A Method of Integration over Matrix Variables

Commun. Math. Phys. 79, 327-340 (1981)

Mathematical Physics

© Springer-Verlag 1981

A Method of Integration over Matrix Variables

M. L. Mehta

Commissariat/t l'Energie Atomique, Division de ta Physique, Service de Physique Thdorique, CEN Saclay, F-91190 Gif-sur-Yvette, France

Abstract. The integral over two n x n hermitan matrices

( P " --1"~

Z ( 9 , c ) = ~ d A d B e x p J - t r ] A Z + B 2 - 2 c A B + ~ ( A * + B 4 ) I ~

is evaluated in

L L " ' ~ J

the limit of large n. For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight. The above integral arises in the study of the planar approximation to quantum field theory.

1. Introduction

In their study of planar diagrams some authors [1, 3] have discussed integrals of the form

Z=S~dM~i)exp{-~V(Mti))+~CijtrMti)M(D }

(1.1)

• i < j

V(M)

= tr M 2 + 9_ tr M 4 (1.2)

n

where M (1), M(2), ... are hermitian matrices of order n x n. The integral is taken over all independent real parameters entering the matrix elements,

n

S dM = 7 "" [, 1-[ dMu Y[

d(Re

Mij)d(Im Mij ).

(1.3)

- o o i=1 l < i < j < n

The case of one matrix is the simplest. There are no cross terms containing

Cij.

The integral reduces to that over the eigenvalues [4],

Z(g) = dM exp{ - tr M2 - g- tr M4

= const.~exp

- x z -b [Ix IA(X)l¢I-Idxi,

(1.4)

i = l Yl

0010-3616/81/0079/0327/$02.80

328 M . L . M e h t a

where

A(x) = I-1 (x~- xj) (1.5)

l < i < j < n

and fl = 2. It is now known [1, 2] that

1 1 Z(g) nl~El(g) 0(n-4), (1.6)

n2 n ~(~ = Eo(g) + +

with

eo(g)

= - ½ In a 2 + l ( a Z - 1)(9 - aZ), (1.7)

24

El(g)

= 112 ln(2 - a2), (1.8)

1

a 2 - a2(g) = ~fl-~ { - 1 + w / t + 6flg},fl = 2. (1.9) In stead of hermitian matrices one could have taken matrices which are real symmetric or which are quaternion self-dual. The corresponding integrals reduce again to Eq. (1.4) where the parameter fl is 1 for real symmetric matrices and it is 4 for quaternion self dual matrices. These integrals can again be evaluated in the large n limit, and give the same E0(9) except that fl is now 1 or 4. The correction term E1(9) may be different. The details of this calculation being of no interest are omitted.

The next difficult case of two matrices was discussed by Itzykson and Zuber [3]. They reduced the integral to that over the eigenvalues. However, the expres- sions given by them are too complicated. Below we will reinvestigate this case

Z(g,c)= SdAdBexp~-tr(A 2 +

B 2 ) - 9 - t r ( A 4 + B4)+

2ctrAB~

(1.10)

n )

where A and B are n x n hermitian matrices. We will show that

1"Z(9'c) --! 1 { cx } dx + O(n

-2) (1.1 t) n 2 m ~ ( l - x )

l n f ( x ) - l n 2 ( l _ c Z )

wheref(x) is given by an algebraic equation of the fifth degree

g - 2

and the root to be taken equals

½cx(1 - c2) - 1

when g = 0.

2. The Method of Diffusion Equation Consider the partial differential equation

04 1 D ~2~

0 t - ~ i~xg

^(2.1)

where the constants D i may be unequal for different directions. The unique solution satisfying the initial condition ~(X ; 0 ) = t/(X) is known to be [5]

~(X ;t) = ~

K(X, Y

; r)t/(Y)d

Y,

(2.2)

N o w let A be an n x n hermitian matrix with elements

Ai~.

