Transitive ensembles of random matrices related to orthogonal polynomials

mmiB ELSEVIER

Nuclear Physics B 530 [PM] (1998) 742-762

Transitive ensembles of random matrices related to orthogonal polynomials Taro Nagao a, Peter J. Forrester h a Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan b Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia

Received 16 January 1998; accepted 26 June 1998

Abstract

Transitive correlations of eigenvalues for random matrix ensembles intermediate between real symmetric and hermitian, self-dual quatemion and hermitian, and antisymmetric and hermitian are studied. Expressions for exact n-point correlation functions are obtained for random matrix ensembles related to general orthogonal polynomials. The asymptotic formulas in the limit of large matrix dimension are evaluated at the spectrum edges for the ensembles related to the Legendre polynomials. The results interpolate known asymptotic formulas for random matrix eigenvalues. © 1998 Elsevier Science B.V. PACS: 05.20.-y; 05.40.+j; 02.50.Ey; 02.50.Sk Keywords: Random matrix; Fokker-Planck equation; Orthogonal polynomials;Legendre ensemble

1. Introduction

Random matrix ensembles were first introduced to physics as models of the energy levels of heavy nuclei and then generally applied to energy level statistics of quantum systems (see, e.g., Ref. [1] ). In the early days of physical applications, the bulk properties of the eigenvalue distribution were mainly of interest, because the bulk correlations of the energy levels show universal features. Recently new applications, such as two-dimensional quantum gravity [2], mesoscopic conductors [3] and QCD Dirac operators [4] have been developed. These new applications invoked considerable interest on the eigenvalue distribution at the spectrum edges. One of the most powerful tools in the study of spectrum edges are random matrix ensembles related to orthogonal polynomials. Using the knowledge of the corresponding orthogonal polynomials, the spectrum edges were extensively studied and their universal properties and applicability to physical problems were demonstrated (see the above 0550-3213/98/$ - see frontmatter (~) 1998 Elsevier Science B.V. All rights reserved. PIISO550-3213(98)OO501-X

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cited review articles and references therein). Besides the contributions to physics, connections to mathematics of non-linear differential equations were also revealed [5]. Among others, ensembles related to classical orthogonal polynomials are of particular importance, partly because they allow detailed exact analysis and partly because they include Wishart and multivariate beta distributions, which are major research subjects in the field of multivariate statistical analysis [6]. Another topic of contemporary interest is the theory of parameter dependent random matrices [7]. Applications have been found in the description of the crossover between the different symmetry classes - orthogonal, unitary and symplectic, which correspond to real symmetric, hermitian and self-dual quaternion random matrices respectively. The crossover between orthogonal and unitary symmetry is relevant to the response of a chaotic quantum with time reversal symmetry to a magnetic field (see, e.g., Ref. [ 8 ] ). In this paper we will combine the topics of the spectrum edge and parameter dependent random matrices, and consider the exact calculation of distribution functions for the parameter dependent Legendre ensemble at the spectrum edge. Although this ensemble is the only one we treat explicitly, the formalism we provide is applicable to all the classical random matrix ensembles (Gaussian, Laguerre and Jacobi). In Sections 2 and 3 some results needed in subsequent sections are revised. These results are the Fokker-Planck equations formalism of parameter dependent random matrices, the calculation of correlation functions in terms of a Green function, and theorems from the theory of quaternion determinants and skew orthogonal polynomials. Attention will be focussed on the case in which the equilibrium distribution distribution corresponds to the eigenvalue probability density function (p.d.f.) of an hermitian matrix, while the initial state is either the eigenvalue p.d.f, for a real symmetric, self-dual quaternion or an antisymmetric random matrix. In Section 4 the general formulas of Section 3 are made more explicit in the case of the antisymmetric-hermitian transition. In particular, formulas are presented for the general parameter dependent distribution in terms of quantities known from the evaluation of the single-particle Green function. In Section 5 application is made of our general formulas to the calculation of the parameter dependent n-point distribution for the Legendre ensemble at the spectrum edge.

2. Fokker-Planck equation formalism The problem of computing the p.d.f, for a parameter dependent random matrix was first addressed by Dyson [9]. Dyson considered Gaussian parameter dependent random matrices, which have joint distribution for their elements

P(X(°);X;r) =ABexp(-/3Tr{(X-e-~X(°))2}/2]l-e-2~l).

(2.1)

Here X is a real symmetric (/3 = 1), hermitian (/3 = 2) or self-dual quaternion (/3 = 4) N x N matrix, and X (°) is a prescribed random matrix which must belong to a subgroup of the symmetry class of X. For r ~ c~ the p.d.L (2.1) is independent of X (°) and is

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T. Nagao, PJ. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

that of the standard Gaussian ensembles (see, e.g., Ref. [ 1] ). The eigenvalue p.d.f, is then known to be proportional to e -#w~"~ with N

w(H) = l ~_,x2_ 2 j=l

Z

1oglxk--xjl

(2.2)

l<.j
(the superscript ( H ) is used because of a connection with the Hermite polynomials). For general r Dyson proved that the eigenvalue p.d.f, p = p(xl . . . . . xN; z) satisfies the Fokker-Planck equation 1 u c9 -aw O nw C = -~ ~.= --eox.i ~ --e'oxJ

__

OPor= £'p'

(2.3)

with W = W (H), subject to the initial condition that p agrees with the eigenvalue p.d.f, of X (°) at r = 0. It is well known that there exist two other ensembles of random matrices with real eigenvalues which share many features in common with the eigenvalue p.d.f. (2.2) of the Gaussian ensemble. These are the Laguerre and Jacobi ensembles, which have eigenvalue p.d.f.'s proportional to e -#w with 1 N

at

W ~1~---- ~ x j - - ~2

N

y~ logx 2j bt

log sin 2 xj

j=l

l ° g l x 2 - x ~ I'

(2.4)

l ~.j
N

W' J' : - a2 Z 2

Z

j=l

j=l

-

2"

