Logarithmic universality in random matrix theory

Logarithmic universality in random matrix theory

B ELSEVIER Nuclear Physics B 548 (1999) 613-625 www.elsevier.nl/locate/ npe Logarithmic universality in random matrix theory K. Splittorff 1 The Nie...

563KB Sizes 2 Downloads 95 Views

B ELSEVIER

Nuclear Physics B 548 (1999) 613-625 www.elsevier.nl/locate/ npe

Logarithmic universality in random matrix theory K. Splittorff 1 The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark

Received 1 December 1998; accepted 25 January 1999

Abstract Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac operator in QCD. In addition to the well established universality with respect to the choice of potential, we prove that microscopic spectral correlators are unaffected when the matrix in the determinant is replaced by an expansion in powers of the matrix. We refer to this invariance as logarithmic universality. The result is used in proving that a simple random matrix model with Ginsparg-Wilson symmetry has the same microscopic spectral correlators as chiral random matrix theory. (~) 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Aw; ll.15.Tk; ll.30.Rd; 70.10.Fd Keywords: Random matrix theory; Universality; Chiral symmetry; Ginsparg-Wilson relation

1. I n t r o d u c t i o n Random matrix theory ( R M T ) deals with universal spectral properties of ensemble averages o f large random matrices [ 1 ]. This universality is revealed in the microscopic limit. In the microscopic limit we consider the spectral properties of eigenvalues A of matrices o f dimension N in the limit as N --, c~ with x ~ NA fixed [2]. Studies of universality in R M T can be divided into two categories. In one category the models studied have unitary invariance (see, e.g., Ref. [3] ) while in the other this invariance is violated (see, e.g., Ref. [4] ). In this paper we consider the former category. The partition function defining the ensemble, hence, can be expresses in terms o f the eigenvalues of the random matrices rather than in terms o f the matrices. We consider random matrix models 1E-maih [email protected] 0550-3213/99/$ - see frontmatter (~) 1999 Elsevier Science B.V. All rights reserved. PII S0550-3213 (99) 00038-3

K. Splittorff/Nuclear Physics B 548 (1999) 613-625

614

where the partition functions involves a general potential V(') (~). This general potential is a sum of a regular and a logarithmically singular part. This paper introduces and proves logarithmic universality. Logarithmic universality expresses the invariance of microscopic spectral correlators with respect to deformations of the logarithmically singular part of V(~) (A). The logarithmically singular part in the eigenvalue representation originates in the determinant of a matrix D. Two random matrix ensembles will be studied: The unitary ensemble (UE) relevant, e.g., for QCD in three dimensions [5-7] and the chiral unitary ensemble ( x U E ) relevant, e.g., for QCD in four dimensions [2,8,9]. In the applications of RMT to QCD the matrix, D, in the determinant is analogous to the Dirac operator. The microscopic spectral properties of D are therefore of special interest due to the Banks-Casher relation [ 11 ]. For the x U E the logarithmic universality includes deformations which violate both the chiral symmetry condition {D, Ts} = 0 and the hermiticity. As an example we will show that a simple random matrix model which satisfies the Ginsparg-Wiison relation {D, Ys} = DTsD [ 10] shares the microscopic correlators of the xGUE. In the first part of this paper, we introduce and prove logarithmic universality in the unitary ensemble. In order to do this we make use of a method established by Kanzieper and Freilikher [ 12]. In the second part, we introduce logarithmic universality in the chiral unitary ensemble and use the universality of the UE to prove the logarithmic universality of the microscopic spectral correlators of the xUE. Finally, we consider the example with the Ginsparg-Wilson symmetric model.

