Chaotic behaviour in one-matrix models

Volume 26 l, number 3

PHYSICS LETTERS B

30 May 1991

Chaotic behaviour in one-matrix models J. J u r k i e w i c z

l

Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Received 28 February 1991

We study numerically the recurrence relations in the orthogonal polynomial method for a one-matrix model with a six-order even potential. We show how the phase structure derived in the saddle point approximation is reflected in this method. We find a new type of solution responsiblefor the phase transitions observed in the saddle point approach.

1. Introduction In the last year matrix models were extensively studied in the attempt to u n d e r s t a n d the critical properties of the two-dimensional gravity (coupled or not coupled to the matter fields). We feel unable to give at this point even an approximate list of important contributions. The most interesting information comes in these models from the neighbourhood of their critical points, where we expect to have a possibility of defining a continuous theory. An important and as yet not fully resolved problem is therefore to understand the phase structure of the matrix models. In our earlier paper ref. [ 1 ] we studied the phase of the one-matrix model

Z= f ~Mexp[-NU(M)]

die point approach the system described above was shown to be in one of three possible phases. These phases, denoted P~, P2 and P3 are characterized by spectral density functions having support on one, two or three arcs. The phase diagram obtained in ref. [ 1 ] is shown in fig. 1. The critical line (a) corresponding to the m = 2 multicritical one-matrix theory (or a standard c o n t i n u u m limit of the one-matrix model as

.18 / / / / / / / / / / / / /

.12

(1)

in the saddle point approximation. In ( 1 ) the integral is over the space of N × N matrices M and U ( M ) = t r ( ½M2+ ¼GhM4+lh2M 6) .

(2)

In (2) we use the parametrization from refi [ 1 ]. With this parametrization, integral ( 1 ) is well defined for all values of G. In ref. [ 1 ] it was argued that the sixorder term cannot be used as a regulator of a fourthorder theory, since the resulting theory has no standard m = 2 multicritical c o n t i n u u m limit. In the sad-

U

.06

k /

/

-4.0

-2.0

-3.0

-I.O

G Permanent address: Institute of Physics, Jagellonian University, ul. Reymonta 4, PL-30059 Cracow 16, Poland. 260

Fig. 1. The phase structure in the {G, h} plane. The region G< Gc is shaded. The measured points are on the dashed line G= -2.

0370-2693/91/$ 03.50 © 1991 - ElsevierSciencePublishers B.V. ( North-Holland )

Volume 261, number 3

PHYSICS LETTERS B

a boundary line of the Pl phase) cannot be reached, because the system undergoes first a phase transition to the P3 phase. The only exception is the m = 3 multicritical point G=Gc= - x f ~ , h = 1/x/270 [21. The saddle point method has a big disadvantage: it is very difficult to obtain the subleading terms in the 1 / N expansion. In this respect the alternative, orthogonal polynomial method was found to be much more successful, at least to describe the one-arc solution PI- It would therefore be very interesting to understand how the phenomenon of a phase transition described above looks in the language of this method. In ref. [3] it was argued that solution P~ can be recovered in this approach if we make an ansatz about the largeN b e h a v i o u r of the recurrence coefficients rn, namely that of a unique scaling function r(x) with x = n/N, satisfying for potential (2)

hx=w(r) =r+3Gr2+ lOr 3 .

(3)

Similarly in refs. [ 1,4 ] the two-arc solution P2 was shown to correspond to a two-band ansatz for the large-N behaviour of r~. According to this ansatz there are two different scaling functions r(x) and s(x) for even and odd n respectively. As was shown in ref. [ 5 ] the two-band ansatz holds only in the limited x region, for 0 < x < ) e , where )e is an explicit function of the coupling constants and 2 = 1 gives the existence domain o f the P2 phase. In ref. [5] the possibility of realization of solutions with more bands was also discussed. As was already observed in ref. [ 1 ] none of these correspond to a three-arc solution P3. The multiband solutions were also studied in the unitary-matrix models [6]. In this paper we show that even in the one-arc domain there exists a range of parameters {G, h}, where eq. (3) has no unique solution, at least for some values of x. This indicates that even in this simple case a standard scaling assumption cannot be satisfied for all x and a more complicated behaviour should be expected. Closer inspection ofeq. (3) shows that this happens for G < Gc = - x / ~ (see fig. 1 ). We study numerically the exact recurrence relations for rn, for large but finite N, in the range of parameters corresponding to the transition from Pj to P3 and back to PI for G = - 2 and varying h. We discover a solution of a new type, with partly regular and partly chaotic behaviour, which is responsible for this transition. This solution approaches a smooth one-band solution both