N

Z log cos 2 xj j=l

-

Z

l°g I sin2 x i - sin 2 Xkl,

l<~.j
(2.5) where in (2.5) 0 <~ xj ~< 7r/2. For both these ensembles, parameter dependent generalizations have been considered [10-14,7], with the result that the corresponding eigenvalue p.d.f, again satisfies the Fokker-Planek equation (2.3), where W is given by (2.4) or (2.5). At the special coupling /3 = 2, corresponding to a final distribution with unitary symmetry, the Fokker-Planck operator/2 can be transformed into an operator in which all the variables are separated. Thus by considering - I(H

- Eo) := e#W/2£e-~w/2,

(2.6)

/3 one finds that for/3 = 2 N 02

N (2.7)

.'=

j=l

Z 0 7 + ,=1 t,

+

,

(2.8)

T. Nagao, P J, Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

N 02 H(J':-Y~

k(a'(a'-l)b'(b':l)) +

j=l

\

j=l

sin----ix--j +

cos2x,

,]'

745

(2.9)

where E(on), E(oL) and E (J) are constants independent of xj's.

2.1. Green function solution As the above operators are Schr6dinger operators for non-interacting particles, which are required to be fermions due to the vanishing of the p.d.f, e -#w at coincident points, the Green function initial value problem can be solved in terms of a determinant. First recall that for the imaginary-time Schr6dinger equation -2~b

= n0,

(2.10)

we say that 0 =: Gs(xl °) . . . . . x_(o). N , Xl . . . . . XN; r) is the Green function solution if it is the solution which satisfies the initial condition N v) ~=o _(o) ). g,(x, . . . . . xN; = H a ( x, °)) (xl °) < . . . < *N /=l

It follows from (2.6) that GFp(XI0)

~ ' ' ' ~ A N ~ ( 0 ) . " a~' l ~ " "" ~ XN; 7")

= eVE0/2

e -W(xl'''''x~)

.... (o) ..(o)~GS(Xl °) . . . . . "~N _(o) ;xl . . . . . XN;Z)

e-Wtxl

(2.11)

,-.-,-~N ,

is the Green function solution of the Fokker-Planck equation (2.3) with fl = 2. It remains to determine Gs. Since the operators (2.7)-(2.9) describe free fermions, this is given in terms of the corresponding Green function solution of Eq. (2.10) with N = 1, g s ( x , y ; z ) , by

GS(x(O), . . . . a..(o). N , xl . . . . . XN; r) = det[gs(x J o),. xk; r) ]j,k=l,...,N.

(2.12)

Furthermore, the method of separation of variables gives that the solution of (2.10) with N = 1 can be written in terms of the eigenfunctions {~n},---o3 .... and corresponding eigenvalues {v,},=o.1 .... according to (°)) e j (u) e_~ ~ gs(u(°);u;r) = ~ ,Pj(u(~)lej} j--o with yj := vj/2 and (fig) : = f~ f(u)g(u) du. For the particular operators (2.7)-(2.9) the eigenfunctions can be expressed in terms of the Hermite, Laguerre and Jacobi polynomials respectively. In fact a unified presentation of the these three cases can be given by introducing the new variables yj = xj in (2.7), 3~i = xj2 in (2.8) and ½(1 - yj) = sin2xj in (2.9), and defining the function g by

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Z

g( y(O), y; 7")dy(O)dy = gs( x(O), x; 7")dx(°)dx.

(2.13)

We then find that

y~O~;Yl . . . . .

a(y~O) . . . . . =

GFp(xIO),.

YN; 7")@! ... dylv

~ . . , - x_(0). N ,-~1, .

. , XN; . . 7 " ).d X l.

dxN,

where

G(y[O) . . . . . y(O); Yl . . . . . y~v) ,v [

w(y))

= e e ° r / 2 H Vw(Y~ 0)) j=l

(Yk - Yj) - (0) .(0) det[g(yJO)'yk;7")lj,k=l,..., lv l<~j
(2.14) with oo

g(y(O), y; 7") = V/w(y(°))w(y) Z p j ( y ( ° ) ) p j ( y ) e j---o

-~'jr.

(2.15)

The weight function w(y) is given by

w(y) = e -y2,

w(y) =yae-Y,

w(y) = (1 - y)a(1 + y ) b ,

(2.16)

where a = a r - 1/2, b := b' - 1/2, for H equal to H (n), H (z) and H (J) respectively, and the polynomials {Pj(Y)}j--o,1 .... are the corresponding orthonormal polynomials, and so proportional to the Hermite polynomials, Laguerre polynomials and Jacobi polynomials respectively.

3. Correlation functions and quaternion determinants In terms of the variables yj introduced above, the p.d.f, p(Yl . . . . . y~; 7-) is calculated from the Green function according to the formula p(yl .....

dye°) . . . f dy~°) po(y~°), ... , y(/v0) )

y N ; r ) = ~.~ If 11

1~

x G (y[°) . . . . . y~0); Yl . . . . . YN; 7-),

(3.1)

where po(y[ °) . . . . . y(N°) ) denotes the prescribed initial p.d.f, in terms of the variables yj, and 11 is the transformed domain. Hereafter the weight function w(y) and the domain 1' are supposed to be general (w(y) is not restricted to one of the forms (2.16)). The corresponding parameter dependent n-point distribution function is then calculated from the formula

P(,)(Yl . . . . . Yn;7-) - C~ (N---n)!

dyn+l.., I~

/ 1~

dyNp(Yl . . . . . YN;7-),

(3.2)

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747

with the normalization constant CN defined as CN = ft' dyl ... fI' dyN p (Yl ..... YN; 7"). Hereafter we omit the domain index I' and the prefactor e e°r/2 for simplicity. Our interest is in the calculation of (3.1) and (3.2) for G of the form (2.14), and with P0 given by one of the functional forms N