The A u are real, while the real and imaginary parts of A~j for i < j are denoted by Re A~ and Im A~j respectively. Similarly for the matrix B. Then

~dB =-

satisfies the equation

~(A ;t) = S

K(A, B

;t)t/(B)

dB,

(2.4)

K ( A , B ; t ) = ( 2 ~ t ) - " 2 / 2 e x p { - l t r ( A - B ) 2 }

-- (2rot)-n2/2 exp - 2t ~ (Au - Bu)2 + 2 ~ (Re A~j - Re

Bi) 2

• i < j

i < j

... [. l~ dB. [[ d(Re Bi)d(Im Bi), (2.6)

- - o D i i < j

04

1 2

~?t - 2VA~' (2.7)

V2A = X + )2 4 aIIm A 0 2 '

(2.8)

and the initial condition

~(A ;0) = o(A). (2.9)

As A and B are hermitian, we can choose unitary matrices

U A

and U B such that

A = U A X U A , +

B = U B YUB,

+ (2.10)

where X =

[xibij ]

and Y =

[yi6~j]

are diagonal matrices. The

x i

are the eigenvalues of A and the

Yi

are those of B. Changing the variables from matrix elements to the n eigenvalues and

n ( n -

1) angle parameters on which

U A

and U B depend, we have [4]

n

de = A2(y)dYdY2B, elY = I71 dy,.

(2.11)

i = 1

M. L. Mehta 330

SO that

~(A ;t) = ~(X, ~ ;t)

= ( 2 n t ) - " 2 / 2 ~ e x p { - ~ t r ( X - U + YU)Z}q(Y, OB)Az(Y)dYdO,,

(2.12) where

U = UBU ~

(2.13)

Observe that if t/(B) is independent of £2 B, then ~(A;t) is also independent of f2 A as can be seen by a change of variables from £2 B to f2 (depending on U),

~(X ;t) = const,

t -"~/2

~exp - ~- t r ( X

- U + YU) 2 tI(Y)A2(Y)dYdfL

(2.14) Seperating ~ the Laplacian into parts depending on X and on

UA,

one sees that ~(X ;t) satisfies the (diffusion) e q u a t i o n

34 1 1 ~. c~ (2.16)

c~t - 2 -A-~X) . a i A e ( X ) O ~ { , 0~ - a x i, and has the initial value

Set

~(X ;0) =

tl(X )

(2.17)

F ( x ;t) =

A(X)~(X

;t) T h e n

(2.18)

(2.19)

(2.20)

(2.21) Z a{F = Z {AO~{ + 2(01A ) (a~{) + ¢0~A }

i /

T h u s

F(X

;t) satisfies the (diffusion) e q u a t i o n

~F 1 _ c~2F

and has the initial value

F(X

;0) =

A(X)~(X

; 0 ) =

A(X)~(X).

t The Jacobian of the transformation from matrix elements to the eigenvalues and angle variables for a hermitian matrix is

Az(X)f(OA),

wherefis independent of the x i [4]. Therefore the Laplacian is given by Eq. (2.15) ([5], end of Chap. 1)

Therefore

F(X

;t) is given by [5]

F(X;t)=c°nst't-"/2~exp{ - l n ~ i ~=1 (x~ - yi)2}A(Y)tl(Y)dY.

^(2.22)

choose

t / ( B ) = e x p { - V ( B ) + c t r B

2}=exp

- V ( Y ) +

yi ,

where

V(M)

depends only on the eigenvalues of M. Setting c : ~ in Eqs. (2,4)-(2.6), 1 (2.18) and (2.22) one gets

A(X)~dB

^{exp{ -}

V(B) + c

^tr B 2 - - c tr(A - B ) 2 }

\2cJ ~dY

exp -

V(Y) + c2Yzi

-- CZ(Xil -- yi)2

A(Y).

(2.24) The constant is obtained by choosing

V(B) = c

^tr

B 2

and performing the gaussian integrals on both sides. Therefore

exp { -

V(A) - V(B) + 2c

^tr

AB}dAdB

=const. ~ dXdOAA2(X)exp { - V(X) + c ~x { }

i

• ~dB

^{exp{ -}

V(B) ÷ c

tr B 2 - c tr(A - B) 2}

=const.~dXdYA(X)A(Y)exp{-V(X)-V(Y)+2c~ixyi}.

^(2.25)

This is essentially the result of Itzykson-Zubar [3] expressed in a simpler form. The constant can be fixed by considering

V(A)

= tr A 2 = ~ x 2. The gaussian integral on the left hand side is then straight forward, while that on the right hand side is given in the appendix. As a result the unknown constant is

n,(,- 1)(2c)-(1/2)n(n- l) i! (2.26)

3. Orthogonal Polynomials Revisited

To get the asymptotic behavior of

Z(g, c)/Z(O, c).

where

Z(g'c)=~exp{-tr(A2+B2)-gtr(A4+B4)+2ctrAB} d A d B n

= const.~exp{ -

~(x~+yZ)-~(x~+y~)+ni 2c~xy~}

. I ] ax,dyl,

i

we will use orthogonal polynomials with a non-local weight.

