The size of random surfaces

Volume 138B, number 1,2,3

PHYSICS LETTERS

12 April 1984

THE SIZE OF RANDOM SURFACES David J. GROSS 1 Laboratoire de Physique Thdorique de l'Eeole Normale Sup~rieure z, 24 rue Lhomond, 75231 Paris Cedex 05, France Received 17 January 1984

The large-scale properties of non-interacting random surfaces embedded in an arbitrary dimensional continuum space are shown to be related to the infrared behaviour of two-dimensional massless free fields. The mean square size of free random surfaces is shown to increase logarithmically with increasing area; thus their Hausdorff dimension is infinite. A large (embedding) dimension expansion is derived and compared to recent Monte Carlo simulations.

It is widely believed that an understanding of the properties of random surfaces would have important applications in many areas of physics [ 1]. One might envisage consequences in statistical mechanics (interface physics, crystal growth, etc.), in the study of gauge theories (whose infrared behaviour can be related to a theory of interacting strings [2] ) and in the theory of relativistic strings. Much of the work devoted to the theory of free strings or non-interacting surfaces has dealt with ultraviolet problems that arise in formulating path integrals over surfaces or in quantizing such systems [ 1,3]. In contrast little is known about the infrared behavior of free surfaces, which is expected to be universal and independent of the precise short-distance regularization required to define the measure for continuum random surfaces. The simplest and most important property one might ask of a theory of random surfaces is the Hausdorff (or fractal) dimension (H) [4] of non-interacting surfaces. This number describes many of the large-scale properties of random surfaces. For example, a typical closed surface of area A will have a mean square size, Cg 2), of order A 2/H for large area * ~. We expect that random surfaces, that can fold and self-intersect, will have a Hausdorff dimension which is larger than their topological dimension, which equals two. In the case of random walks the Hausdorff dimension is well known: HRW = 2. It is indeed universal (i.e. independent of the precise manner by which the measure on random curves is defined), it describes a purely geometrical property of random curves which is independent of the dimension of the space in which they are embedded. Consequently, the size of a typical closed random curve of length L grows as L 1/H = N/~ for large L. The value of H has important implications for field theory, due to the fact that free scalar fields can be rewritten in terms of noninteracting random walks. The trivial infrared behaviour of spin systems (as well as the triviality of relativistic scalar field theory) when the dimension of space (space-time) is greater than four is essentially a consequence of the fact that two random walks never intersect when embedded in a space of dimension greater than 2HRw = 4

[5] 4-2

On leave from Princeton University, Princeton, NJ 08544, USA. 2 Laboratoire Propre du Centre National de la Recherche Scientifique, associ6 fi l'Ecole Normale Sup6rieure et fi l'Universit~ de Paris-Sud. 4-1 We shall in fact use a precise version of this property as a definition of the Hausdorff dimension. This concides, in all but pathological cases, with the rigorous mathematical definition of H [4]. ,z Parisi [6] has given heuristic arguments that, for random surfaces, H = 4 and consequently the upper critical dimension of interacting surfaces is 2H = 8. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 138B, number 1,2,3

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The traditional approach to the study o f free surfaces is to consider the canonical partition function (the propagator) of non-interacting surfaces (usually defined on the plaquettes o f a cubic lattice) Zc=

~ exp[-/3(area)], surfaces

(1)

where one sums over all surfaces, embedded in D dimensions, with given boundaries and a fixed topology. The entropy - the number of surfaces of given area A - grows as n(A) ~ eOAAb, where ~ is regularization dependent [7]. b is universal, is related to H, and can be determined by studying the critical surfaces at/3 ~ ~, whose mean area (in units o f the lattice size) diverges. However few definitive conclusions have emerged from this approach. Recently a microcanonical approach has been proposed [8], in which one considers the measure for random surfaces of a fixed area and determines H by evaluating the mean size of the surfaces as a function of area. To do this one defines the surfaces in terms o f a fixed number, N 2, o f triangles o f fixed area embedded in continuum space in the limit a s N 2 ~ oo. This procedure has proved to be extremely efficient for numerical simulations. In this paper I shall show that it is equally advantageous from an analytic point o f view. It is very instructive to consider the analogous procedure applied to random walks or curves. We can take the following measure N

=const.fi~=ldDxi

Zq

N exp(--/3~i=l

]Xi-Xi+112q)

'