N

1-I ~ H j=l N/2

j
lYj - YlI, N/2

I I w(yj)6(yj -- Yj+N/2) I I lYj -- Y/I 4 j=l j
(assuming N even),

[N/21

Po(Yl ..... yN) OC

(3.3)

H w(yj)8(yj + Y/+[(N+I)/2]) j=l

iN/21

~ 1, Neven,

x 1-I lYY - Y/212 j
iN/Z]

H YY' N odd. j=l

For the final of these initial conditions, it is assumed that the weight function w(y) and the interval I t are symmetrical about the origin. Each choice has significance in the theory of random matrices: they correspond to the original matrix being real symmetric, self-dual real quaternion and antisymmetric hermitian, respectively. For each choice (3.3) ofpo the integral over y~0). . . . . y(N°> required to compute (3.1) can be computed in terms of a Pfaffian [ 15-17]. The result depends on the parity of N. For N even, performing the integration in (3.1) gives N

N

P (Yl . . . . . yN; 7") = H ~ j=l

where the function function (2.13) by

1-I (yj -- yt)Pf[ F(yj, Yt; ~')]j,l=t,2,...,N,

(3.4)

j>l

F(x, y; 7-) is given in terms of the transformed single-particle Green t Z

F(x, y; 7-) = f

dz'f

dz {g(x, z;r)g(y, z'; 7-)- g(y, z;7-)g(x, z';7-)},

F(x,y;7")=

/ dz{ g(x,z;7-)-~-zzgCy, o z;7-)-gCy, z;7-)-~-zzg(,z;7-) Ox}

F(x,y;7-) =

/ dz~--z{g(x,-z;7")g(y,z;7-) , -g(y,-z;7-)g(x,z;7-)}

,

(3.5) (3.6) (3.7)

0 for the real symmetric, self-dual real quaternion and antisymmetric hermitian initial conditions respectively. For N odd, performing the integration in (3.1) gives

T. Nagao, P.J.Forrester/NuclearPhysicsB 530 [PM](1998) 742-762 N N

748

P(Y, . . . . . YN; 7") = H ~

I I (YJ -- Yt)

j>l

j=l

xPf[ [F(yj'yl;'r) lj'l=l'2"''N-[f(yl;7")ll=l,.,Notf(YJ;r)]J=l'""lV 1 ,

(3.8)

where F is given by (3.5) or (3.7) (the initial condition (3.6) corresponding to a self-dual real quaternion initial condition is only defined for N even), while f(y;~') =

fg(y,z;,)dz,

(3.9)

f(y;~') = ½g(y, 0;r)

for the real symmetric and antisymmetric hermitian initial conditions respectively.

3.1. Quaternion determinants The functions (3.4) and (3.8) have representations as a quaternion determinant [ 18,11 ] which are of crucial importance to the calculation of the distribution function (3.2). First we introduce the quaternion determinant. A quaternion is a linear combination of four basic units {1, el, e2, e3}:

q = qo + q" e = qo + qlel + q2e2 + q3e3.

(3.10)

Here q0, ql, q2 and q3 are real or complex numbers. The first part ql is called the scalar part of q. The four basic units satisfy the multiplication laws

1.ej=ej.l=ej,

1.1=1,

j=1,2,3,

(3.11)

e~=e~=e~=ele2e3= -1.

The multiplication is associative and in general not commutative. The dual 0 of a quaternion q is defined as 0 = qo - q" e.

(3.12)

A matrix Q with quaternion elements qjt has a dual matrix Q = [00]. We can express the quaternion units as 2 x 2 matrices 1~[10~],

e2---+[?i;i],

e,~

[0;11,

e3---~ [ ; ? i ] .

(3.13)

A quaternion determinant Tdet of a self-dual Q (i.e. Q = Q) is defined as I

TdetQ = '~-'~( - 1 ) N-t 1-[(qabqbc... P

1

qda)O,

(3.14)

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749

where P denotes any permutation of the indices ( 1,2 . . . . . N) consisting of l exclusive cycles of the form (a --, b -+ c -+ ... -+ d ~ a) and ( - 1 ) N-t is the parity of P. The subscript 0 means that we take the scalar part of the product over each cycle. Expressions can be given for (3.4) and (3.8) in terms of a quaternion determinant for general F(x, y; r) and f(x; r).

Theorem 1. The p.d.f. (3.4) can be written as the quaternion determinant 0/2 p(yl ..... YN) = Hrj-l(r)Tdet[f(Yj,Yt;r)],

j,l= 1,2 . . . . . N.

(3.15)

j=l

The quaternion elements f ( x , y; r) are represented as

[ S(x,y;r) l(x,y;r) ] f ( x , y ; r ) = LD(x,y;r ) S(y,x;r) "

(3.16)

The functions S(x, y; r), D(x, y; 7-) and I(x, y; r) are given by N/2 S( x, y; r) =

1

{¢b2k-2 (x; r) v / - w ~ g 2 k _ 1(y; r)

(¢) k=l r k - I

-~2k-1 (x; r) X/w(y) R2k_2(y; r) },

N/2 D(x,y;r) = Z

1

(r) k=l rk-1

(3.17)

V/w(x)w(y){R2k_2(x;r)R2k_l(y;r )

--R2k-I (x; r)R2k-2 (y; r)},

N/2 l(x,y;r) = _ X-'/_~ k=l

1

rk_l (T~----

{t~2k-2(X; r)~2k-1 (y; r)

(3.18) - t~2k- 1(x; r)~2k-2 (y; r) }

-F(y,x;r).

(3.19)

Here 4~k(x; T) = / F(y, x; r) ~ R k ( y ;

r)dy

(3.20)

and the R k ( y ; r ) = yk + . . . are arbitrary polynomials of degree k with the coefficient of the highest order term equal to 1.