(3.1)

332 M. L. Mehta

Writting

A(X) = I-I (xi - x j)

as the Vandermonde determinant, one sees that

i < j

A(X)

= det [-x{- 1] = det [Pj_ t(xi)] io =1,2 ... ,,, (3.2) where

j - 1

Pj(x) = x j + ~ akx k,

(3.3)

o

is an arbitrary polynomial of degreej with the coefficient o f x J equal to 1. Similarly,

A(Y)

= det [Qj_ l(Yi)]ij= 1,2,...,,,. (3.4) where

Qj(x)

is another set of similar polynomials.

Since

is symmetric in x and y, we will choose Pi(x) =

Qi(x)

and such that

S w(x, y)Pi(x)Pj(y)dxdy = hiaij

(3.6)

- c o

where the Kronecker symbol 6 u is 1 or 0 according as i = j or i (=j. Such a choice is possible. In fact

mo, iI

m l l . . . m l i

Pi(x) = const, detl mlo (3.7)

~l

( i - 1)o m ( i - i ) i "'" m(i-i

X . . . X i

where

m,j-- Sw(x, y),& dxdy, (3.s)

- - ¢ t a

are the moments of

w(x, y).

In particular, since y~ can be expressed as a linear combination of

Pk(Y)

with k < j, one has

~ w(x, y)Pi(x)yJdxdy

= 0, for i > j. (3.9)

- 0 3

With such a choice of Pi(x) we expand the two Vandermonde determinants, multiply and use the orthogonal property (3.6) to integrate various products.

The only terms which contribute have equal indices of the polynomials in x and in y, they contribute the same quantity, and they are n ! in number. Thus

n - - I

Z(g,

c) = coust, t7! [ I hj(,q, c), (3.10) o

and we need to know the asymptotic behaviour of the product of h r. F o r this purpose, we proceed as with the usual orthogonal polynomials.

As w( - x, - y) =

w(x, y), mi;

= 0 for i + j odd, and

P~(x)

has a definite parity,

Pi(- x) = ( - 1)iPi(x)

(3.11)

Let

xPi(x) = Pi+ 1(x) + RiPi-1(x) + SiPi-3(x),

(3.12) where R~ and S~ are certain coefficients. Iterating thrice we get

x3Pi(x) = Pi +

a(x) +

(Ri q- Ri+ l + Ri +

2)PI+ :t(x)

W {Ri(Ri_I-I- Ri-I- Ri+I)-]-(Si-I- Si+I-[- SI+2)}Pi_I(X)-I- ....

(3.13) Thus expressing

xkp~(x)

as linear combinations of

Pj(x),j <= i + k,

and using equa- tion (3, 6) we get

Also integrating on x by parts, the left hand side of the above equation is

11 _~ dPi-dx l(x) Pi(Y)W(X' y)dxdy

= 0, (3.15)

because of equation (3.9). F r o m the last two equations we get

hi{1 +2~gn(R,_ 1 +Ri+R,+l)}=cRihi_ 1.

(3.16) Similarly by integrating

and

Pi- a(x)Pi(Y)(X - cy q- 2g x3 ) w(x' (3.18)

in two different ways, we get the relations

chi=hi_t - ~ + R i +n(Ri_~+Ri+Ri+~) + (Si+Si+~+S~+2) ,

(3.19) and

2 g- h i = cSih i_

3" (3.20) n

4. Asymptotic Evaluation of Z(g, c)

Let us write fi

= hi~hi_l,

so that Eqs. (3.16), (3.19) and (3.20) can be rewritten as

334 M . L . M e h t a

2g } - 1

Ji=cRi I + ~ ( R ~ _ I + R i + R i + I ) ,

(4.1)

c~=-i-+Ri~12 ( + n ~ ( R '

+ 2g(s i + Si+ 1 + Si_z),

(4.2)

n

and

cSi

= ~-f~f~-lf~- 2, 2g (4.3)

For large i and n, the f~, Ri and S i can be replaced by continuous functions.

Thus

f~ ~ n f ( x ) , f~ +_ ~ ~ n f ( x + ~),

R i ~ nR(x), Ri+_ 1 "~ nR(x +_ e),

Si ~ n Z S ( x ) , Si+_ 1 ~ n 2 S ( x +-- 8), (4.4)

i 1 (4.5)

/ I ' n

Making these substitutions, we get to the leading order,

f(x) = cR(x){1 + 6gR(x)}-" 1,

cf(x)

= - ~ + R(x)(1 + X

69R(x) + 6gS(x ), cS(x) = 29f 3(x).