2. Logarithmic universality in the unitary ensemble The unitary ensemble is defined by the partition function (see e.g. Ref. [ 13] ) ZUE -- / dM det2'~M e -NTrV(M)

( 1)

oo N

"~ / I-i (dai A~'~exp{-NV(ai) }) I,a(a)12,

(2)

--oo i=1

where the Ai are the eigenvalues of the hermitian N x N matrices M and V(M) is an expansion in even powers of M, p V ( M ) = ~ - ~ -g2k M2k , g 2 k G R w i t h g 2 p > 0 . (3) k=l

The Vandermonde determinant, N A(/~) ~

H (•i -- a j ) , j>i=l

(4)

is the Jacobian describing the change to angular coordinates [ 14]. Finally, the measure of the matrix integration is the Haar measure.

K. S p l i t t o r f f / N u c l e a r P h y s i c s B 5 4 8 ( 1 9 9 9 ) 6 1 3 - 6 2 5

615

The eigenvalue correlators of this ensemble have been studied in detail and have been shown [ 3 ] to be independent of the coefficients g2k in the potential V in the microscopic limit. The bulk [ 15] and soft edge [ 12] correlators also enjoy this universality. We now introduce a new set of deformations and show that the microscopic spectral correlators are invariant with respect to these deformations. We shall prove that the microscopic spectral correlators of the partition function

zuE = f dMdet 2" ( f ( M ) ) cx~

exp{-NTrV(M)}

(5)

N

1--[ (dAi[f(ai)12~exp{-Nv(Ai)}) IA(a)12

/

(6)

i=1

--oo

are independent of the coefficients f k C N in the polynomial C2(3

f ( M) = ~

f2k+lM 2k+1

(7)

k=0

with f l 4: O. The expansion defining f is assumed to have suitable convergence properties. We denote this independence as logarithmic universality. In the proof of this assertion, we shall follow, and extend where necessary, the argument of Kanzieper and Freilikher [ 12]. At present, Ref. [ 12] provides the most far-reaching proof of the range of the universality class. Without loss of generality, we can choose f l = l.

2.1. Proof of logarithmic universality in the UE For notational simplicity, we will absorb the f-dependent part of the integrand in the partition function of Eq. (6) into a general potential P

V(~)(a) = V(A) + Kln7 (A) = Z

g 2 k /~ 2k

~-

Ol

9

- ~ log[f(a)]-

(8)

k=l

In order to determine the kernel, KN( A, A~), from which all spectral correlators follow,

p(hl,A2 . . . . . A , ) =

det

I <~a,b<~ s

Ku(ha, ab),

(9)

it is sufficient to find the set of polynomials P,(,~) orthonormal with respect to the measure dlz(a) = da~o(a) - d A e x p { - N V ('~) (,~)} on the real axis OQ

J dtz(A)Pn(A)Pm(A) = ~3.,n. -- 00

Along with the polynomials P. (,~), we will require the functions q~n(A)

(10)

K. Splittorff /Nuclear Physics B 548 (1999) 613-625

616

~Pn( A ) - - P n ( , A ) e x p { - N v ( a ) ( , A ) } ,

(11)

which are by construction orthonormal on the real axis with respect to the measure d,t. The kernel can be expressed in terms of the ~ok as N--I

KN( A, A') = E

¢k( A )¢k( A')

(12)

k=0

=CN

~ N ( at)~PN_l ( a ) -- ~pN(,~.)(PN_I (/~.#) ,~l _

A

,

(13)

where the latter equality is derived using the Christoffel-Darboux formula [ 16]. The three term recursion relation

(14)

a e n - l ( /~) = cnen(,~ ) -- C n _ i e n _ 2 ( a )

determines the coefficients cn of which we have just encountered cN. The aim is to derive a differential equation for the orthonormal functions ~Ok(A) [ 12]. In order to do so, we write the derivative of P,(A) as dPn(A) - - An(A)Pn-I (/l) - Bn(A)Pn(A), dh

(15)

where the functions An(A) and B,(.~) are [12] oo

An(a) =Nc,

~

P2,(t).

(16)

--00

Bn(/~)=Ncn i dit6(~ (dV(~(t) dV(a)(A)~ \ dt da ] P.(t)Pn-l(t).

