Numerical study of field theories coupled to 2D quantum gravity

Nuclear Physics B (Proc. Suppl.) 25A (1992) 69-86 North-Holland

PROCEEDINGS SUPPLEMF.NTS

NUMERICAL STUDY OF FIELD T H E O R I E S COUPLED TO 2D QUANTUM GRAVITY

S.M. Catterall, J.B. Kogut, R.L. Renken Loomi~ Le/*orzfor~ of Pk~t~ie~ Uni~er:itl~ of Illinois at Urbaaa-Ghampaign 1110 Welt Greevt Street Urbana, lllinoi~ 61601

We study the scaling properties of modeh which, being formulateAon dynamical I.~ttices, indude a couplinb t( a fluctuating metric. The first of these deals with a set of bosonic Adds whose &ction nmy be written in terms *,f xa extrlmic curvature. We present evidence for a 'crumpling' trmas'tion, e~tlmate certain critical expomnts and dlscu~a p~ible continuum limits. The secattd study e ~ ferlmoni~ field~,r ~ t e d here ~mIsing ~pi~s, A careful finite size sealing ~tudy is performed yielding exponents in agree~-ent w~th analytlc/d pvediztlons. We further study th,, equation of state and the intrlr~sic spin-spin correlation rune,ion.

C R U M P L I N G TRANSITION

Introduction String theory in a Euclidean space can be thought of as a e/aegry of ,ree bosons ccapled to 2d quantum gravity. Whilst claxsicall, the dimension of the embedding space is a free paranteter, the quantum theory is ot~ly tractable analytically in the critical ,limension O = 26 where the gravity sector effectively deeouples, To i,ry to analyse the structure of t,~e sub.critical theory various autl:ors [I-3] have prop,~sed the u ~ of a regulated model based on dyaamically triangulated random surfaces. The worm sheet of the string is approximated by a simplic~al triangulation T with a finite number of elemet,'s. Metrical fluctuations ar~, included by requiring the partition function of the model to sum overall such triangulations consistent with sore. fixeG t o p d ogy. The embedding ~s accomplished by o.qsigning coordinate (matter) fields r u to thp nodes of tile graph. It is theu p,~ssible ~o write down

a discrete versiot~ cf the Polyakov action which couples only neare, t n,dghhour fields x ~ on 7' and study the resrltin 5 model via Mont~ Carlo techniques [4-7]. The conclusions, of both these sludies and cmnplimentary z,talytical work [8] were unfort~,nately rather wgative. It api~eared that the s0,faces g~,n~rical'y d,*geneca,.e hl~o bra.ched polymeis (at le&-t for large elnbcdaJug dimension) which corr~ [~ond in the coiltinuum iimit to free scalar ~eld theory, Tlas has Leen attributed to a .ol~-scalin& of the lattice rLriug tension [9]. One way to try to e w d e these problems was suggesl.,.d in [10] with the addition of a new term to the :tction - the square of the extrinsic curvature 1","of the surface.

If = f d2~xlthh°'a~an i • V s n i The covariant derivatives act on a set ~f fields n~, corresponding to the D - 2 normals to the sLrface at a point. The induced metric fi.;~ is del~rlllit~.,M via the embedding %a = 0 . z - 0az

70

$.M. Cattera//et a/. / NrmericaJ Study of Field Tbeor~ Coupled to 2d Quantum Gravlgy

Such a term preserves the reparametrisation invariance of the continuum action and clearly favours at least locally smooth extrinsic geometry. Sinee its coupling )t is d'mensionless it should b~ ~dded in on the grounds of generality. In addition, it has beet, suggested t h a t such a term is induced in the effective bosonic action when the fermions are integrated out in superstring theories [II]. Previous numerical work [12-14] has provided strong evidence for a continuous phase transition in such a m,"del at a critical coupling of ,Xc ~ 1.5. For A > ,go the model is in a smooth phase ~ith Hausdorff dimension du -- ~ and sm~ll extrinsic curvature, whilst for ,X < ,gc the model pOssesses a large Hwzsdorff dimension and is crumpl~d. Whilst the ini'ial simulations were confined to a hulk space dimension of D = 3, these conclusions have been sl:own to remain qualitatively the same for larger J: [15]. The crucial question of the behaviour of the string tension close to this new critic~l point w'~s addressed in [IG] and more recently in [17] with encouraging results. However all of tb~ previous numerical work h ~ been hampered by the lack of an efficient way to ge ta larger .¢arf~.ces (in D > 0). The flt'id nature of the fat ~ce and the varying local coordination numb- ,,-prohibits standard methods of vectorisat i o d o r parallel processing. In this paper we propose a new algorithm which works not with triangulations, but their dual graphs - vacuum diagrams of a ~ scalar field theory. We show t h a t it is po~slble to pipeline both the node mow*s and local graph updates. The resulting c~de is used to investigate the crumpling transition on lattices with uIc, to 1000 nodes. Model