30 May 1991

for small and large x. In the intermediate region it has a pseudo-oscillatory, chaotic behaviour which cannot be interpreted as a solution with any integer number of bands. No simple scaling assumption seems to apply for this solution. The critical behaviour of the one-matrix model in the neighbourhood o f the m - - 3 critical point was investigated in ref. [ 7 ], where the flow from the m = 3 to the m = 2 critical behaviour was studied numerically. The conclusion was that there is a nonperturbative instability in this definition of nonperturbative pure gravity. The present study shows the origin of this instability: the nonperturbative effects shield completely a system from the m = 2 critical domain. Instead we have a new type o f solution, which surprisingly resembles the oscillatory behaviour shown in ref. [7]. The important difference is that in our case there seems to be no simple scaling function u(x), the frequence of oscillations being an increasing function of N. The paper is organized as follows: In section 2 we recall the saddle point solution ofref. [ 1 ], in section 3 we describe the recurrence relations in the orthogonal polynomial method and discuss the numerical method used to find their solution. The solutions are presented in section 4 and in section 5 we discuss their qualitative behaviour.

2. Saddle point solution

The integration over the angular variables of ( 1 ) can easily be performed giving Zoc

~ d~i I-[ ( 2 i - 2 a ) 2 exp

- N i ~= U(2i)

i
1

= e x p ( - N2..~) ,

(4)

where 2j, j = 1, ..., N, are the eigenvalues and U(j.)___l~2_+_ aGh2 | 4 +gh i 22 6 .

(5)

Introducing the spectral density function p(2), 1 x

p ( 2 ) = ~ Z1: d ( 2 - 2 , ) ,

(6)

!

we solve a maximization problem for 261

PHYSICS LETTERS B

r F = J d2 d/tp(X)p(it) log12-/~]

30 May 1991

or

- ~ d2 (p(2)U(2) +7 1 d2p().) ,

(7)

with respect to a positive function p(2), where 2 is a Lagrange multiplier, a "chemical potential" introduced to enforce the normalization condition [ 8 ] f d~p(2)=l .

(8)

Performing variation of (7) with respect to p(2) we find that either p ( 2 ) = 0 or it satisfies the integral equation 2 ; d#p(/t) log[2-/~[ - U(2) + 7 = 0 ,

(9)

L

where the integral is over one or more arcs L, where p (2) # 0. Differentiating (9) with respect to 2 we obtain the standard form of this equation [3]

h>h, = ~ o [ G 3 - 5 G + 3 x / 6

(G2+~)3/21 .

(15)

For h = h,, wl (y) has a zero at y2= 4a 2. This is a standard m = 2 continuum limit critical line. For h=hr, w~(y) has a double zero for 0 < y 2 < 4 a 2. This line corresponds to a phase transition to the P3 phase. Curves h+(g) and h,(G) are plotted in fig. 1 and are denoted respectively by a and b. Phase P2 has a support on two arcs 2e [-2A1, -2A2] and 2e[2A2, 2A~]. Introducing again the rescaled variable y=X,,/h and denoting a=Aj,~h, b=A2x/h we obtain the rescaled spectral function u2(y) =p2(A)/x/~ in the form

u2(y) = ~1 ~ _ y 2 ) ( y 2 _ 4 b ~ w 2 ( y

),

where

w2(y)= [ ½ G + ( a 2 + b 2)]y+½y3. P ; dlt p(/~) + ½U' (2) = 0

(10)