(17)

--(30

These expressions for An(A) and B,(A) can be obtained by expressing the left-hand side of Eq. (15) through an expansion dPndA (/l) - S dtz(t)---h-7-z_.,Pk(A)Pk(t), "dP,(t) -(x3

(18)

k--0

and then rewriting this integral using partial integration and the Christoffel-Darboux formula. Using Eqs. (16), (17), and the three-term recursion relation (14), one finds the following identity:

B , ( A ) + Bn-l(~-) -+- N dV(a)(A) dA

-

-~-~ An-1 (~-). Cn- 1

(19)

We are now able to derive a differential equation for the orthonormal functions ~'n. Differentiating Eq. (15) with respect to A and making use of the recursion relation (14) and of the defining equation ( l l ) for ~Pn, we obtain [12]

K. Splittorff /Nuclear Physics B 548 (1999) 613-625

dZ~on(A) d,~2

d~on(A) dA

~.(A) -

617

(20)

+ ~ n ( a ) ~ o n ( a ) = 0,

where (21)

.T'n(h) -- d-~ log A.(A) and ~.(a)

-

cn A n ( A ) A # - I ( A ) -

Cn--I

N dV ('~) ( A ) ) 2 Bn(A) + 2 dA

d ( NdV(~_~)(_h)~ + - ~ B.(A) + 2 dA J

dlogAn(A)(Bn(A)+

N dV(~) (A)

dA

2

dA

}

o

(22) It is straightforward to obtain the kernel, provided that we can solve these equations for ~PN and ~oN-i, cf. Eq. (13). In accordance with Re(. [ 12] we choose the following convenient representation of the functions An(A) and Bn(A): (n) A , ( a ) = A~ng) (A) + aAsing(A),

(23)

_D(n) (.~.), Bn ( A) = B(reng) (A) + ~Dsing

(24)

where the regular parts depend only on the polynomial terms in the general potential, Eq. (8), and the singular part depends on the term log[((A)] 2. Since we have not changed the polynomial part, we are led to the same expressions for the regular parts of AN (A) and BN(A) as obtained in [ 12]. As our interest is the microscopic limit, we evaluate the differential equation, Eq. (20), in this limit. In doing so it is essential to keep track of the N-dependence of the various terms. We will assume that the recursion coefficients c~ and the functions A~g~ and B}eng ) approach smooth functions in the large-N limit so that

CN+I =CN + O ( 1 / N ) , A(N+I) _ A(N) 1/N) reg -- "-reg -~- O (

(25) ,

B(N-41) = R(N) + O(1/N) reg --reg

(26) (27)

With these conditions, the function Areg (N) is related to the macroscopic spectral density [ 13] p(A)=

lim ~ 1 A ( N ) ( A ) v / 1 - ( A / a ) z,

N----~oo N~"

reg

(28)

where a -- limN--.oo 2CN can be identified with the endpoints of the spectrum. From this, we infer that A reg (N) (A) is of order N in the limit N --~ cxD,provided that A is sufficiently far from the spectral endpoints.

618

K. Splittorff /Nuclear Physics B 548 (1999) 613-625

Using the assumption of Eq. (27) to evaluate Eq. (19) in the large-N limit, we obtain

a(B(N-I) _ B(N)] 2c~ A N ( / ~ ) -- "2sing sing-"

BN(a) + N - d V 2( dA ~)(a)

(29)

(N)(/~) depend on f ( A ) . This f dependence can The singular parts A(N)( sing , a ) and Bsing carry through to the differential equation (20). We will now prove that the f-dependent terms in Eq. (20) are smaller than the f-independent terms by at least a factor of I/N in the microscopic limit. (N) ) and Bsing (N) (A). The singular part First, we determine the A dependence of Asing(A (N) Asing (A) reads ,.