(2)

to generate piecewise linear closed curves defined by N points, Xi, embedded in D dimensions (XN+ 1 = X1, and II' means that we have eliminated the trivial translational mode by setting zN= 1 Xi = 0). This measure is concentrated on curves of length ocN, with fluctuations of order X / ~ . Thus as N ~ oo it is equivalent to a microcanonical measure for random curves o f fixed length. The Hausdorff dimension of random walks can then be determined by the dependence o f the mean square size o f the curves on N N

= 1 ~ <~2> ~ N2/H. -N i=l N~,,o

(3)

Random walks, being markovian processes, are quite trivial, and can be solved by a variety o f techniques. The simplest method is that o f the transfer matrix, from which one can easily deduce that = (NIl 2) <(A~') 2 >, where <(z2tX)2 > is the mean square size o f the individual random steps. Thus HRW = 2, independent o f the precise form of the measure (for example H is independent o f q) as long as <(&X) 2 > is finite. For example, for (2): <(ZkX)2> = f

dDx

exp [-/3(x2)q]x2/fdDx exp [-/30(2) q ] = 13- l/q

r((O + 2)/2q) £(D/2q)

(4)

In particular we could choose q = 1, in which case we are simply dealing with a one-dimensional theory o f Dcomponent free fields on a lattice o f spacing 1/x/'fand length N/X~. The linear dependence o f (X 2> on N is simply a reflection of the linear infrared divergence o f one-dimensional massless fields: ~ f l / N dp/p 2 ~ N. For q 4= 1 we can derive this result by rewriting Z_ as a measure over free massless (D component) one-dimensional fields. We introduce a variable gi = (Xi - Xi+l)2 ~by means of N

N

N

i=

i=1

"=

Zq=const.f~ d)tidgil-I'dDXiexp(ii~=l In this form it is clear that 186

X i is a

N

~i[gi-(Xi-Xi+l) 2] -/3 ~ g i q ) .

free field, whose propagator

i=1

Mi~ 1,

(5)

Volume 138B, number 1,2,3

PHYSICS LETTERS

12 April 1984

Mi] = 6ij()~i + ~'i- 1) - 8i,j- l )ki - 8i, j+ l ~j ' has h dependent weights. Since the constant ~. mode (i independent) has non-zero weight we conclude, correctly, that the infrared divergence of (X2), in the infinite volume (N) limit, is universally proportional to N and thus H=2. in the case of a random walk the explicit form of (X 2) is easily calculable, since the integration over Xi, in eq. (5), can be performed with the result

o (

1 _ exp x/det M

In ),7. - -~ In

.

(6)

i=1

For large N, the second term in the exponential is negligible (O(ln N / N ) ) compared to the first. Therefore the 9~., gi integrations factorize. We then derive that

(X2)=~_I/qD

N-1 ~ 1 (1/2),) ~ ~-l/q N p _ 1 4 sin2(pTr/N) N--+oo

ND(1/),) 24

(7) '

where the mean value of 1/),/is independent of i, and given by < I / ) , ) = M _ 1/Mo ,

(8)

Mn = f dXdgexp(i)~g_g q - ~D 1 In ),) ~ F(n + 1 - D / 2 ) P ( ( D - 2 ) / 2 q ) .

Therefore we recover (l/X) = (2/D)F((D + 2)/2q)/F(D/2q) in accord with (4) If I have belabored the treatment of the trivial case of a random curve, it is only because I wish to argue that random surfaces are equally trivial. We shall rewrite the microcanonical measure for random closed surfaces in terms of free massless two-dimensional fields and conclude from the logarithmic volume divergence of the free propagator that their Hausdorff dimension is (logarithmically) infinite. In order to compare with the numerical results of ref. [8], and for convenience, we shall treat the case of a torus. Thus the points on the surfaces are labelled Xi,/, where i, j = 1 .... N form a hexagonal lattice (see fig. 1), and Xi,/+ N = Xi+N, ] = Xi, I. Each point, Xi,/, is a vertex of six triangles, which we label Ag., a = 1..... 6. For the measure we take

(1

N

Z=const.f i,/=1 Fl doXi/exp '

-5

N

6

i,]=la=l

(9)

where A (A) is the area o f the triangle A and thus the action equals the total area of a triangulated torus with N 2 points. This measure is concentrated on tori of area = D ( N 2 - 1)/2, with fluctuations of order N, and is therefore

~

_

i

i*l'j Fig. 1. The hexagonal lattice used to discretize the two-dimensional parameter space of a random toms. This lattice serves to label the point Xi] in D dimensions, as well as the triangles that make up the surface. 187

