Z ∞ −∞ dr ′ φ ( r ′ ) r − r ′ = w 2 2 r − ( m − 1) r , Z + ∞ −∞ dr φ ( r ) = 2 (21) When m = 1, this in tegral equation ha s t he w ell kno w Wigner distribution as a solution: φ ( r ) = w 2 2 π r 8 w 2 − r 2 , − √ 8 w ≤ r ≤ √ 8 w W e can now write Φ( ρ ) = w 2 4 π r 8 w 2 ρ − 1 , 0 ≤ ρ ≤ 8 w 2 The symme tr ic solution of (21) for m > 2 has been discussed in [32] [33]. It is a t w o cut solution, with generating functional G ( z ) = w 2 2 z − ( m − 1) z − w 2 2 z p ( z 2 − r − 2 )( z 2 − r + 2 ) . The cuts are in the in terv als [ − r + , − r − ] and [ r − , − r + ], with r + > r − > 0. The asymptotic condition and the absence of a p ole at z → 0 fix r 2 ± = 2 w 2 ( m + 1) ± 4 w 2 √ m , in p erfect agreemen t with (17). The densit y is then [33 ]: φ ( r ) = w 2 2 π | r | p ( r + 2 − r 2 )( r 2 − r − 2 ) , r − 2 ≤ r 2 ≤ r + 2 . This agrees with (18), recalling that 2 r Φ( r 2 ) ≡ φ ( r ) ≡ φ ( − r ). 11 6 Density of eigenv alues and zeros o f p olyno - mials It turns out that it is p o ssible to relate the densities o bt a ined in the previous section to the densit y of zeros of certain Laguerre or Hermite p olynomials [33] [34]. W e use the results of Calogero’s w ork [35], ba sed on the classical results of Stieltjies [36]. They show tha t the zeros of the Laguerre p o lynomial L α N ( x ) satisfy N X j =1 ,j 6 = i 1 x i − x j = 1 2  1 − 1 + α x i  (22) In terms of eigen v alues, equation (16) ta kes the form: N X j =1 ,j 6 = i 1 ρ i − ρ j = w 2 4 − ( m − 1) 2 ρ i (23) Comparison o f these equations show s that the solutio ns of (23 ) are the zeros of L m − 2 N ( w 2 2 ρ ) , m ≥ 2 . It has also b een established [35] [36] that the zeros of the Hermite p oly- nomial H 2 N ( r ) satisfy N X j = − N ,j 6 = i 1 r i − r j = r i (24) F or simplicit y , w e ha v e ch o sen an ev en p olynomial. Comparison o f the ab ov e equation with (21), considered when m = 1 and is express ed in terms of eigen v alues, sho ws that its solutions are the zeros of H 2 N ( w √ 2 r ) [34]. The usual relationship betw een Hermite and La guerre po lynomials is ob- tained, in the discrete, by noting that, with x i > 0, and since x − j = − x j , the left ha nd side of (24) can b e rewritten as N X j = − N ,j 6 = i 1 r i − r j = N X j =1 1 r i + r j + N X j =1 ,j 6 = i 1 r i − r j = 1 2 r i + N X j =1 ,j 6 = i 2 r i r 2 i − r 2 j , (25) 12 and comparing with (22). 