L

For given values of the couplings in U(2) we have to findp(2) and its support L. As we already mentioned the potential discussed in this paper admits a three-phase structure with one to three arcs. We shall denote these phases by P~, P2 and P3. Phase PI is characterized by p1(2) nonvanishing on one arc 2e [ --2Ab 2A~ ]. Introducing the rescaled variable y = 2x/k and denoting a =A i x/h we obtain the rescaled spectral density u~(x)=p~(2)/x/h in the form U, ( y ) = ~

1

4 x / 4 a 2 - - y 2 iv, ( y ) ,

(

11 )

where

w~(y)=(}+Ga2+3a4)+(½G+a2)y2+~y 4

(12)

(16)

(17)

The normalization condition reads

h= [G+4(a2+b 2)](a2-b2) 2

(18)

and we have a self-consistency constraint

½+G(a2+b2)+2(a2+bZ)2+(a2-b2)2=O.

(19)

As before the solution exists if w2(y) is positive for 2a] o r d u e to (17) if w2(2b)>0. Simple algebra shows that the existence domain is

ye[2b, G<-2,

0 < h ~ < ~ , f 3 ( G 2 - 4 ) 3/2.

(20)

The boundary (20) is represented by curve c in fig. 1. Phase P3 has a spectral density function P3 ()~) nonvanishing on three arcs: 2~ [ -A~, - 2A2 ], 2E [ - 2A3, 2A3] and 2e [2A2, 2A~ ]. As before we introduce rescaled quantities a=Al,,/h, b=A2xfh, c=A3,~h and y = a x / h and the rescaled spectral function u3(y)= p3(2)/x/h. It has the form

with the normalization condition u3(Y)

h = a 2 + 3 G a 4 + 10a 6 .

(13)

[Note that u~ (y) is normalized to unity.] The existence domain of the phase P~ is obtained from the requirement that the spectral density function is positive for 0 < y 2 < 4a2. This is equivalent to 1 3 O
262

2

- T10 )

3/2

]

(14)

_ 1 xf(4a2_yS)(y2_4b2)(y2_4c2) ~rh

w3(y ) ,

(21) where w3(y)= [½G+

(a2+bZ+c2)]+½y 2.

(22)

Volume 261, number 3

PHYSICS LETTERS B

In (21) we have chosen the square root to be real positive on the right arc. It is therefore negative on the central arc and again positive on the left one. Function W z ( y ) must have the same property if the spectral density is to be positive. To obtain parameters a, b and c we have the normalization constraint

h= [G+4(a2+b2 +c 2)]

30 May 1991

the standard notation of ref. [9 ] we consider polynomials ; ~ ( y ) =yn+..., orthogonal with respect to the measure d/~(y) = d y e x p [ - (U/h)(~y2+ ]Gy4+~y6)]. (26) Normalization conditions are H=6nm = f d/z(y) ~,,(y)~,,(y).

(27)

× (a4+b4+c4-2a262-2a2c2-262c 2) + 16a2b2c2 .

(23)

Y'~n (Y) = "~J~n+1 (Y) "31-rn 'C~n--I (Y) ,

We have also the self-consistency constraint

(24)

r==H=/H=_l,

ro=0.

(29)

In terms of rn's the free energy .~ (4) can be expressed as

Finally requirement 2b

f dy u3(y) = 0

(28)

where

½+G(a2+b2+c 2) +3(a4+b4+c 4) + 2( a2b2 + a2c2 + b2c 2)=0.

The polynomials .:~n(Y) satisfy the recurrence relation

(25)

1 (~

(N-n)logrn+NlogHo

2C

guarantees the equality of the "chemical potential" y on various arcs (cf. ref. [ 1 ] ). The transition P2~P~ can be realized in two ways: either c-~b, in which case we obtain the P~ spectral density function with support ye [ - 2 a , 2a] and with wt (y) having an extra zero for y2 = 4b 2, or b~a, when the P~ support is ye [ - 2 c , 2c] and w~ (y) has a zero for y 2 = 4 a 2 [and a second one for 4c2
3. Orthogonal polynomial method The most powerful method used to study the properties of the matrix models is the orthogonal polynomial method, which in principle permits a direct calculation of Z for arbitrary N. We introduce as in section 2 the rescaled variable y = 2 x / h and following

~ N 2 log

h+const.).