(N)

asing(A)=--CN

f

d/z(t)

[-,s,ng(t)

~ - A ~k

dt

--00

=--c':~"(t)J

d Vsing (,~)

p2N(t)

dA

dl°g[f(A)]2)

--00

oo

"--2CN f --00

dt~(t)A (f'(t)f(/l)~f(t)f'(A)'~P~/(t ). \ f(t)f(A) ]

(30)

We now introduce the defining expansion of f ( a ) and expand 1/(t - A) in h: OG

(') --2CNfd.(t) l(A1 - f - t - " " ) Asing(A) = A + f3A3 + . . . ~ --OO

x (f'(t) (A + f3 A3 + . . . ) -- f ( t ) ( 1 + 3f3 A2 + . . . ) )

p2(t)

o0

-2C. f d/x(t) {- t I- -~tA + tAf'(t) } f(t-----~+ " " p2N(t)

= A + f3/~3 + . . .

--OO OO

1 1 ff(t___~+...}p2N(t). (t) (31) --2CN f d/x(t) {__.+t

= 1 + f3 a2 + . . .

--(X)

In the final equality we have used the fact that both the weight w(t) of the measure and P~(t) are even in t. PZ(t) is even in t since Pn(t) has parity Pn(-t) = (-1)"Pn(t), as can be shown using the three term recursion relation (14) iteratively. The singular part of BN(A) is evaluated in an almost analogous fashion. The only extra ingredient is that one must use the recursion relation (14) in order to perform the integrations. We find that the term proportional to 1/a is independent of f . In addition to this term, which is identical to the one obtained in [ 12], we obtain terms proportional to positive powers of A. Denoting the sum of the additional terms by TB(N, A), we have B~inNg)(,~) _ 1 -- ( _ _ | ) N

a

+TB(N,A).

(32)

619

K. Splittorff / N u c l e a r Physics B 548 (1999) 613-625

To complete the proof, we determine the differential equation in the microscopic limit. Inserting Bsing (N) (3.) into Eq. (29), we find for the large-N limit of Eq. (19)

N dV(a)(3.) BN(3.) + 2 d3.

--

3. AN(3") 2c~

OL

--

Ol

--(Ts(N-

(--I)N-~

1,3.) - TB(N, 3.)).

2

(33)

Inserting this into the defining equation (22) for G., we can determine its asymptotic behaviour for large N 3.2) ( _ 1) NO/_ if2 A~(3') 1-~-$ + Az +(--I)N--~AN(3. )

GN(3')=A~(3')

-[-AN(3') ( 2 + (--1)NoI )

-

-

or2 --~(TB( N - 1,3.) - TB( N, 3.))2

CN

+ ~a ( T B ( N - I , 3 . ) - T B ( N , o~d -----(Try(N2 d3.

3.)) (C~ AN(a) - 2 ( - i ) Nor -~+ AIN(3")) AN(3")J

1,3.) -TB(N, 3.)).

(34)

In the microscopic limit, two terms in GN(3') are dominant (i.e. of order N2). To see this we consider the N dependence in the microscopic limit of the various terms involved. As noted, Eq. (28) implies that OC N . A(N)(3") reg

(35)

(N)(3'), Eqs. (31) and (32), From the evaluation of the 3.-dependence of A~iNn~(3') and Bsing we have A~inNg ) (3.) o( O( 1),

(36)

B~in~(3') oc ( l -

(37)

(--1)N)N + O ( 1 ) .

Finally, the macroscopic density and, hence, --leg A (N) through Eq. (28) is independent of 3. for 3. ~ 0. We can thus use Eq. (31) to obtain A~v(3.) ~x O(1).

(38)

Keeping only terms of order N 2 in Eq. (20) results in dA~pn(x) dx 2

+

{ (A~(e~)(0))2 (--1)Ha-- o~2 } N2 + x2 ~pn(x) = 0.