Theorem 2. The p.d.f (3.8) can be written as the quaternion determinant (N+I)/2

P(Yl ..... yN;r)=

H

rJ-t(r)Tdet[f°dd(YJ'Yl;r)]J'l=l'"U"

j=l

The quaternion elements f°dd(x, y; r) are represented as

(3.21)

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750

[s°dd(x,Y; r) l°dd(x,y;r)] f°0d(x, y; 7") = LDoOd(x, Y; r ) s°dd(y, X; 7.)

"

(3.22)

The functions S °dd, etc., are given in terms of S, etc., in Theorem 1 according to

s°dd(x, y; 7.) : S( x, y; r)[Rk~..~R~kd"N~--'N--1 Jr

1

r(N-l)/2( 7.)

f ( x; 7.) ~ R ~ d l

(y; 7.),

D°dd(x, y; 7") : O ( x , y; T)IRk~_~RO~d,N~-+N-I'

(3.23) (3.24)

/°dd(x, y; 7") = I ( X, y; 7")IRk~_+R~O,N~-*N-I

q- r(N--1)/2(7.)1

(CrPN-I(X;7.)f(y;7.) -- f(x;7.)qbN-l(y;'r)).

(3.25)

Here R~dd(x; 7-) is an arbitrary polynomial with the coefficient of the kth order term equal to 1 such that N H(xj

- Xl) = det[ R~dl (x j; T)]j,~=l,...,N

j>l and t/'k (x; r ) = /

F ( y , x; r) X / - ~ - ~ R ~ d (y; 7-) dy.

(3.26)

Now let us revise how, by an appropriate choice of the polynomials {Rk(X, 7.)} and {R~dd(x,7-)} in Theorems 1 and 2, the integrations required by (3.2) to compute the n-particle distribution can be carried out.

Theorem 3. Let the quaternion elements qjz of a self-dual N x N matrix QN depend on N real or complex variables xl, x2 . . . . . xN as qjl = f ( x j , Xl; 7.).

(3.27)

We assume that f ( x , y; 7.) satisfies the following conditions:

f ( x , x; r ) d / z ( x ) = c, f(x,y;7-)f(y,z;7-)dtz(y)=f(x,z;7-) + Af(x,z;7-) - f(x,z;7-);t.

(3.28) (3.29)

Here dlz(x) is a suitable measure, c is a constant scalar, and A is a constant quaternion. Then we have

f TdetQs d/z(x/v) = (c - N + 1)TdetQN_l,

(3.30)

where QN-l is the ( N - 1) x ( N - 1) matrix obtained by removing the row and the column which contain xN.

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751

By Schmidt's orthogonalization procedure, we can construct the monic polynomials Rk(x; 7") of degree k so that they satisfy the following skew-orthogonality relation: (R2m ( x; 7"), R2n+l ( Y; 7-) ) = - (R2n+l ( x; 7"), R2m (y; 7") ) = rm (7") ~rnn,

(R2m(X;7"),R2.(y;7"))=O,

(R2m+t(x;7"),Rz.+l(y;7")) = 0 ,

(3.31)

where

(f(x),g(y)) = ~

dx

dyv/w(x)w(y)F(y,x;7")[f(y)g(x)

- f(x)g(y)]. (3.32)

Then it is straightforward to show that f ( x , y; 7") in Theorem 1 (N even case) satisfies the condition on f ( x , y; 7") in Theorem 3. For f ( x , y; 7") in Theorem 2 ( N odd case) to satisfy the conditions of Theorem 3 one requires Rn°da(x; 7") = R,(x; 7") r(N--l)/2( 7") = SN--I (7"),

s.(7")

, , RN-1 (x; 7"),

r(N--1)/2~7")

n = 0 . . . . . N - 2,

(3.33)

R °ad N-1 ( x ; r ) = RN-I(X;7"),

where I" Sn(7") = J d x

X/w(x)f(x;7")R.(x;7").

This means that for both N even and N odd the n-point distribution function (3.2) can be written

P(,) (Yl . . . . . Yn; 7") = Tdet[ f (yj, Yk; 7") ] j,k=l,...,n.

(3.34)

3.2. Transformed summations Expressions (3.17)-(3.19) for S(x,y;7-), D(x,y;7-) and I ( x , y ; r ) require knowledge of the polynomials Rk(x; 7"). For 7" = 0, and the first and second initial conditions in (3.3), these polynomials are known for the Gaussian, Laguerre and Jacobi ensembles [ 19,20], and for the third initial condition in (3.3) they have been given in [17] for the Gaussian ensemble. In these works, the polynomials Rk(x; 0) have been written in the form n

R,,(x;O) = E

a.jCj(x),

an. = 1,

(3.35)

j=0

where the Cj(x) are monic polynomials of degree j, related to the orthogonal polynomials pj(y) in (2.15) by

pj(y) = h~/zCj(y), so that

(3.36)

752

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Nagao, PJ. Forrester/NuclearPhysicsB 530 [PM](1998) 742-762

f dxw(x)Cj(x)Ck(x)

hj6jk.

(3.37)

With {a,y} defined by (3.35), it is straightforward to verify that we have n

Rn(x; r)e r'~ = E a~jCj(x)en~' j--o

(3.38)

which is done by showing from the definition (3.32) that

{Rm(x; r), R,(y; r) ) = e-(r'+r")~(Rm(x; 0), R,(y; 0))l~_0-

(3.39)

It is useful to also consider the inverse of the expansion (3.38). Thus, if we defined [fijt]j,t=L..,n as the inverse of [ajl]Zl=l,...,n, then we can write n

C,(x)e r'r = EflnjerJ~Rj(x;r), finn= 1. j--o

(3.40)

One immediate application of (3.40) is towards rewriting the integral formula (3.20) defining ~ , ( x ; r ) . Consider the first initial condition in (3.3). Then the function F(x, y; ~) is given by (3.5). For the single-particle Green function occurring in (3.5) we substitute (2.15) with the pj therein replaced by Cj according to (3.36). The resulting expression for F(x, y; r) is then substituted in (3.20). Use of the expansions (3.38) and (3.40) then allows us to rewrite (3.20) as qbn(X;7")er"r = X / ~

E

Cv(x)e-r~r hv

v--O

fl~,j(Rn(z;O),Rj(z;O))lr=o. (3.41) j--O

Repeating the same procedure with F given by (3.6) and (3.7) shows that (3.41) again holds in these cases. The skew orthogonality relation (3.32) then yields

E

¢2k-l(X;r)er2k-'z=--~

C~(x)e-rvr h~ f v 2k-2rk-l(O),

v=2k-2 oo

(I)2k--2(X;T)eZ2k-21"= ~

E 1 C~(x)e-r~r v=2khv fly 2k-lrk-l(O).