The pactlticn func*ion we are considering takes the form

n Gm

i----1

The term S c is a discrete version of the P Jlyakov action ij where the adjacer.cy matrix Cij is unity when i and j are connected by a link on the g r a p h and zero otherwise, it plays the role of the metric ' ensor in the continuum formulation. The part,tion function requires a sum over all ~b3 grapLs G,, with i, nodes (we are working in a microcanonicol ensemble) and an integration over all out one field coordinate (which is fixed to eliminate the contribution of the translational zero n~ade). We further have restricted ourselves to the case of spherical topology and to a bulk space of dimensio,'~ D --- 3 . For the discrete c v r v a t m e K we employ the following form

I," = ~

~ i

(1 - nj ( i ) , , k (i))

j
The variabIes nj (i), j = 1 . .3 are unit norreal vectors associated with the node i. Each is constructed from tile vector product of two of the three link vectors attached to the node. Previous diseretisations of tile extrinsic curvature have been ba.~d on triangulations of the manifold and the normal vectors 1hen associated with tile ele:nel,tary triangles. !u contrast In re these degrees of freedol, are tied to the nodes of the graph. Thus in principL ~:~: a, tion is different to t h a t studied in [12-14]. llowever, by appeallug to universality one would expect this to be mfimportant ill tile vicinity of ~o;ne critical re giou. The u~mlerical results we shall present later are consistent with this. Exploiting the scale invarianee of A" allows us t~ set tile ba,e string tension P0 to unity (A~ is il~tlependen! of P0) and determines {So)

S.M. C'atterM/et M. / Nmnerical Study of Field Theories Coupled t o 2d Quantum Gravity ( s ~ ) = ¥D ( . - 1)

fl

I,

Mot,itnring the expectation value of the gaussian term provides one check on the numericM procedure.

Aloorithm There are two main difficulties encountered in devising an efficient vector (or indeed parallel) Monte Carlo algorithm for simulating these models. The proof of detailed balance (whi.:h is sufficient to guarantee that the configurations have the correct Boltzmann weights) rests on identifying independent subsets oi the lattice which may be updated in parallel --- to: a dynamicM mesh these subsets have to bc continually recalculated. In addition it is imporLant t h a t the local environment of each elemen~ of the lattice be the same - - this is not true for the triangulat,'d models as the cooraination e f a node is not fixed. Tire use of the dual $s graph with our choice of di~.crete curvature K, immediately circumvents the eoor,Anation problem. Whilst a complete solution to the ~ty~tamical sublattice problem would b¢ extremely difficult, we show later how a rather simple algorithm wifi generate suitabVe se;.s of sites arm links. The Ioc:,l graph update we employ derives d;reetly fro n the csrresponding link flipping operatio:~ on the dir,:¢t lattice [3]. This guarantees us ergodic;ty in H,e space of all such graphs. Fig. 1 :~:c:'.':: . ; ~ : [.~i ,~:: c f a ~ g, aph centred about link (,~d) wit'.* ncighl,out node~ labelled b, c, ~, a n d / . On ti,e direct lattice the operation of fl~pping the link dual to (ad) could result in one of two possible reconnections of the Os graph. One of .hese itgal possibilities is shown in fig. I fthe 91 aer is vlerely its mirror reflection in link (ad)L links {dr) and (oh) are deleted aqd l~,.k~ {¢~f), ,idb) are added. Such a move affects the lengths