12 April 1984

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Volume 138B, number 1,2,3

equivalent for large N to a microcanonical measure. We shall determine the Hausdorff dimension, H, by evaluating ()( 2} = N - 2 EN] = 1 Xi~' which should scale as Area 2/H ~ N4/H, as N -->oo. We shall, once again, rewrite Z as a measure over free, massless (D component) two-dimensional fields, by introducing a symmetric 2 X 2 matrix, g~., which we equate to the induced metric tensor, G~., on the triangle A~.. In other words Z is rewritten as N

Z=const.f

N

[I

i,j=l

a=l ..... 6

N

d Z .tIC ' . dt Ig...~. F I ' d D X i / e x p ( i t,l=l

N

~

tr -Aifl'giJ-~ G~') - - 61 a=l~..... 6 ~

i,f=l

(10)

)

i,j=l

a=l ..... 6

where [ (Xi+I ,] G.I.= ~,l t (Xi+I,1

Xi, j) 2

(Xi+I ,j - Xi, j)(Xi, j+ 1 - Xi,]) ~

XL/)(Xi,j+I-Xiff)

(Xi, j + l _ X i , ] ) 2

),

etc ....

The above action is now quadratic in the X i ]'s, and clearly Xi/is a free massless field whose propagator has X dependent weights. We therefore conclude that (X 2 } contains the infrared divergence, in the infinite volume limit, of a massless two-dimensional field. Thus (X 2) ~ f l / N d2p/p2 ~ In N and H = oo Random surfaces are more complicated than random curves. The evaluation of the X dependence of the X integration is not simple, and the X and g integrals do not factorize, even in the N - + oo limit. Thus the evaluation of the value o f the coefficient (which is not universal) of the log N term, is non trivial. To assure ourselves that nothing strange occurs, and to compare with ref. [8], we shall evaluate the properties of the surfaces in the large D limit. In the limit of infinite D the above measure is dominated by a saddle point. The saddle point values of k i! and g~. are clearly, due to the translation and rotation symmetries of our lattice, independent of a, i, andj. A little algebra then yields N2

a,i,f

tr(~,~.G~) = 2()tl1 + )t22 + ~kl2) /'~1 ((Xi+I 'j - Xi 1.)2 + (Xi, j+ 1 _ Xi/.)2 + (Xi_ 1,f+l - Xi j)2}.

i, "=

'

'

'

The X integration can then be easily performed, providing us with the action S = N2x/-d~ g - 6iN 2 tr(~, -g) + ½DN 2 ln(~ll + X22 + X12).

(I 1)

We therefore find, as D -+ oo. a saddle point g l l =g22 = 2g12 =D/N/~,

Xll = X22 = -2X12 = - i / N / ~ 6

(12)

about which a systematic large D expansion can be carried out [9] ,3. Here we shall simply report the results for infinite D. First we check that the total area is indeed equal to N2D/2 (which is derived directly from (9) by scaling). This requires that N 2 D / 2 = N 2 d~/-d-~, so that det g = D 2 / 4 in accord with (12). We can now calculate (X 2 ) precisely, it is given by: N, D~oo2N 2 2(Xll + X22+X12) p ~ inN 2 =0.0791nN 2. N ~ 47i" D-.oo

l 4{sin2(prr/N)+sin2(qrr/N)+sin2[(p+q)Tr/N]

} (13)

.3 There is no reason why the 1/D expansion should not converge for all D/> 2. Certainly the log N dependence of
Volume 138B, number 1,2,3

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12 April 1984

This result is in excellent agreement with the results o f ref. [8], where the logarithmic dependence D - 1 (X 2) was discovered and the coefficient o f l n N 2 was found to be 0.138 +- 0.003, 0.114 -+ 0.002, 0.101 -+ 0.001 and 0.092 -+ 0.002 for D = 3, 4, 6 and 12. Extrapolating these results to infinite D yields a coefficient of 0.083 -+ 0.004. The agreenrent with these, finite D, measurements is even better if we express (X 2} in terms of ((2U02), the mean square size o f the side of the triangles. The above saddle point implies that, ((~)2)

~

gll=D/x/-~,

(14)

D--+m 1

as if all the N 2 triangles were equilateral (this is also evident from the fact that g12 = ~ g l l ) . Thus we predict that for infinite D the mean triangle is equilateral. In ref. [8] the mean square triangle side was measured for D = 3, 4, 6, 12 with the result (the predicted value of (14) is in parenthesis) 3.96 (1.73), 3.96 (2.31), 4.78 (3.46), 8.02 (6.93). Thus the triangles are becoming, with increasing D, equilateral. The ratio of (13) and (14),