(3.42)

The formulas (3.42) can be substituted in the definition (3.17) of S(x, y; ~-), thereby giving an alternative expression for that quantity. We find

N-I C~(x)C~(y) S(x, y; r) = V/w(x)w(y) E hv /,=0 oo N--I

+X/w(x)w(Y) E E Cv(x)e-"r h~ fvkRk(y;r)erkZ' v=N k=O

where use has been made of the fact that

(3.43)

T. Nagao, PJ. Forrester/NuclearPhysics B 530 [PM] (1998) 742-762 rio(0)/rk(7") = e O'2k+r2k+')~,

753 (3.44)

which follows from (3.39). Notice that this formula separates off the 7"~ c~ behaviour of S(x, y; 7"). In fact this term can be simplified by the Christoffel-Darboux formula N-1

V/w(x)w(Y) E v--O

= V/W(X)w(y )

C~(x)C~(y) h~ 1 CN(X)CN-I(y)--CN-I(X)CN(y) hN-l x -- y

(3.45)

It is also possible to rewrite F(x, y;7") and thus, using (3.19), to rewrite l(x,y;7"). This can be done by first noting from the completeness and orthogonality of {Rt(x; 7")} that for an arbitrary function f ,

f(x) = E l--o

(f, e2/+l ) cx~ (f, R2l) R21+I (x; 7"). rl(7") R2l(x;7") -- EI__.O rl(7"-'--'--T

Substituting t~(x - y) for f ( x ) in this formula gives

a(x - y) =

\/'w-(-~(~=o ~21+l(Y;7")R2t(x;7")_~-:~,crP2t(Y;7") - R2t+l (x; 7")). _ rl(7") l=O rt(7")

Replacing x by xl, multiplying by F(x, xl;7-) and integrating over xl gives [~2k-2 (x; r)@zk-l(Y; 7-) -- ~2k-l (X; 7-)~2k-2 (Y; 7") ], F(x, y; 7-) = E - k=l rk-l(r) (3.46) which is the desired formula. Substituting (3.46) in (3.19) then shows oo

I(x, y; r) =

- -

k=(N/2)+t

rk-l(r)

[t~2k_2 (X, T)~2k_ I (y; T) -- qb2k_I (X; 7")qO2k-2(y; 7") ]. (3.47)

In the final section application will be made of (3.42), (3.43) and (3.47) in the calculation of the n-point distribution for the Legendre ensemble at the spectrum edge.

4. Antisymmetric initial condition In this section formulas involving the normalizations h~ will be given for the coefficients flnj in (3.40) in the case of antisymmetric initial conditions, which corresponds to the third p.d.f, in (3.3). This allows (3.42) and (3.43) to be similarly re-expressed. As remarked earlier, when the initial distribution is antisymmetric, because of the symmetrical distribution of the eigenvalues with respect to the origin, the weight function satisfies

754

Z

Nagao, P.J. Forrester/NuclearPhysics B 530 [PM] (1998) 742-762

w(x) = w ( - x ) .

(4.1)

Then we can assume that the orthogonal polynomials Rn(x; r) are either even or odd polynomials: Rn(X;~-) = ( - 1 ) n R n ( - X ; r ) ,

n=0,1,2 ....

(4.2)

The skew orthogonality relations for the skew orthogonal polynomials Rn(x; O) read

f dz W(---~Z){R2m(--z;O)R2n+I(Z,O) Z 0

--

R2n+I(-Z,O)R2m(Z;O)} = rm(O)t~mn, (4.3)

dz - ~ z ) {R2m(-Z;O)R2n(z;O) - R2n(-Z;O)R2m(Z;O) }

=0,

(4.4)

o

f dz--~z w(z){R2m+l(-Z;O)R°n+l(Z;O)

- e2n+l(-Z;O)e°m+l(Z;O)} = 0.

(4.5)

o Because of the relation (4.2), Eqs. (4.4) and (4.5) automatically hold. Now we assume that R2,+1 (x; O) = xR2~(x; 0).

(4.6)

Then Eq. (4.3) is rewritten as

1 fdz

W(z)R2m(Z;O)R2n(z;O)

(4.7)

= rm(O)t~rnn.

This is the orthogonality relation for ordinary orthogonal polynomials. Comparing (4.7) with (3.37), we obtain a general useful formulas

R2n(x; O) = C2n(X),

R2n+l ( x ;

O) = xC2n(X),

rn(O) = lh2n.

(4.8)

Furthermore, it is well known that there exits a three-term recurrence relation for general orthogonal polynomials with a definite parity,

xC~(x)=C,+l(x)+.hn

C,_I(X),

n=0,1,2 .....