(39)

Here we have changed variables to x = 3.N according to the microscopic scaling limit and divided by N 2. The important feature is that there are no references to the expansion coefficients fk in Eq. (7). Making use of Eq. (13), the kernel of the UE follows, cf. Ref. [ 12]. Since every orthonormal functions which enters in this kernel is universal

K. Splittorff/Nuclear PhysicsB 548 (1999) 613-625

620

with respect to deformations in both the potential V and logarithm f we infer that the microscopic correlators obtained from this kernel by Eq. (9) are universal. This completes the proof. The restriction to an odd polynomial in Eq. (7) was necessary because the weight in the measure need to be even and the lowest order term in Eq. (7) has to be linear. Following the lines of the proof given above, it is simple to prove that if one replaces f(Ai) in Ztjz with f([Ai[), then without affecting the microscopic spectral correlators one can introduce even powers in the definition of f . That is, microscopic spectral correlators of the partition function oo

Z

/ --oo

N

1-I (d'~i[f(lail)]Z"exp{-NV(Ai)})la(a)l

(40)

z

i=1

are independent of the coefficients fk E IR of the polynomial co

f(a) =

~--] k a k

(41)

k=l

with f l =/= 0. While this partition function to our knowledge does not have any explicit matrix analogue it will nevertheless prove useful as we turn to the chiral unitary ensemble.

3. Logarithmic universality in the chiral unitary ensemble Chiral unitary random matrix ensembles were originally introduced [2,8,9] to model the consequences of chiral symmetry in the spectrum of the Euclidean Dirac operator of QCD. The structure of the random matrix representation, M, of the Dirac operator was determined by the vanishing anticommutator of M and 3'5. This anticommutation has one particularly simple spectral consequence: The eigenvalues come in pairs of opposite sign. In extending the universality of the chiral ensemble, we shall again replace the matrix, M, in the determinant by an expansion in powers of the matrix. However, we must be somewhat careful because we will need to retain a remnant of chiral symmetry. Here we show that the condition for universality of the microscopic spectral correlators is complex conjugation symmetry of the spectrum and not chiral symmetry as established through the anticommutation with 75. To this end, we will prove the assertion that:The microscopic correlators of the

partition function ZxUE = / dM det'~ ( f ( M ) ) exp{-NTr~'( WWt) },

(42)

where & is an integer, are independent of the coefficients f k c IR of the polynomial oo

f(M) = ~ k=l

fkM k

(43)

K. Splittorff/Nuclear Physics B 548 (1999) 613-625

621

when ft ~ O. By definition the hermitian 2N x 2N matrix M is block off-diagonal: (

0

M =_ iW t

iW) ,

(44)

where W is a complex N x N matrix. The non-singular potential is given by P

~'(WW+) = Z

? (WW+)k'

~k E R with ~,p > 0.

(45)

k=l

Again, it is sufficient to consider f l = 1.

3.1. Proof of logarithmic universality in the xUE The strategy of the proof is first to write the xUE partition function in an eigenvalue representation and then to relate this partition function to that of Eq. (40) introducing appropriate shifts of the arguments. This will allow us to express the kernel of the x U E in terms of the orthonormal functions of the UE and, hence, carry over the logarithmic universality of the UE to the xUE. We rewrite the M integration in the generating functional in angular coordinates W = VAU t, where V and U are unitary matrices and A is a diagonal matrix with real and positive entries Ai, i = 1. . . . . N. This is useful since ZxUE is invariant under the transformation of W --~ U t WV. This invariance is obvious because the transformation of M and hence of f ( M ) is unitary. Under the unitary transformation of M, the measure dM changes to [14] N

dM = U dA] [dV] [dU] [A(A2) [2, i=1

where A(A2)

=

H(A 2

- -

A.~).

(46)

t
The eigenvalues A2 of WWt are related to the eigenvalues of M, which are +i,~. It follows from the reflection symmetry of the spectrum of M and the choice of real coefficients fk that the spectrum of f ( M ) has complex conjugation symmetry. The determinant factor can therefore be written N

det ( f ( M ) ) = H f(iA/)f(iaj)*.