I1 / i [ /

rl

r;~

//Q

b,~--¢-(" b2 Fig. I. ~3-Kea~h of tile four loops or 'rings' in the ,,#-graph which are neighbour to the link (see fig. 1) The neighbonr rings r l and r2 both los,., a side whilst rS, r4 gain one. The graphs are restricted so that a given pair of nodes are connected by at most one iit~k and no link can have the same endpoints. This is equivalent to elimi~ating all self-energy apd tadpole in~rtions i:~ the scalar field theory and ensures that ~he minimum ring length is three. T,~ effect this we keep track of which rings are c,trrently touching (have one link in common). The potential graph i~pd ate of fig. I is then prohibited if r3 and r,I arc already touching. Notice that the comput'ttion of the change in the extrinsic curvattlre term will uec~sitate knowledge of some of ti~ hext nearest neighhour nodes such as f l and f 2 as well. ()ur d a t a structures contain not < nly the identiD' o~reach 3 f t h e thr,,e links incido! t on a given :~xle, but ai,~o the labels of the rings which they Lorder. Furthermore since every hnk tan be ref:re,wed through either of iis end~oints we store a i m the information necessary to pass u~tween

S.M. Catterall ~ a$. / lqumeric~ Study of Field Theories Couple-./to 2d Quantum Gravi~

72

these two representv.tious. By using a cyclic ordering ¢,f the link d a t a we can unpac': all of the intorma,~,ion relevant to a link update in a manner independent of position in the lattice. With this information the change in action can be computed a i d the flip update subjected to a standard M~:tropolis test. A node updat~ i. t-ivial - one simply adds a random vector to the field coordinate z~ on site i, unpacks the local information in a way similar to the graph update and uses another Metropolis test on the resultant change in action. The step size is tuned with A to maintain an acceptance rate close to -~0%.

oi o

t2~

t~o

I~

2

A Fig. 2. Link Acceptance rate No such tuning is available with the graph update and fig. 2 shows t h a t the acceptance rate A for local graph updates falls with increasing A. Notice t h a t close to the transition Ac ~ 1.5 the link acceptance is only O (1%). In principle this low acceptance cuuld lead to a situation where the motion through the space of all graphs is too slow (compared to the node update) to simulate the true dynamical lattice. To check for this we

experimented with a situation ~here f sweeps of the links are made for every node sweep. We ran at a variety of coup!ings near )% a n d for lattice sizes n = 50, n = 100, and n = 200. With runs of typically 2 x 106 sweeps we detected no systemat~¢ shifts in any expectation values which were statistica!!y sigrd.qcant. For the n = 500, lf~0 node m::,~bes we adopted f = 3 close to the critical point. lit priaciple it would seem t h a t it should be possible to adopt a modified form of link update chosen in such a way as to increase its acceptance rate. An obvious Way to proceed would be to incorporate not only a local reconnection about tile link (ad) but also a shift of the coordinates a and d..'Tot example, the coordinates of a and d call be subjected to an al, proximate gradient descent on the trial reconnected lattice This final "cooled' configuration can then be passed into tile Metropolis step. This is analogous to llybrid Monte Carlo techniques. Unfortunately we found tile resultant method prone to 'overcooling', leading to systematic errors in the results. We are currently investigating this further. Since both the node a n d graph updates are independent of position in the lattice we can att e m p t to pipeline the update. To ensure t h a t detailed balance is not violated we need to find sets of links and nodes whicl" do not influence each other under the Into[ updetes, For the case of the links we adopt the following procedure. A fixed number of links ( t r i a l ) are selected initially at random anti their neighbouriug rings r l - - r4 unpacked. Then the list is scanned to progressively eliminate links which have any of their rings in common with the flagged rings of previously accepted links. A similar approach is used for determining potential independent node updates. Again t r i a l nodes are selected at rahdom altd their nearest neighbours computed. This array is then compared with a dynamic

S.M. Callera/I et al. ! Numer/cat Study of Field Theories Coupled to 2d Quanluta Gravity array of accepted n o d e / n e i g h b o u r combinations and new ones added only if there were no overlap. Both of these m e t h o d s being recursive are inherently scalar operations. T h e trick to using t h e m successfully is to tune the p a r a m e t e r t r i a l so as l.o achi-v,- a m a x i m a l n u m b e r of a t t e m p t e d u p d a t e s aer unit C P U time. If t r i a l is too small the time spent in the scalar portion of the code is small, but the resultant vectors are short and little gains are m a d e from veetorisation. Equally if t r i a l is s,~.t too large then although the vector portion of the code is executing efficiently, a large time is spent in the search routines and the u p d a t e becomes slow.