(4.9)

]'/n-- 1

where we define C-1 (x) = 0 and h - i = 1. From (4.8), (4.9), (3.35) and (3.38), we can see that

R2n ( x; ~) e r2"r = C2n(x) er2"r, R2n+l (x; r ) e r2"+lr = C2n+1( x ) e r2"÷17+

h2n C2n-I (x)e r2"-~7. h2n- 1

(4.10)

The simple nature of (4.10) allows Ck(x) to be expressed in terms of {RI(X;O)} and thus from (3.40) the coefficients flkt to be determined. We find

T. Nagao, PJ. Forrester/NuclearPhysics B 530 [PM] (1998) 742-762

fl2nj =

755

j=2n

1, O,

otherwise,

/3(2n+l)(2t) = 0

(4.11)

h2------L-PI-I h2q-I

(4.12)

and /~(2n+1)(2l+1) = ( - 1 ) t - " 1 2 I p=l

h2p-1 q=l hzq "

Thus, according to (3.42) we can write the q~n as t~2n_ 1(x; ~-)e~'2"-'~ =

-½~e-'/2"-~'C2n_2(x), n-l ( h2u-l ~ V / ' ~ 1-I \ ~ ]

qb2n--2 (X; 7)eY2"-2r = l ( - 1 ) n - l h 2 n - 2

v--o

' ( ×

-1)/1-I l= -

)

\h~a-lJ h~l+l C2t+'(x)"

(4.13)

A--0

Now substituting (4.11) and (4.12) into (3.43) gives

N-I Cv(x)Cv(y) S(x, y; 7") = V/w(x)w(y) Z h~ u---O

+~/w(x)w(Y)(-1)(N/2)-I

(N/2)-1 (h2~,-1 ~ CN_l(y)erU_l , I I \h-~z~/ p=O

x

(-1)tII

l=

\h--~a-l/ h~t+l C2t+l(x)"

(4.14)

A=0

Thus in the case of the antisymmetric initial condition, the correlation functions are generally expressed in terms of the corresponding ordinary orthogonal polynomials. There is no need to explicitly use the skew orthogonal polynomials.

5. Legendre ensembles Now we turn our attention to the calculation of the parameter dependent distribution function (3.34) in the neighbourhood of the edge of the Legendre ensemble (the special case a = b = 0 of the Jacobi ensemble). The limiting case r = 0 was treated in [ 19,23,24] and we generalize the results in those works. As our interest is in the N --* to limit, we restrict attention to the real symmetric and self-dual quaternion initial conditions (the antisymmetric initial condition at the edge is locally identical to an hermitian initial condition in the N ~ (xz limit, so in this case the distributions will be ~" independent). In this case, the orthogonal polynomials Cn(x) are given by the Legendre polynomials P,(x):

756

T.

Nagao, PJ. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

~n , F ( n + l) ,~ ~ , Cn(x) = z n~ ~--~n--+ -()rn~x)

(5.1)

with corresponding normalization hn = 2 2 n + l

(n[)4

F ( 2 n + 1)F(2n + 2)"

(5.2)

Also, from Ref. [22] [Eq. (4.24.2)], the eigenfunctions of (2.9) with N = 1, a = a t - 1/2, b = b' - 1/2 are (sin x) a+t/2 (cos "~3 v~b+l/2o(a,b) " n (cos 2x) with corresponding eigenvalues 4 (n + (a + b + 1)/2)2. Thus for the Legendre ensemble y. = 2(n + ~1 ) .2

(5.3)

5.1. Real symmetric initial condition

Since we are interested in the N --* c<~ limit, it suffices to suppose N is even. Now, the skew orthogonal polynomials Rn(x; 7) are known to be (see, e.g., Ref. [21] ) 1

d

R2, (x; 0) - 2n +-~ dx C2~+1(x) =2

2n+l vF(2n + 2) ~ F(4k + 2) ( 2 n ) ' F ( 4 n + 3 ) k---o2 2 k ( F ( 2 k + 1))2C2k(X)'

R2n+l (x; 0) = C2,+1 (x)

(5.4)

with the corresponding normalization rn(O)

:

¢r F ( 2 n + 1 ) F ( 2 n + 2) 24n+------q F ( 2 n + 3 ) F ( 2 n + ~)"

(5.5)

Thus, according to (3.38), n

R2, (x; r)e r2,'~ = 22n+l (2n) I F ( 2 n + 2) y ~ ( 4 k + 1)er2krP2k(x), • F ( 4 n + 3) k--O R2n+l (x; ~-)er2"+lr=

2 2n+l

F ( 2 n + 2) -2,, lr (2n + 1) ! ~ e " + P2n+l (x),

where use has been made of (5.1). From the expressions for Rn in (5.4) and the definition (3.40) of the immediately that 1~2n,2./+I : O ,

1

j=n,

(2n) (2n -- 1) ( 4 n + 1 ) ( 4 n - 1)

fl2n,2j---0

j = n - 1, j
(5.6)

~kl it follows

T. Nagao, P.J. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

757

(5.7)

/~2n+ l,j = 62n+l,j.

Substituting (5.7) into (3.42), we find 22k-2 F ( 2 k - 1) ~ 1) [e-Y2kTP2k(x) -- e-Y2k-2~P2k-z(X) ]'

t~2k_ 1 (X; 7")ez'2k-~ = rk_l (0) - ~

22k-I F ( 2 k + i) ~2k_2(X;T)er2k-Zr=rk_l(0) - ~ F(2k) 2 e--~'2k-~rP2k_l(X).

(5.8)

Furthermore, substituting (5.7) along with (5.1) and (5.2) in (3.43) gives N PlV(X)Plv-1 (y) - Ply-1 ( x ) P N ( y ) 2 x-y

S( x, y;1") =

-le-~'NrP1v(x)

Z

( 4 k + 1)e~'2rrP2k(y).