(X 2} ~ V~ = 0 . 1 3 8 , In N 2 ((zXX) 2) D~o~ 4n

(15)

N-+

is likely to have smaller finite D corrections. The results o f ref. [8] are 0.105 -+ 0.002, 0.115 -+ 0.002, 0.127 + 0 0 1 , 0 . 1 3 8 -+ 0.003 f o r D = 3, 4, 6 and 12. Many other properties o f the critical random surface can be easily determined for infinite D. These are o f some interest since it is clear that the surface must be highly folded and self-intersecting. In D = 2 in fact the surface can do nothing but fold upon itself. For example, an immediate consequence o f the existence of a translationally invariant saddle point at D = oo and the vanishing o f the total curvature o f the surface. (The Euler character o f a torus is zero), is that the mean surface for infinite D has zero local gaussian curvature. One can also easily verify that the angle between two adjacent triangles, which share a common side, is c o s - 1 ( _ 1/3) = 109.5 o, compared to zero angle if the surface was planar. A complete analysis, as well as the finite D calculations will be presented elsewhere [9]; In conclusion, we have found that non-interacting random surfaces, o f a given area, can be understood in terms of massless two-dimensional free fields. Their mean square size increases logarithmically with area as a consequence of the logarithmic infinite volume, infrared, divergence o f two-dimensional massless fields. Therefore random surfaces have 0ogarithmically)infinite Hausdorff dimension. If one were to define a random object o f arbitrary dimension d, similar arguments would imply that the Hausdorff dimension w a s H d = 2d/(2 - d), for 0 < d < 2. For d > 2 we would find no infinite volume, or infrared divergence and thus Hd>2 = oo. Thus random closed three-dimensional volumes, embedded in dimension ~>3, would have an extent that remained finite in the limit o f infinite volume! The dynamics o f such objects would presumably depend solely on their, non-universal, short-distance regularization. An immediate consequence of the infinite Hausdorff dimension is that the "upper critical dimension" of random surfaces is also infinite. The dimension o f the generic set of intersections of two random surfaces is (using d(A ^ B) = d(A) + d(B) D) infinite in any dimension D. Thus one would expect that the infrared behaviour of interacting surfaces is always non trivial, for all D < co. This suggests that string theories, and perhaps gauge theories as well, are non-trivial in all dimensions. Therefore in order to carry out an "e expansion" one would have to expand, at best, about D = co The infinite value of the Hausdorff dimension has implications for many o f the critical exponents of the canonical sum over surfaces. By standard arguments (see ref. [ 10] ) one can relate H to the coefficient in the entropy of random surfaces and to the critical exponent of the l o o p - l o o p correlation function. I f H = oo one finds that b = --2 and that the "glueball" mass vanishes logarithmically at the critical point. In the above treatment we have discussed only non-interacting, closed surfaces with the topology o f the torus. It is evident that the same infinite Hausdorff dimension would result for surfaces with boundaries and with arbi189

Volume 138B, number 1,2,3

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trary Euler character. On the other hand interactions are bound to have a dramatic effect on the value of H, pre. cisely because it is infinite for free surfaces. It would be very interesting to explore the effect of interactions (curvature dependent weights, self-avoiding surfaces, ...). However this is beyond the scope of this paper. The author thanks A. Billoire, J. des Cloizeaux, E. Marinari, and especially E. Br6zin, for useful conversations.

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

[6] [7] [8] [9] [ 10]

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A. Polyakov, Phys. Lett. 103B (1981) 207. V.A. Kazakov, Phys. Lett. 128B (1983) 316. See J. Schwarz, Phys. Rep. 89 (1982) 223, and references therein. B. Mandelbrot, The frac~al geometry of nature (Freeman, San Francisco, 1977). K. Symanzik, J. Math. Phys. 7 (1969) 510; M. Aizenman, Phys. Rev. Lett. 47 (1981) 886, Commun. Math, Phys. 86 (1982) 1; D. Brydges, J. Frohlich and D. Spencer, Commun. Math. Phys. 83 (1982) 123. G. Parisi, Phys. Lett. 81B (1979) 357. D. Weingarten, Phys. Lett. 90B (1980) 280; T. Eguchi and H. Kawai, Phys. Lett. ll0B (1982) 143, 114B (1982) 247. A. BiUoire, D.J. Gross and E. Marinari, to be published in Phys. Lett. 139B (1984). D.J. Gross, in preparation. B. Durhuus, J. Frohlich and T. Jonsson, Nucl. Phys. B225 (1983) 185 ; preprint (1983).