7 Summary and discus sion In this comm unication, w e considered ga ussian ensem bles of m complex N × N matrices and identified a closed subsector that is naturally asso ciated with the radial sector of the theory . In the large N limit, the ensem ble is describ ed in terms of the densit y of radia l ev en v alues, and these hav e b een obtained: m = 1 m ≥ 2 Φ( ρ ) = w 2 4 π q 8 w 2 ρ − 1 Φ( ρ ) = w 2 4 π q ( ρ + ρ − 1)(1 − ρ − ρ ) 0 ≤ ρ ≤ 8 w 2 ρ − ≤ ρ ≤ ρ + ρ ± = 2 w 2 ( m + 1) ± 4 w 2 √ m . Extending to the full line with Φ( ρ ) dρ = φ ( r ) d r , ρ = r 2 , the densities tak e t he form: m = 1 m ≥ 2 φ ( r ) = w 2 2 π q 8 w 2 − r 2 φ ( r ) = w 2 2 π | r | p ( r + 2 − r 2 )( r 2 − r − 2 ) − √ 8 w ≤ r ≤ √ 8 w r − 2 ≤ r 2 ≤ r + 2 r 2 ± = 2 w 2 ( m + 1) ± 4 w 2 √ m . A (restricted) Wigner distribution is presen t only for m = 1. The existenc e of this closed sector is related to an enhanced U ( N ) m +1 symmetry , and the measure in this subsector has b een obtained, with a result that generalizes the w ell kno wn single hermitean matrix V an der Monde determinan t. There are sev eral further areas of study that arise naturally from the study presen ted here. P erhaps o f most in terest is an inv estigation of fur- ther systmes , suc h as Hamiltoniand and/or in teracting systems, where this symmetry is presen t or where the subsector with t his symmetry ma y pro- vide phy sically relev an t truncations. I t is also of intere st to estabish if the results describ ed in this comm unication extend smo othly to an o dd n um b er of matrices. 13 8 Ac kno wl edgemen t s W e w ould like to thank Rob ert de Mello Ko c h fo r in terest in this pro ject and for commen t s, and G. Cicuta for p oin ting out to us references [27], [28 ] and [29] after submission of this communic a t io n’s e-prin t to the Phy sics arc hiv e. References [1] E. P Wigner, Ann. Math. 62 (3), 548 (195 5). doi:10.230 7/197007 9. [2] M. L. Meh ta, ”Random Matrices ”. Amsterdam: Elsevie r /Academic Press (2004) . ISBN 0-120- 88409- 7 . [3] M. Masuku, “Matrix Polar Co ordinates”, MSc D issertation, F aculty of Science, Unive rsity of t he Wit w a tersrand, No vem ber 200 9 . [4] M. Masuku and J. P . Ro drigues, arXiv:0911.284 6 [hep-th]. [5] G. ’t Ho o ft, Nucl. Ph ys. B 72 , 461 (197 4). [6] T. Eguc hi and H. Ka wai, Ph ys. Rev. L ett. 48 , 10 63 (1982). [7] G. Bhanot, U. M. Heller and H. Neub erger, Ph ys. Lett. B 113 , 47 (1982). [8] D. J. Gross a nd Y. Kitazaw a, Nucl. Ph ys. B 206 , 440 (19 82). [9] J. Polc hinski, Ph ys. Rev. Lett. 75 , 472 4 (19 95) [arXiv:hep-th/951 0 017]. [10] T. Banks, W. Fisc hler, S. H. Shenk er and L. Susskind, Ph ys. Rev. D 55 , 5112 (199 7) [arXiv:hep-th/9610043]. [11] J. M. Maldacena, Adv. Theor. Math. Ph ys. 2 , 231 (199 8) [In t. J. Theor. Ph ys. 38 , 1113 (1 999)] [arXiv:hep-th/971 1200]. [12] S. S. Gubser, I. R. Klebano v and A. M. Poly ak o v, Ph ys. Lett. B 428 , 105 (1998 ) [arXiv:hep-th/980210 9]. [13] E. Witten, Adv. Theor. Math. Ph ys. 2 , 253 (1998) [arXiv:hep-th/9802150 ]. [14] S. Lee, S. Min w alla, M. Ra ngamani and N. Seib erg, Adv. Theor. Math. Ph ys. 2 , 697 (19 98) [arXiv:hep-th/9806 074]. 