(5.9)

k=0

The quantity S(x,y;~') is thus explicitly given, while from (3.47) and (3.18) the quantities I ( x , y; 7") and D(x, y; r) are specified in terms of q~n(x; r) and Rn (x; 7) and thus can be computed by use of (5.8) and (5.6). We seek explicit formulas for S(x,y;~'), l ( x , y ; r ) and D(x,y;~') for x and y appropriately scaled in the neighbourhood of the spectrum edge (x, y ~ - 1 ) . Now for x ,-~ - 1 the Legendre polynomials have the asymptotic behaviour [22] Pn(x)

~ (-1)nJo(n('n'-O)),

x=cos0,

"n'-O=O(1/n),

(5.10)

I1"-"~ 0 0

where Jo(x) is the Bessel function. Use of (5.10) in (5.9) shows that

s(x,

lim

- 1+

+

2(N

71/2)2

X1/2 J1 ( X1/2 ) jo ( y1/2 ) _ yl /2 Jo ( Xl /2 ) j I ( yU2 ) 2 ( X - Y)

e--' 4 rJ°(X1/2) f

1

se~'2J°(Yl/2s)ds"

(5.11)

0

Use of (5.10) in (5.8), together with the formulas (5.5) and (3.44) relating to rk(r), shows that rk-11('r) '/~2k-2 - l + 2 N 1

---kse-

~.s2

1""

2 2(N_~1/2)2 d

~2k-I

Jo(X /~s) --ds(e-rS2j°(yl/2s) )'

- l + 2 N 2 2(NQS1/2)2

k, N --* oo,

s = 2k/N fixed. (5.12)

Substituting this into (3.47) shows that

T. Nagao, P.J. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

758

i(X,y;r) := lim i ( _ 1 +

X

N---,~

Y

z

~-ff, - 1 + 2N----~;2(N q: 1/2) 2

)

OO

[

d s e -2rs2

].

[Y1/ZJo(Xl/2s ) J1 (Y1/Zs) - x l / Z J o ( y l / Z s ) J1 ( x l / 2 S )

1

(5.13) Also, using the expressions (5.6) and the asymptotic formula (5.10) together with (5.5) and (3.44), we obtain

1

(1÷

X

rk_l(r) R2k-2 --

"r

----5; 2N 2 ( N + 1/2) 2

)R

(

Y

~-

2k-1 - 1 + 2N2; 2(N_~1/2)2

)

1

-4k3erS~ Jo (Y1/2s) f du u ers~u:Jo ( xl/2su),

k, N --~ co, s = 2k/N fixed.

0

(5.14) From the above formula, the limiting behaviour of the function D(x, y; 7) is easily evaluated as

D(X,Y;r):= lim

D

N-~e~ 2 - ~ 1

-1+

2-~' -

1+

• 2N 2' 2(N + 1/2) 2

1

i f f du s3ue-(,+ul

= 1--6 ds 0

0

x [ Jo(XI/2s) Jo(yI/2su) - Jo(yl/2s) Jo(X1/2su) ].

(5.15)

Define the scaled parameter dependent distribution by

.....

:=

Lm Xp(n)

( - 1 + 2N--' Xl-':. . . . . _1+Xn • ) -2N - 5 ; 2 ( N q : 1/2) 2 ' (5.16)

where on the RHS P(n) is given by (3.34). Then

[ S(Xj'Xk;g) I(Xj'Xk;7") ] P(,) (X1 . . . . . X~.;r) = Tdet D(Xj, Xk; ~') S(Xk, Xj; z) j,k=l,...,n'

(5.17)

where S, I, and D are given by (5.11), (5.13) and (5.15) respectively. In particular, this formula gives for the parameter dependent density at the spectrum edge

p(1)(X) = S(X,X; r) = ¼(( Jo(X 1/2) )2 ÷

( Jl(Xl/2) )2)

l

e~4rjo(Xl/2) f serS:Jo(Xl/2s)ds. 0

(5.18)

T. Nagao, PJ. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

759

We note that for large X integration by parts of the integral using the fact that d -~ss(SJo(s)) = J l ( S )

shows that the integral decays faster than the other terms, which in turn give the behaviour p(t) (X)

1

~ x~oo 2~rX1/2 '

(5.19)

independent of the parameter r.

5.2. Self-dual quaternion initial condition For the case of the self-dual quaternion initial condition, the skew orthogonal polynomials R, (x; 0) are written in terms of the Legendre polynomials as (see, e.g., Ref. [ 21 ] )

v/-~F(2n + 1) R2,(x;0) =C2n(x) + 22nF(2 n + ½) , R2.+l (x; 0) = C2.+~ (x) -

n > 0,

2n(2n + 1) C2n_l(X), ( 4 n - 1)(4n + 1)

n

> 0

(5.20)

with corresponding normalization

•rF(2n + 1)F(2n + 2) rn(O)=24n_lF(2n+l)F(2n +3),

n>0,

ro(0) =2.

(5.21)

Thus, according to (3.38),

R2n (x; ~-)e y2"r = 22n (2n) vF(2n + 1) v/-~F(2n + 1 ) ~or . F ( 4 n + l ) e ~ ' 2 " ~ ' P z n ( X ) + - - r - - en + ~) ,

n>0,

R2n+l(x;r)e~2"+'r=22n+~(2n+ 1 ) t F ( 2 n + 2)e~2"~rP~ (x) • F(4n + 3) ~n+l _22n(2n)T F ( 2 n + 2 ) ~2 ~ • F(4n + ~-~e "- P2n-l(x),

n > 0.

(5.22)

A straightforward calculation allows these relations to be inverted according to (3.40), thus giving the matrix Bit as

1

j=2n, v/-'~F(2n + 1) 22nF(2 n + ½)

~2n,j =

0

J = 0 (n > 0), 0
32n+1,2j = O,

/32,+1,2j+1=

22j 2 n F ( 2 n + 2) F ( 2 j + 3) - F(2j+2) F(2n+3) "

(5.23)

760

T. Nagao,

PJ. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

Substituting (5.23) along with (5.20) and (5.21) in (3.43) yields

S(x, y; 7") =

N PN(x)PN-I (y) - PN-1 (x)PN(y) 2 x--y oo

- e :'°r ~ ( 2 k + ½)e-r2k~'P2k(X) k=N/2 oo

3 --~2k+l't" +erN-'TP'v-l(Y) Z ( 2 k + ~)e P2k+l(x). k=W2

(5.24)