Eliminating

R(x)

and

S(x)

from the last three equations, one gets

(4.6) (4.7) (4.8)

f ( x ) { ( 1 - 6~ f(x))- 2 - cZ } +12g2f3(x) = ½cx.

(4.9)

When g = 0, the value off(x) will be denoted byJo(x). From (4.6) and (4.7)

fo(x) = ½ cx(1 - c2) - 1.

Now from Eq. (3.10) we have

Z(g,c) ,~1 hi(g,c )

n - - - - In

z(o,c)

,=o ~ 7 ' . - 1

~, I n h i = n in h o ^{+ (n - i)} lnf/,

i = 0 i = 1

(n - i) In f/(g, c) = S(1 - x)In

(nf(x))dx + O(n-2 ).

t't i = l 0

(4.10)

(4.11)

(4.12)

(4.13)

Therefore

But

Hence

1 l n Z ( g ' c ) 1 h o ( g , C) 1 . , f(x)

- - - m - - - + t(1 -

x)m77x-~dXjot)

+0(n-2)"

n 2 Z(O,c) n ho(O,c) o (4.14)

- 1-3-g(l n -c2)-2+°(n-2)

(4.I5)

1 h°(g' c) n . . . 0(n- 1). (4.16)

ho(0, c)

Eqs. (4.14), (4.16), (4.9) and (4.10) together give the result announced in the intro- duction, Eqs. (1.11) and (1.12).

5. S o m e Remarks

5.1. Formula (2.25) looks trivial, but it is not. To be honest, we have no shorter way to derive it.

Itzykson and Zuber [31 derive a formula equivalent to (2.25) in a different way as well. They introduce the decomposition of unity into characters of irreduci- ble representations of the unitary group. Using the orthogonality of these charac- ters they can perform the angular integrations. The final result is a series containing eigenvalues of A and B, characters of irreducible representations of the unitary group, their dimensions and the number of times an irreducible representation occurs in various Kronecker powers of the initial matrix. This method can be adapted to deal with real symmetric or quaternion self-dual matrices; one has only to replace the unitary group by the orthogonal or the symplectic group.

The formulas however, do not seem to be simple.

5.2. The same method adapted to evaluate the integral over a chain of matrices Sexp - V(M(i))+2 ~

ciM")M (i+1) I~dM (o

i = 1 i = 1 1

in the limit of large n will be considered elsewhere [71 5.3. An expansion in powers of g gives

f(x)

- 1 - 6#x(1 - c 2 ) - 2 + 392x2(1 --

ca)-¢(c 4 +

8c 2 + 15) + 0(# 3) (5.1)

fo(X)

so that

1

- Eo(g ) = j'(1 - x ) i n f ( x )

o f-~)) dx

= - g(1 - c2) - 2 + 1g2(1 _ c2)--4(c 4 + 8c 2 + 9) + 0(g 3) (5.2)

336 M.L. Mehta 5.4. D e n o t i n g by ( ) the average with respect to any positive measure, one has the inequality [16].

( e r ) => e (v) T a k i n g

o r

and

( . . . ) = ~dAdB exp { - tr(A 2 + B z - 2cAB)} .... (5.3)

( . . . } - f d X d Y A ( X ) A ( Y ) exp - (x~ + y~ - 2cxy~) ... (5.4)

n

F = - 9-tr(A4 + B 4) = - g-~(x~ 4 + y~), (5.5)

n n 1

we get the inequality (see the appendix)

Z( 9, c) _ exp{ - n2 Eo (g) - El (g ) + 0 ( n - 2 ) } z ( 0 , c)

> exp { - 9(1 - c2)-Z(n 2 + ½)} (5.6) T h u s one sees that in agreement with Eq. (5.2),

Eo(g)

= + g(1 - c2) - 2 + o ( f ) ,

E l ( g ) = + 30(1 - 1 cZ) - z + O(g2). (5.7) In general, let D be the p x p matrix [5~ - C~s ] and D k the same matrix with its k th r o w and U h c o l u m n removed. Observe that

~exp - 2 t r A / 2 + 2 2 CistrAiAs d A 1 . . . d A k - l d A k + l . " d A p

i = 1 l <-_i<j<=p

= 2" a k exp( - b k tr A~),

with

a k = (det Dk)-"2/2, akb k- n2/2 = (det D)-n2/2.