14 [15] S. Corley , A. Jevic ki and S. R a mgo olam, Adv. Theor. Math. Ph ys. 5 , 809 (2002 ) [arXiv:hep-th/011122 2]. [16] D. Berenstein, JHEP 0407 , 018 (2 0 04) [arXiv:hep-th/0403 110]. [17] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, JHEP 0204 , 013 (2002) [arXiv:hep-th/02 02021]. [18] N. R. Constable, D. Z. F r eedman, M. Headrick , S. Min walla, L. Motl, A. P ostnik o v and W. Skiba, JHEP 0207 , 017 (20 0 2) [arXiv:hep-th/0205089 ]. [19] N. Beisert, C. Kristjansen, J. Plefk a a nd M. Staudac her, Ph ys. Lett. B 558 , 229 (2003) [arXiv:hep-th/0212269 ]. [20] R. de Mello Ko c h, A. Donos, A. Jevic ki and J. P . Ro drigues, Ph ys. Rev. D 68 , 0 6 5012 (2003) [a r Xiv:hep-th/0305042]. [21] N. w. Kim, T. Klose and J. Plefk a, Nucl. Ph ys. B 671 , 359 (2003 ) [arXiv:hep-th/0306054 ]; T. Fisc hbac her, T. Klose and J. Plefk a, JHEP 0502 , 039 (2005) [arXiv:hep-th/0412331 ]. [22] N. Beisert, Phy s. Rept. 405 , 1 (200 5) [arXiv:hep-th/04072 77]. [23] A. Jevic ki and J. P . Ro drig ues, Nucl. Ph ys. B 421 , 278 (1994 ) [arXiv:hep-th/9312118 ]; R. de Mello Ko c h a nd J. P . Ro drigues, Phys . Rev. D 54 , 7794 (1996) [a rXiv:hep-th/9605079]; S. R. Das and A. Je- vic ki, Ph ys. Rev. D 68 , 044011 (2003 ) [arXiv:hep-th/0304093]. [24] A. Jevic ki and B. Sakita, Nucl. Ph ys. B 165 , 5 11 (1980 ) ; A. Jevic ki and B. Sakita, Nucl. Ph ys. B 185 , 89 (1981). [25] J. Ginibre, J. Math. Ph ys. 6 , 4 40-449 (1965). [26] Y. Kim ura, S. Ramgo olam and D. T urto n, JHEP 1005 , 052 (2010) [arXiv:0911.4408 [hep-th]]. [27] G. M. Cicuta, L. Molinari, E. Montaldi, F. Riv a, J. Math. Ph ys. 28 , 1716 (198 7). 15 [28] A. Anderson, R . C. My ers, V. P eriw al, Ph ys. Lett. B254 , 89-93 ( 1 991); A. Anderson, R. C. My ers, V. Periw al, Nucl. Ph ys. B 360 , 4 63-479 (1991); R. C. My ers, V. Periw al, Nucl. P hys. B390 , 71 6 (1993). [hep-th/9210082]. [29] J. F ein b erg, A. Zee, J. Stat. Ph ys. 87 , 47 3 (1997). [cond-mat/96091 9 0]. [30] R. C. P enner, Bull. A mer. Math. So c. 15 , 73 (1986); J. D iff. G eom. 27 , 35 (1 988); J. Harer and D. Z agier, In ve nt. Math 85 , 457 (1986) . [31] E. Brezin, C. Itzykson, G. P arisi and J. B. Z ub er, Comm un. Math. Phy s. 59 , 35 (1 978). [32] C. I. T an, Ph ys. Rev. D 45 , 28 6 2 (1992). [33] R. de Mello Ko c h and J. P . Ro drigues, Ph ys. Rev. D 51 , 5847 (1995) [arXiv:hep-th/9410012 ]. [34] A. Jevic ki and H. Levine, Phys . Rev. Lett. 44 , 1443 (1980 ) [Erratum- ibid. 46 , 1546 (1981) ]. [35] F. Calogero, Lett. Nuo v o Cim. 19 , 5 05 (1977 ); Lett. Nuov o Cim. 20 , 251 (1977 ). [36] S. Szego, “Ort honormal P olynomials”, V ol. 23 , New Y ork, NY., 193 9 . 16