(30)

Following ref. [ 3 ] we obtain the recurrence relations satisfied by rn:

nh =rn[ l +G(r,,_l +r=+rn+l)+r,,+lr~_l N k-rn_l (r=_2 +r=_~ +r~) +r,(rn_~ +r, +r~+~) +r=+~(r=+r=+, +r,,+2) ]

.

(31)

In principle these relations are sufficient to determine all coefficients rn from the first two ( r o = 0 and rl = {y2}u). In practice, however, it becomes impossible, at least for large n, since recurrence (31 ) requires an exponentially accurate value for r~. To obtain coefficient rk we must know in rj at least a number of digits proportional to k. Therefore, the standard method to proceed is to make an ansatz about the large-N behaviour of coefficients r=. The standard one is a single-band scaling assumption rn-~r(n/ N)=r(x), x=n/N. Neglecting terms O ( 1 / N 2 ) , rn can be determined from the equation

hx.=w(r) =r+3Gr2+ 10r 3 ,

(32)

where the solution r(x) should satisfy r ( 0 ) = 0 . Neglected terms become important near the critical 263

Volume 261, number 3

PHYSICS LETTERS B

point, where w' ( r ) = 0 , giving rise to the so-called double scaling limit. It is important to observe at this point the wellknown relation between eq. (32) for x = 1 and eq. (13) for the arc length in the P~ phase:

30 May 199l

For the P2 phase it was shown in refs. [ 1,4,5] that it can be reproduced with a two-band ansatz. We require that even and odd rn scale differently, i.e., r,,-,r(x), n even and rn-~s(x), n odd with r(x) ~ s ( x ) . Functions r(x) and s ( x ) satisfy

(33)

hx=r[1 + G ( r + 2 s ) + 3s2+6sr+r 2] ,

which suggests that the one-band ansatz reproduces the P, phase. In fact a closer inspection of eq. (32) shows that this is true only if G > Gc= - x / ~ , where w(r) is a monotonic function ofr. For G < Gc we have the situation shown in fig. 2 (for G = - 2 ) . Function w(r) has in this case a nontrivial structure. We have plotted also values of h corresponding to the phase transition points denoted (b), (a) and (d) in accordance with fig. 1 ). From this plot we see that even in the P, phase we have a range in x, where eq. (32) has no unique real solution. For the P, case this range happens to be for 0 < x < 1. If we demand that we choose for r(x) a solution of (32), which is real for small x it will become complex for x large. This cannot happen because by definition all rn must be real [cf. ref. ( 2 9 ) ] . We must conclude that for G
h x = s [ l + G ( s + 2r)+ 3rZ+6sr+s 2] .

a2=r(1),

.[0

.....................................

hb

,05

0

0.0

I

0.2 r

0.4

Fig. 2. Function w(r) versus r for G= - 2 . The values hb, ha and h~ correspond to the phase transition lines (b), (a) and (d), respectively.

264

(34)

As we already mentioned this ansatz can hold only for 0 < x < 2, where X~3 ( G 2 - 4 )

3/2

(35)