(47)

j=l

Eq. (47) shows that the determinant is a function of the ,~2. Thus, the complex conjugation symmetry of the spectrum of f ( M ) is the remnant of chiral symmetry which we have retained to ensure that the determinant can be written in terms of the eigenvalues of WWt. Due to the cyclic property of the determinant and the trace, the integrations over V and U become trivial, producing irrelevant factors in ZxUE. The partition function now reads

K. Splittorff/Nuclear PhysicsB 548 (1999)613-625

622

oo N

ZxOE~ f H (da21f(/a~ )12aexp{-NfZ(a2) })Izl(a2)l 2 0

(48)

.j= 1 oo

N

f H (dzjlzjllf(izj)12a exp{-NfZ(zf)}) IA(Z2) 12.

(49)

--cx~ j = l

This partition function is not of the same form as that of the unitary ensemble Eq. (6). However, inspired by Ref. [13], we can carry over the universality argument of the preceding section. Given the set of coefficients fk defining the polynomial f , we introduce polynomials Pt(La)(z 2) orthonormal with respect to the measure on the real axis. Notice that it is possible to replace ]A(z 2) 12 by a determinant involving only the even polynomials Pt(f'a) (z2). The idea is

dzlzllf(iz)lZ~exp{-Nf/(z2)}

to identify the polynomials/sy,,~) (z 2) with the polynomials P(2f'~) (z) for the partition function (40). We will now show that this is justified if ce = ti + 1/2 and the expansion coefficients of f in Eq. (41) are related to those of f in Eq. (43). This can be done provided that we identify V(z 2) ~ 2 V ( z ) .

(50)

The proof is then completed by expressing the kernel of the x U E in terms of the orthonormal functions ~ot of the partition function (40). Since these orthonormal functions are universal in the microscopic limit, the kernel is also universal in the microscopic limit. In order to find the set of coefficients {fk} in Eq. (41) such that Iz[If(iz)l 2~ = f(Iz [)2(,~+~/2) is satisfied for a given set {f}, let us first expand the left-hand side of this identity. Using the fact that f l = 1, we find

Izl[f(iz)l 2~

oo 2 oa 2 = 'Z'l ( k~_Of2k+l(--1)kz2k+l ) -I- ( k=~O(--1)k+l.f2k+2z2k+2) }&

k,U=O oo

-1

Z

/

gt

<-l>Z:,*'+':H

kl ,k I ,...,k,~ ,k~=0

= IZ[2~+~) + . - .

4_ ~

~ ,

2

kj+k~)+2,i+)

j=l

(51)

Since only odd powers of ]z] appear, it is appropriate to compare this expansion with that of f(Izl)2<~+l/2) of Eq. (41) where f2k+2 = 0.

K. Splittorff /Nuclear Physics B 548 (1999) 613-625

[f(Izl)12(~+½) -

623

f2k+llZl 2~+1

=

~

f2ko+l I I f 2 k j + l f 2 k j + l l z l 2 ( k ° + ~ ' k i + k : ) + 2 & + l j=l

ko,kl ,ktl,...,ket,k~--'O

= Izl =(~+@) + . . .

(52)

The proof that it is possible to choose the f2k+l for an arbitrary choice of coefficients fk such that the expansions of Eqs. (52) and (51) are identical to all orders can be made iteratively: First, note that the lowest-order terms in the two expansions are identical. Assume that the identity holds for each term up to power 2n + 2a + 1 of Izl. Now, the next term in the x U E expansion yields some coefficient for Iz Iz(n+l)+2'~+l, and this coefficient has to be matched by

E

S~o+,1-[ S~,+lS~+,.

ko+kl+k'l +...+k~+k~=n+l, kiENo

j=l

Since f i = 1, we find that 2~ + 1 of the terms in this summation are f2(n+l)+l. As f2(n+l)+l does not appear in any lower order term, one can choose it to and match the x U E expansion of Eq. (51) and the UE expansion of Eq. (52) to order 2n + 2a + 1 in [zl. This completes the proof by induction. In order to determine the kernel of the xUE, we introduce orthonormal functions (tOt as in Eq. (11)

(o~L~) ( zE) -- lz lll2lf ( iz ) l~ exp {-N ri ( z2) } ,IL~) ( zE).