Use of (5.10) then shows that

S(X, Y; r) =

XI/2 J1 ( X1/2) Jo(yl/2) _ y1/2 jo( X1/2) j I ( y l / 2 )

2(X - Y) oo

+¼(erJo(V 1/2) - 1) f se-rS2jo(X1/2s)ds,

(5.25)

1

where S(X, Y; r) is defined in (5.11). Substituting (5.23) along with (5.1) and (5.2) in (3.42) gives 22k-ZF(2 k _ ½) ~b2k-l(X;~')eY2k-tr= --rk-l(O) ~ - - ~ - ~ 1) P2k-2(x)e-Y2k-2r'

k > 1,

oo

22k-II'(2k _ ½) Z ( 2 v -- ½)P2~-l(x)e -~2"-'r. ~2k-2(x;r)e r2k-2*=rk-l(O) ~ ~ ~=k (5.26) From these formulas, (5.10), (5.21) and (3.44), it follows that

(

1 ~2k-2 -- 1 + __XX. 7" ](P2k-l 1 2 / rk-l(~) 2N 2 ' 2 ( N + 7)

(

--

1+

"

" 2N 2' 2 ( N +

1)2

)

oo

16k3Jo(yl/2s) e -rs2 f ue-rS2"2Jo(X1/Esu)du,

k,N ~ cxz, s = 2k/N fixed.

1

(5.27) Substitution in (3.47) then yields

I(X,Y;r):=lim

~ O(3

I

-1 + ~,-1

+ 2N2; 2 ( N + ½)2

OO

s3ue-rS2(l+u2) = - - l f f d uds 4

1

1

x [Jo(X1/2s)Jo(yI/2su ) - Jo(y1/2s)Jo(XI/2su)].

(5.28)

T. Nagao, P.J. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762

761

Similarly, use of (5.10) in (5.22), together with (5.21) and (3.44) gives

,

(

x

R 2 k - 2 r)k _ , ( r

(

1 + 2 ~ 2 2 ( N ~ ½ ) 2 R2k-,

s[ Jo( X1/2s)e *s2 + 1]-~s(erS2Jo(YVZs)),

, 1+2N2

2(N~-~

k , N --* oo,

s = 2k/Nfixed, (5.29)

which leads to

D(X,y;T):= lim D ( _ I + N--+oo

X

~-~,-1

Y

7"

+ 2N~; 2 ( N + ½)2

)

1

1 f eZrs2[ y,iZjo(Xll2s ) j,

(yllZs) _ x11Zjo(yllZs) j, (X,iZs) ]d s

, I

o

1

~.

+-~e ( Jo(X 1/2) - Jo(y1/2) ).

(5.30)

The scaled parameter dependent n-point distributions are given by (5.17) with S, I and D given by (5.25), (5.28) and (5.30), respectively. In particular, the scaled parameter dependent density at the spectrum edge is given by

PC,) ( X) = S( X, X; ~) = ¼( ( Jo( X '/2) )2 + ( J1 ( X '/2) )2) 0<) g*

+¼(erJo(X 1/2) - 1 ) / s e - * s E j o ( X l / 2 s ) d s .

(5.31)

t /

1

As in the corresponding formula for real symmetric initial conditions (5.18), the term involving the integral does not contribute to the leading large X behaviour, which is thus again given by (5.19). The independence of this asymptotic behaviour on the initial condition and parameter r indicates a type of universality.

References [1] [2] [3] [4] [5] [6] [7]

M.L. Mehta, Random Matrices, 2nd edition (Academic Press, New York, 1990). P. DiFrancesco, P. Ginsparg and J. Zinn-Justin, Phys. Rep. 254 (1995) 1. K. Slevin and T. Nagao, Int. J. Mod. Phys. B 9 (1995) 103. J.J.M. Verbaarschot Nucl. Phys. B (Proc. Suppl.) 53 (1997) 88. C.A. Tracy and H. Widom, Commun. Math. Phys. 177 (1996) 727. R.J. Muirhead, Aspects of Multivariate Statistical Theory, (Wiley, New York, 1982). P.J. Forrester, Random matrices, log-gases and the Calogero-Suthefland model, to appear in Jpn. Math. Soc. Memoirs, Vol. 1. [8] G. Lenz and K. Zyczkowski, J. Phys. A 25 (1992) 5539. [9] EJ. Dyson, J. Math. Phys. 3 (1962) 1191. [101 A.M.S. Mac~do, Phys. Rev. B 53 (1996) 8411. [11] K. Frahm and J.L. Pichard, J. Phys. I France 5 (1995) 877. [12] T. Akuzawa and M. Wadati, J. Phys. Soc. Jpn. 65 (1996) 1583.

762 [13] [14] [151 [16] [17] [18] [19] [20] [21] [221 [23] [24]

T. Nagao, P.J. Forrester/Nuclear Physics B 530 [PM] (1998) 742-762 T. Guhr and T. Wettig, J. Math. Phys. 37 (1996) 6395. A.D. Jackson, M.K. Sener and J.J.M. Verbaarschot, hep-th/9605183. A. Pandey and M.L Mehta, Commun. Math. Phys. 87 (1983) 449. M.L. Mehta and A. Pandey, J. Phys. A 16 (1983) 2655. P.J. Forrester and T. Nagao, J. Stat. Phys. 89 (1997) 69. F.J. Dyson, Commun. Math. Phys. 19 (1970) 235. M.L. Mehta, J. Math. Phys. 17 (1976) 2198. T. Nagao and M. Wadati, J. Phys. Soc. Jpn. 60 (1991) 3298; 61 (1992) 78; 61 (1992) 1910 T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (1995) 401. G. Szeg6, Orthogonal Polynomials (4th edition, American Mathematical Society, 1975). T. Vo-Dai and J.R. Derome, Nuovo Cimento B 30 (1975) 239. T. Nagao and M. Wadati, J. Phys. Soc. Jpn. 62 (1993) 3845.