~ = 36h The structure of the solution for x > Y will be discussed elsewhere. In ref. [ 5] other multi-band solutions were constructed, but as was already observed in ref. [ 1 ] none of these can be interpreted as a P3 solution or the P, solution for G < - x f ~ . To understand this point better we decided to perform a numerical study of the recurrence relations in the interesting region of the parameters. We recall again that following relation (33) the size of the arc in the P, is simply r( 1 ). From this we guess that, at least in the P~ phase, eq. (32) should give the large-n (or large-x) behaviour of the recurrence coefficients rn. This is the starting point of our study. We look for a solution of the exact recurrence relations for large, but finite N (typically 50-400) which has an asymptotic behaviour for large n given by a solution of (32). The solution has to satisfy ro=0. We perform an iteration of the recurrence relation (31 ) in which a new value of rn is computed from the old values o f r,,_2, r,~ h rn+l and rn+ 2. The starting configuration is a set of rn's obtained by solving eq. ( 32 ) for values of x, where it has a unique solution and a simple interpolation in the intermediate region. We have taken a system big enough (up to n = 2 0 0 0 for N = 50 ) to be sure that 1/N 2 corrections can be safely neglected for the endpoints. We have convinced ourselves that this simple numerical procedure indeed leads to a convergent process (in all the cases presented here) and that the accuracy is a function of time. In all our calculations the final error for (32) was below 10 -~° for all n. We have checked that a solution of this kind exists for all values of h and that its form is responsible for a transition PL ~P3 and then back to P, when we move along the G = - 2 line. The

Volume 261, number 3

PHYSICS LETTERS B

solutions will be discussed in the next section. To compare the numerical solution obtained this way with the predictions of the saddle point approximation we compute the (finite-N) spectral density function p(y) (cf. ref. [3] ). It can be expressed in terms of the polynomials -~n(Y) as

30 May 1991

0.8

0.4

xb 0

P(Y)= U.=o H,,

Xexp[-(N/h)(½y2+ ~4Gy4+~y6)]

.

0.8

(36)

The polynomials ,~n (Y) can be iteratively computed using numerically obtained values ofr,, eq. (28) and P0(Y) = 1. Using numerical values ofrn we cannot get the normalization of p(y). This is also computed numerically.

04 x~ v

o o8

4. Numerical results

As we mentioned earlier our numerical study was made for the points lying on the G = - 2 line for various h (see fig. 1). The first example is the point h = 0 . 5 . From fig. 1 we see that it is well in the P~ phase, but rather far from the phase transition line (b). Figs. 3a-3d show the distribution ofrn for N = 50, 100, 200, 400 respectively. On all these figures we plot with a dotted line points satisfying eq. (32). Numerical points are joined with a solid line. This example shows a characteristic behaviour o f the solution. For large x it approaches the dotted line [solution of ( 3 2 ) ] . The same property can be seen for small x. In the intermediate region, near to the x values, where (32) has three real solutions, coefficients rn show a chaotic, pseudo-oscillatory behaviour. The period of oscillations does not depend on N on the n scale, so they become more dense on the plot r,, versus x. This might indicate that the solution presented above can be viewed as a combination of the one-band and multi-band solutions. The analysis of the pseudo-oscillatory domain shows, however, that it cannot be interpreted as a multi-band solution with any integer number o f bands. We were not able to guess an ansatz that could reproduce this behaviour, although as will be discussed in the next section some qualitative aspects can be understood. In figs. 3a-3d we have also marked values o f x corresponding to xa = ha/h and x b : hb/h, where hd and hb are the

0.4 /

o

o.s

~

:

o

i

>:

I

/,

Q2

0.4 x

Fig. 3. Coefficients rn versus x for G = - 2 ,

h=0.5.

The saddle

point solution is marked with the dotted line. (a) N=50, (b) N= 100, (c) N=200, (d) N=400.

critical values of h for G = - 2 (cf. fig. 2 ). We observe that the onset of the oscillatory behaviour coincides with the point xd. As we shall argue in the next section we expect that for large Nalso the point Xb marks a transition to the smooth behaviour. Fig. 4 shows the spectral density function p(y) calculated with the help of (36) respectively for N = 50 (dashed line) and N = 4 0 0 (solid line) together with the saddle point solution ( N = ~ ) marked with the dotted line. We note that already for N-- 50 the saddle point solution is a very good approximation. As we observed earlier one can except that the same type of a solution for rn will appear for all values o f h 265