73

of approximately an order of m a g n i t u d e from the scalar limit (trial = 1). Typically we set trial ~ 3Vfff, with a fraction o being accepted (c~ ~ 0.3 at small n, tending to unity as n ~ oo). To test the vector algorithm further ~ e ran at D = 0 (corresonding to the ease of pure gravity) and eolnpar,~,i ' h e distribution of ring length~ with tile analytical result [3].

t~-~

I' (I)-

(1) = 1 6 ( 3 ) i (I - 2 ) ( 2 1 -

, I

i ol

l)

()

°°•.° I I:ig. 4. Ilin I l)islllbuli,m • '1 e data ,~o is in fig. 4 d,>rivc from aT00 no~le illesh alld lhe ~x~lid curve froiii the pr+'diclion for

Fig. 3. Attempted node ulxlates per second Fig. 3 shows a plot of how tile n u m b e r of at-

n ~

~ . There is no sign o f a disagrcem,-ot. To

che.~k tile coue out of D = 0 we ran on a ,r)U .od," latlice and compared tll~" ~,~'ctor p r o g r a m

t e m p t e d node updates per second, i/, varies with

reslllls against tho~e produced by a scalar cod*.

t r i a l for n = 200 at )l = 1.0. In practice one

The res,il! s o f I lie two codes were ctJulparcd over

can sit close to the peak (say trial = 40) in this curve with tile fraction of time spent in the scalar routines heing O (10%). Furthermore t.llt~,e conclusions are only very weakly )t dependent. One

a wi,le rang,' ill )t and wl're always ill agrecri~:nt withiu errors. In addition larger !altwes wf-te ranln both at A = 0 and close to 2~, and no systematic difl'erelio's b~'tween scalar aud w'ctor p r o g r a m s

call ¢oeet h a t close to tile m a x i m n n i there is a gain

¢,'c"re appareut.

74

S.M. Catterall et at. / Numedc~J Study of Field TIneories Coupled to 2d Quantum Gravity

An alternative way of achieving a vector code flight be to simply replicate the h t t i c e a number of tillles. Provided each latiic,, w ~ g;~ a a thenmalisod, independent start configuration l lte s,,h~e,l.~en! Molit,. (~ar]o evohltion would be ,'quivah'nt to a single long run on one lattice, Furth,'ru~or,' it. is the,t: trivial to I,ipdine scalar ~$,datos on the set of lattices. T h e drawback to litn~ a i preach is i~h;slrated in fig. 5 which shows a portion of a typical ruu for K on +l = 200 at

of M o l t " Carlo sweeps to thermalise the lattice.

atsutts Fig. 6 shows tile expectation value or tile extrhlsic c u r v a t a r e K on lattice sizl~ 50, 100, 200 and 500 for a range of cou01ings A.

A :: 1.5. T h e autocorrelation time is clearly of . r d , ' r lil e' :.weep: ] "I l us the prol)lenl call now be se,'n to I,,. iJ, the ?~(,vision of a lare,e n u m b e r of dvcorre~atcd start ,'(,nfigurathlns and the necessity of running s,.:~eral autocorretation times on each lattice. We found this to be itopractical.

il

ol ol I ..

"-

A

Fig. 6..Llean Extnlnsic C.irv~ture T h e al,s,'li.=o .)f any discontinuities i . d i c a t e (as il, lit, ,!irect lattice case) the unlikehl.Jod of a lirst order I ransitiorl. As usual, the best. iudkcator for possibl,, phase ~rausitions, is the specific heat C

~÷

M.(' Time

Vig. S '!'yplcM h." evolution N : 200

This is shown for lattice sizes It = 50, n = lOO, u = 200, 1, = 500, n = U)O0 i.I fig. 7

"!~) gcncr~,te ! he graphs wu start from ;tit eh'n,'ultary tetral,*dron a!ld l',i~'khlg nodc~; at ratv l,oltl add all extra link bel~.~2t.ll t ~ u netghbouring links. This process in(rei~.ses the n u m b e r oi nodes it by t w o and the ring count by one whil:,t pre~'erving the plana~ character of the gra,,dl. Sv~h link addhhm.~ are Sellaratvd by s,'queuces

rI,

P.;lk (incrcasin g w i t h latti
uicative ~1' a possible continuous phase transition in the rood ,I. Notice also that tile critical coupling A, appears retnarkably close to its value on tl~e triangulated lat.qcc see e.g. [19]. It would apI,"ar that the nlain effect of using tile ~ba-graph rather than .: triangldation is to renorlnalise the

S.M. CaZterall et al. / Nume~cal

S¢,udy

of Field Theories Coupled ¢0

2d Quantum

Gravity

75

mension for the surface. T h u s we expect C 4

where

t

('

(L) ~ A ' n '¢ + B'

ttT

i 2~ I

wr =

c~

ud

+

! !