A p o w e r series expansion in g gives

tr A/2÷ A -t-2 ~ C i j t r A i A s dA 1...dAp Z ( g ) - ~. exp - i l ~ < j ~ v

= Z(O) 1 - ng~l [dA k tr Ak 4 exp( -- b k tr A2)/~dAg exp( - b k tr Ak 2) + . . .

So that

z ( o ) = _{k = l}

5.5. An obvious upper b o u n d for Z(9)/Z(O) is 1. Another of the form k 1

gk2

^{c a n} be obtained by Schwartz's inequality.

5.6. Note that if V(M) has a term in tr M6(tr M S , . . . ), then xP~(x) in Eq. (3.12) will also have a Pi- 5 (x)(Pi- 7 (x) .... ) present.

5.7. Let us represent the integral in Eq. (1.1) by a graph; the matrices M (° are noted as points and the points i and j are joined by a line if C~j ~ 0. If this graph contains no cycles, the angle variables can be integrated by using Eq. (2.24). The remaining integrations over the eigenvalues, even in the limit of large n, are not simple.

5.8. Examples. F o r a cyclic graph with p points,

Z(g) = ~ 1-IDA i exp - V(Ai) + 2c tr AiAi+ 1 ,

1 i = l i = 1

Ap+ 1 -- A1, with V as in Eq. (1.2).

= =~ 1 - 2 c c o s = thOth ,

Z(g) > Z(0)'exp - (2n 2 + 1)p thOth-~ , where 2c chO = 1.

F o r a p x q square lattice graph with periodic b o u n d a r y conditions,

i

Z(g ) = ~ dA~jexp. - 2 ~, EV(Aij)- 2C~ tr AijA(~ + t)j

i = 1 j = l i = 1 j = l

)

2 C 2 t r AijAi(j+ ~)] ~,

Ai(q+

^{1) ~}

Ail, A(p+ 1)j =- AU,

(D -1) . . . 1

"'" pq

V(A) = T r A 2 + 9_ tr A 4.

n

)1

1 - 2C~ c o s - - - 2C 2 cos 27cj

j=1 P q

~" - - S _{4~ 2} (1 - 2C 1 cos 0 - 2C 2 cos 4))- 1 dOd~

0

= ~ ! { ( 1 - 2C 1 cos 0) 2 - 4 C22} - 1/2d0 for p and q very large. Hence

Z ( g ) >

exp - ~ ( 2 n 2 + 1)pq{1 - 4 ( C 1 - C 2 ) 2 } -1 ( 1 - ~ 2 s i n 2 0 ) - t / 2 d O

z ( 0 ) =

338 M.L. Mehta with ~z = 16C,C2{1 _ 4(C, -

C2)z} -*

F o r a star graph with m branches

Z(9) = ~dAI-[dB ~

exp -

V(A) - 2 V(B~) + 2c

tr

AB~ ,

1 1

where

z(g)

z(o)

^{- - (1 -}

mce)(1/z)"Z(2c-m)(1/2)"("-

1)7[-(1/2)n

(b)

!

\ 1 /

-j" exp { -

V(X) + mcEX 2 }

{det [Fj_~

(xO]/A(X) }~A2(X)dX

~ exp[ - ~(2n2 + l ) ( 1 - mcZ)- 2 { l + m(l + c2 - meZ)2 } l

Fj(x) T z - 1 / 2 ~ e x p { _ ( y _ x ) 2 = ~y4 1 yJdy

A p p e n d i x

Evaluation of the constant in Eq. (2.25).

We will need the

Lemma.

Let F(X) ==- F(xl,..., x ) be a symmetric function of x l , . . . , x and A (X) = l-[ (x i - x j). Then for arbitrary numbers 21~ one has

1 <=i<j<n

y dXA(X)F(X) l-[ (x~ - xj + )@ = ydXA2(X)F(X).

i < j

Proof.

Expand the product l~ (xi - xj + 2i~) in powers of x ~ , . . . , x, and note that

i < j 51 0;2

yx a x 2 ... x~2A(X)F(X)dX = 0

if any two of the % are equal; this is so because if cq = c~j, then integrand is anti- symmetric in the variables xi and x;. Therefore the monomial in x t , . . . , x which will give a non-zero contribution to the integral must have all cq distinct, and its degree is at least

0 + 1 + 2 + ... + n - 1 = ½ n ( n - 1).