Volume 261, number 3

PHYSICS LETTERS B

30 May 1991

0.8

0.4 4 q

0.5

Xb

.0 0.8

q._

04

:f"l

xt; 0 -.<

v

-I.8

-I.2

-0.6

0.0

0.6

08

1.2

y Fig. 4. Spectral density function p(y) for G= - 2 , h=0.5 and N=50 dashed line), N=400 (solid line) and N=oo (dotted

line). in this region. We find that this is indeed the case. Next examples follow the line G = - 2 toward smaller h. In fig. 5a we show the case h = 0.15 and N = 50, 100. This point corresponds also to the P~ phase but much closer to the phase transition line (b) (see fig. 1 ). As before we mark the values Xb and xd to show the onset of the oscillatory phase. In fig. 6a we show the spectral density functions for N---100 together with the saddle point solution. We see that again the agreement is very good. Fig. 5b shows the N = 50 and N = 100 solutions for h = hb= 0.8 and fig. 6b the spectral density function for N = 100. For this case the oscillatory behaviour extends to x = 1. The spectral density function develops a zero for a finite value of y. In fact following (36) we see that for finite N, p(y) can never be zero (or negative) and therefore its value will only be zero up to non-perturbative, exponentially small corrections. Fig. 5c shows the r,'s for h = 0 . 0 6 and N = 5 0 , 100. In fig. 6c we show the spectral density function for N = 100. This point is in the P3 phase in fig. 1 and indeed the spectral density function becomes exponentially small where the saddle point solution is zero. We have plotted the saddle point spectral density function satisfying ( 2 2 ) - ( 25 ) and we see that again 266

04

0 08

04

o 0

0.4

0.8

1.2

1.6

x

Fig. 5. Coefficients r,, versus x for G = - 2 , N = 50 (dashed line) a n d N = 100 (solidline). (a) h=0.15, (b) h=0.08, (c) h=0.06, ( d ) h=O.05.

the agreement is remarkable. From fig. 1 we see that phase P3 corresponds to the situation when we have the oscillatory behaviour of rN for x close to one. Figs. 5d and 6d show the r,'s for N = 50, 100 and p(y) for N = 100 and h = 0 . 0 5 . In fig. 1 we see that this is the point where in the saddle point approximation we have both the Ps and P~ solutions. As was argued in ref. [ 1 ] the P3 solution is the dominant one, as can be easily seen from fig. 6d. For h values below the h = hd the oscillatory region shifts to values x > 1 and consequently we have again the P ~solution. We do not show here the corresponding measurements, which fully confirm this behaviour.

Volume 261, number 3

PHYSICS LETTERS B

30 May 1991

Let us observe that an asymptotic large-x behaviour o f this solution implies that fluctuating corrections should be expressed. To see it better let us assume that for large n the recurrence coefficients r,, can be p a r a m e t r i z e d as

~a ) I0

r,=R(x)+G,

/b)

where R(x)=r(x)+(finite-N corrections) is a smooth function o f x satisfying ( 3 2 ) for large x and G is a small correction. Inserting (37) to (31) and neglecting terms O (ez) we get

i

I0

Z

(37)

A _

(I+4GR+16R2)en+(GR+6R2)(e, ~'R

2 (tn

2 -~ffn+2)--~-0

l +G+~)

.

(38)

In (38) R=R(x), but i f N is big enough we can assume that it is a constant a p p r o x i m a t e l y equal to r= r(x) (x in this a p p r o x i m a t i o n is constant on the n scale). With this simplification we can look for solutions o f (38) in the form 4

en = ~ Ado,~,

(39)

i=1

- 1.0

00

I.O

Fig. 6. Spectral density function p(y) for G= - Z and N= 100 (solid line). (a) h=0.15, (b) h=0.08, (c) h=0.06, (d) h=0.05.