(53)

Using the relation between the potentials imposed in Eq. (50), we see that ( Z). ¢~S:'<~)(Z E) _- 4o(f(f),~+l12) El

(54)

The kernel is obtained by evaluating N--1

KN(Z 2, wE) = ~ ¢~'~)(z2)¢j'~)(wE).

(55)

I---0

Following the derivation in Section 3 of Ref. [ 13], i.e. using the Christoffel-Draboux formula, we find

KN(Z E, w2)

(f(f),#t+½) (f(f),&+½) . . (f(f),&+½) ~D(f(f),dt+½) . . W~2 N (Z)~2N_ l t W) -- Z~2N_ I ( Z ) 2N tW)

= cEsv

zE _ wE

(56) It is immediately seen that the kernel is universal in the microscopic limit since the orthonormal functions ~Pt are independent of both rv" and f in that limit.

624

K. SplittorffTNuclear Physics B 548 (1999) 613-625

4. Example: The Ginsparg-Wilson relation As an indication of the utility of the results obtained above, we consider the partition function Eq. (42) with the choice D ~ f(M)

= al [ 1 - e a M ]

~ M ~ w . ak , - 1

(57)

k=l

where M is given by Eq. (44) and a is a real number. This choice falls within the scope of the results in Section 3 since the expansion coefficients are real. Hence, the microscopic spectral correlators in that case are identical to those obtained for f ( M ) = M which is used when considering QCD in four dimensions with x U E [2,8,9]. That model contains S U ( ~ ) × SU(ge) chiral symmetry since

{M, ~'5} : O,

(58)

where Y5 = diag( 1,1 . . . . . 1, - l , - 1 . . . . . - I ).

(59)

It is readily seen that D t = ysDT5 and that the eigenvalues of D lie on a circle in the complex plane. Further, using Eq. (58) it is easy to show that {D, 3'5} = a D y s D .

(60)

This anti-commutation relation is known as the Ginsparg-Wilson relation [10] and implies a SU(ge) × SU(&) symmetry of the model [17]. This example shows that genuine chiral symmetry and the Ginsparg-Wilson-Ltischer symmetry lead to identical microscopic correlators. The Ginsparg-Wilson relation is of relevance when constructing lattice actions without fermion doubling, see, e.g., Ref. [18]. Regarding a as the lattice spacing, it is clear that the Ginsparg-Wilson-Ltischer symmetry will reduce to genuine chiral symmetry in the limit a --, 0. The present argument show however that the microscopic spectral correlators will be obtained for all a and that, in this regard, there is nothing special about the a ---, 0 limit. The first study of random matrix correlations in a Ginsparg-Wilson symmetric lattice action have appeared recently [19].

5. Summary In this paper we have extended the universality class of the unitary invariant random matrix ensemble and of the chiral unitarily invariant random matrix ensemble to include logarithmic universality. Logarithmic universality expresses independence with respect to deformations of the logarithmic singular part of the general potential. In the case of the xUE, this means that the original matrix structure in the determinant can be deformed by means of a series expansion in the original block matrix without altering the microscopic spectral behaviour provided that the spectrum associated with this series has a complex

K. Splittorff /Nuclear Physics B 548 (1999) 613-625

625

c o n j u g a t i o n s y m m e t r y . This result w a s u s e d to d e m o n s t r a t e that the matrix a p p e a r i n g in the d e t e r m i n a n t o f the x U E partition f u n c t i o n can be c h o s e n to satisfy the G i n s p a r g W i l s o n r e l a t i o n w i t h o u t altering the m i c r o s c o p i c spectral correlators. T h i s p r o v i d e s an i n d i c a t i o n that lattice g a u g e s i m u l a t i o n s can satisfy chiral r a n d o m matrix statistics for finite lattice s p a c i n g s e v e n w h e n g e n u i n e chiral s y m m e t r y has been sacrificed in the interest o f e l i m i n a t i n g f e r m i o n d o u b l i n g .