This is also the degree of

l ~ ( x i - x j ) = A ( X ) .

Hence terms containing any 2i~

i < j

drop out on integration. End of proof.

To calculate the constant in Eq. (2.25) we may choose

V(A)= tr A z = ~ x~.

i = 1

Then

{n }

5exp

_ ~ ( x 2 + y2i - 2exy,) A(X)A(Y)dXdY

1

= Sexp -- ((1 -- cZ)x 2 + (Yl-- cxi) 2 A ( X ) A ( Y ) d X d Y

= ~exp - 1 - c2)x~ + y A ( X ) d ( Y + c X ) d X d Y

= c"/2~"("-~'Sex p - ( 1 - c 2 ) F x ~ A 2 ( X ) d X [ e - '~dg

1

by the lemma. T h e integration over the y~ is elementary. F o r that over x~ change variables to

x' i -- (1 - c2)1/2xi so that

T h e last integral can be evaluated 2 by introducing H e r m i t e polynomials which are o r t h o g o n a l for the gaussian weight. T h e final result is

Sexp{ - X(x~ + y2 _ 2 c x , y , ) } A ( X ) A ( Y ) d X d Y n--i

= c(1/2)"(n- 1)n(1/2)"( 1 ^-

¢2) -(1/2)n2t'1! H

( ~rl/22-/i [)

0 n

= 7"cn(1c)(1/2)n(n- ^{1 ) ( 1 --} C2)--(1/2)n2 H i!

1 F o r Eq. (5.6) we need to evaluate

S 2 ( x ~ + y ~ ) e x p - 2 ( x 2 + y~ - 2xiy i) A ( X ) A ( Y ) d X d Y ,

1 1

= 2 ~ l x ~ e x p - ((1 - c2)x~ + y ~ ) A ( X ) A ( c X ) d X d Y

-- 2nc (1/2)n("-1)(1 -c2)-~1/2)(n2+4)~ ~/2~exp - x x~ A 2 ( X ) d X as above. O n c e m o r e introducing H e r m i t e polynomials, the last integral is seen to be /

S

^{e x -- X X}

\ 1 /

n-1 n-1 ~ x,~ H~ (x)e-X~dx

= ( n - 1)[ y [ ( 2 - ' i ! r d / 2 ) ~ iH2(x)e_X2dx

0 i = 0

2 [4], Chap. 6

340 M.L. Mehta

F i n a l l y , f r o m t h e t h r e e t e r m r e c u r r e n c e r e l a t i o n a n d o r t h o g o n a l i t y o n e g e t s

Sx4H2(x)e-X2dx

= ¼(2i 2 + 2i +

1)SH~(x)e-X~dx.

P u t t i n g e v e r y t h i n g t o g e t h e r o n e sees t h a t

I2(x

+ y ~ ) e x p

- (x~ + y2~ _ 2cxy~) A ( X ) A ( Y ) d X d Y

1

= ( 1 -

e2)-2(,¢ + ½n)fex p

- Y~(x~ + y2~ _ 2x~y~) A ( X ) z l ( Y ) d X d Y ,

1

i m p l y i n g E q . (5.6)

Acknowledgements.

I am thankful to my colleagues, to C, Itzykson who got me interested in this problem, to J, B. Zuber who explained me with great patience their article and to M, Gaudin, J. des Cloizeaux, A, Gervois, G. Mahoux and J.M. Normand with whom I had many helpful discussions.

References

1. Brezin, E., Itzykson, C., Parisi, G., Zuber, J. B. : Commun. Math. Phys. 59; 35-51 (1978) 2. Bessis, D. : A new method in the combinatoric of the topological expansion, Commun. Math. Phys.

69, 147-163 (1979)

3. Itzykson, C., guber, J. B. :The planar approximation (II), J. Math. Phys. 21, 411 421 (t980) 4. Mehta, M. L, : Random matrices. Chap. 3. New York : Academic Press 1967

5. Morse, P.M. : Feshbach, H. : Methods of mathematical physics. Chap. 2.4. New York: McGraw- Hill 1953

6. Hardy, G. H., Littlewood, J. E., Polya, G. :Inequalities. p. 138. Cambridge: University Press 1964 7. Chadha S., Mahoux G., Mehta M. L.: A Method of integration over Matrix variables. II. J. Phys.

A (in press)

8. Bessis D., Itzykson C., Zuber J. B., Adv. Appl. Math. 1, 109-157 (1980) Communicated by E. Br~zin

Received October I, 1979