5. Discussion The numerical results presented above indicate the existence o f a new type o f solution o f the exact recurrence relations (31 ) which combines the smooth oneb a n d b e h a v i o u r with simple large-N scaling, valid either for small or large x, and the chaotic pseudooscillatory behaviour, interpolating between the two smooth sectors. As we m e n t i o n e d before, the period o f oscillations is roughly i n d e p e n d e n t o f N on the n scale and cannot be interpreted as an onset o f the multi-band solution with an integer n u m b e r o f bands. It would be very interesting to find an ansatz that could reproduce this behaviour. We shall return to this p r o b l e m in a future publication. Here we shall only discuss some qualitative aspects o f the new solution.

where ~cisatisfy the fourth-order symmetric equation. The solutions o f this equation will in general be complex. Since for large n we expect e,, to become negligible, coefficients A, corresponding to I x,I > 1 should better be zero. On the other hand if I ~cil < 1 its contribution to e, will be exponentially (in N) d u m p e d on the x scale. Therefore only solutions for which I K,I = 1 will be i m p o r t a n t in the large-N limit. Solutions o f the fourth-order equation can be easily obtained. Introducing a new variable

c=½(x+l/x)

(40)

we reduce the p r o b l e m to solving a quadratic equation for c:

4rZcZ+2r(G+6r)c+(l+4Gr+14rZ)=O.

(41)

We can now easily analyze the solutions for a:: they come in two complex conjugate pairs with modulus one i f e q . (41) has a real solution - l ~ c ~ < l . This happens ifr~< ro where

rc= - ~ ( G - x/6 x[GZZ~) .

(42)

If r(x) is a solution of ( 3 2 ) this value corresponds tO X=Xb,

267

Volume 261, number 3

hxb = h ~ ( G ) ,

PHYSICS LETTERS B

(43)

where h~(G) is given by eq. (15) and is a point on line (b) corresponding to the phase transition between the P~ and P3 phases (fig. 1 ). This analysis confirms the observation made in the previous section that indeed the chaotic pseudooscillatory solution is responsible for the phase structure of fig. 1. We are in phase P3 when the point x = 1 is in the chaotic sector and in phase P~ when it is in a smooth one. The behaviour of the coefficients r, presented in figs. 3a-3d clearly indicates that for x > Xb we see exponentially decaying solutions for ~,,, which become smaller with growing N. This is a typically nonperturbative behaviour [like e x p ( - c N ) ]. For X
268

30 May 1991

Note added

When this article was sent for publication we became aware o f a preprint by Sasaki and Suzuki [ 11 ], where a similar analysis was performed.

References [ 1 ] J. Jurkiewicz, Phys. Lett. B 245 (1990) 178. [2] E. Br6zin and V.A. Kazakov, Phys. Lett. B 144 (1990) 411. [ 3 ] E. Br6zin, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59 (1978) 35. [4] G.M. Cicuta, L, Molinari and E. Montaldi, J. Phys. A 23 (1990) L421; Nucl. Phys. B 300 (1988) 197; L. Molinari, J. Phys, A 21 (1988) 1. [5] K. Demetrfi, N. Deo, S. Jain and C.I. Tan, Brown preprint BROWN-HET-764 (July 1990)~ O. Lichtenfeld, R. Ray and A. Ray, Princeton preprint IASSNS-HEP-90/69 and CCNY-HEP/90/18 (September 1990). [6] K. Demetrfi and C.I. Tan, Brown preprint BROWN-HET769 (September 1990); L. Houart and M. Picco, Universit6 Paris VI preprint, PARLPTHE 90/35 and ULB-TH 90/04 (July 1990). [7] M.R. Douglas, N. Seiberg and S.H. Sbenker, Flow and instability in quantum gravity, Rutgers preprint RU-90-19 (April 1990). [8] J. Jurkiewicz and K. Zalewski, Nucl. Phys. B 220 [FS8] (1983) 167. [9] D. Bessis, C. Itzykson and J.B. Zuber, Adv. Appl. Math. 1 (1980) 109; C. ltzykson and J.B. Zuber, J. Math. Phys. 21 (1980) 411. [ 10] O. Lichtenfeld, On eigenvalue tunneling in matrix models, Princeton preprint IASSNS-HEP-91/2 (January 1991 ). [ 11 ] M. Sasaki and H. Suzuki, On matrix realization of random surfaces, Kyoto University preprint YITP/U-90-29 (December 1990), Phys. Rev. D, to appear.