Acknowledgements The a u t h o r g r a t e f u l l y a c k n o w l e d g e d i s c u s s i o n s with RH. D a m g a a r d , A. A n d e r s e n , J. C h r i s t i a n s e n and A . D . J a c k s o n .

References [11 M.L. Mehta, Random Matrices, 2nd edition (Academic Press. New York, 1991 ). [21 E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules lbr the Dirac operator in QCD, Nucl. Phys. A 560 ¢1993) 306. 13I G. Akemann, P.H. Damgaard, U. Magnea and S. Nishigaki, Universality of random matrices in the microscopic limit and the Dirac operator spectrum, Nucl. Phys. B 487 IFSI (1997) 721. I41 T. Guhr and T. Wettig, Universal spectral correlations of the Dirac operator at finite temperature, Nucl. Phys. B 506 (1997) 589. ]5] J.J.M. Verbaarschot and I. Zahed, Random matrix theory and three-dimensional QCD, Phys. Rev. Lett 73 (1994) 2288. 161 P.H Damgaard and S.M. Nishigaki, Universal Massive Spectral Correlators and QCD~, Phys. Rev. D 57 (1998) 5299. 171 J. Chrisliansen, Odd-flavored QCD3 and random matrix theory, Nucl. Phys. B 547 (1999) 329. [ 8 I J.J.M. Verbaarschot and 1. Zahed, Spectral deusity of the QCD Dirac operator near zero virtuality. Phys. Rev. Lett. 70 (1993) 3852. l egl J.J.M Verbaarschot, Spectrum of the QCD Dirac operator and chiral random matrix theory, Phys. Rev. Lett. 72 (1994) 2531: Spectral sum rules and Selberg's integral formula, Phys. Lett. B 329 (1994J 351 I()1 PH. Ginsparg and K.G. Wilson, A remnant of chiral symmetry on the lattice+ Phys. Rev. D 25 ( 1982~ 2649. I11 T. Banks and A. Casher, Chiral symmetry breaking in confining theories, Nucl. Phys. B 169 (19801

103. 121 E. Kanzieper and V. Freilikher, Random matrix models with log-singular level confinement: method of lictitious fermions, Philosophical Magazine B 77 (1998) 1161. 131 G. Akemann, P.H. Damgaard, U. Magnea and S.M. Nishigaki, Multicritical nficroscopic spectral correlators of Hermitian and complex matrices, Nucl. Phys. B 519 (1998) 682. 141 T.R. Morris, Chequered surfaces and complex matrices, Nucl. Phys B 356 (1991) 703. 151 E. Br6zin and A. Zee, Universality of the correlations between eigenvalues of large random matrices, Nucl. Phys. B 402 (1993) 613. 161 G. Szeg6, Orthogonal Polynomials (American Mathemetical Society, Providence, 1967). 17] M. Ltischer, Exact chiral symmetry on the lattice and the Ginsparg Wilson relation, Phys. Lelt. B 428 1998) 342. 181 P. Hasenfratz, Prospects for perfec! actions, Nucl. Phys. (Proc. Suppl. I B 63 (1998) 53: H. Neuberger, More about massless quarks on the lattice, Phys. Lett. B 427 (1998) 353; K. Splittorff and A.D. Jackson, The Ginsparg-Wilson relation and local chiral random matrix theory. hep-lat/9805018. 191 F. Farchioni, 1. Hip, C.B. Lang and M. Wohlgenannt. Spectrum of the fixed point Dirac operator in the Schwinger model, hep-lat/9809049.