I

z

,,

i,:

,,4

,J~,

,~

,

i:

~

i,,

!

i~

i

A

F i g . 7 . S p e c i f i c I l e a t ,is ,~

gaussian coupling which doe.,, not affect the critical curvature coupling. We haw. a,;sessed our errors by tl,.e usual binning procedure, and collected statistio~ until the error plateaued in the 2 0 - 40 bin region and the specific heat agreed with its average single bin value within one standard deviation. For the n = 500 poit-: lattice this entaih:d runs of 4 x 10~ vector sweep~.. To extract critical exponents for the tranaiticn we attempted a finite size scaling study ef C. The standard scaling hypothesis is

4

hl n

F i g . 8. I n C " v s L , n

I"ig. 8 .,hows a log-log plot of C ; , a t versus n. ,%1 large n this should approach a straight line with gradient w°. A fit to the last three points gives wl :. 0.185(50) witll a , ~ of 0.02. If the hyperscaling relation

C,~,,~(L)~AL'~4. B

vd=2-~ relating the peak in the specific heat as a timetiou of ,~ for a given t, (L is the linear size of the lattice) The exptment w = ~/u where (J: is the exponent governing the divergence of the specific he~t, and v the correlation length exponent. We can extract ~ iin,~ar scale for these dynal,fical lattices by setting L = r~ ~"

The parameter d is an internal II~e,;docff di-

i,~ ,L, s u n , e d ( w h ; e l , h ~ } , , e u simown to hold fol all dynanfical lattice models amenable to analytic .~oh:ion), then it is possible to extract values for vd and ~ separately. O u r values are: o = 0.31(7),

v d - - 1.68{7)

These values are consistent with an earlier stmly on triangulated lattic,,s [14] which yielded tae L~timates a = 0.24(5), vd = 1.76(5).

S.M. Catterall et al. / Numerical Study of FirM Theori~ Coupled Io i'd Quantum Gravity

76

We have also coluputcd tit(' nlean gyration ra-

dil,.~ (,2) I i

with Y# = ¼ ~ i

"rT"For large n we expect tl:,,;

The -xponent dlt is termed tile Iiausdorff dimension. We |ind that for A < A, ~ 1.5 the gyration radius (z 7) increases only very slowly with n rorr,'sponding to a large Ilausdorff dimension, whilst it scales linearly (d/t = ?) with n at large A. A lit to the d a t a at A = 1.5 (which we estitnate is an upper bound on the infinite lattice critical e~mpliug) yields a Ilausdorff dimension

till

= 2.1(1).

'l'lw ex|riusi( curvature also has a l l ell'cot OU

the IIt~rinsic t:eontetty ,)f the lattice. This is most ea.sily seen in th ~ ring length distribution which is essentially exp )n,.ntial at A = 0. Ilowever, by the time we reach large coupling A > A, tile disl ritmtion I~as shifted ~c> peak elo~ to the regular lattice value of six. This qual,t;'.~ i~, the aitalogue (If Ihp In(,.: (':,oi'dlllal;c.;, lUluiller ill tile direct lattice a],(I Ih(. data tl~xls relh'cts an increasing teutJencv :(, sllp~)ress large ~,drinsic curvature w!Yl~ql~.r~,e A. ('olltl~sloo5 "[" Slll)ll~li~rise, we have, g i v e n evidc:::,,: f o r i h j' ('~is|t'llCt, e l ~i c r u l l l p l i l l [ ~

{r:tllhliloU

011 r a u -

dam ~,'rfa:e mo
ent from the conventional bosonic string, based ou Polyakov's action. Ilowever, a 'stringy' continuum liufit would imply a rather special sealing of th- string tension as the critical point is ap0roached. Tantalisingly, the preliminary reSUITS OI IIOJ alHI [ l l J inOlcate t h a t flus s c a l i n g

behaviour may indeed occur. Further work on larger lattices will be necessary to examine this question. With this in mind, the dual lattice fornmlatiou proposed here may well prove very effective.

MEASURING TIlE STRING SUSCEPTIBILITY All of the previous work has been done in the microcanonical ensenlble where the number of andes ill the triaugulatio, is fixed. In this section, we ~ouhl like to present an estimate of')'str, an exponent tlnat gu~.:.m the entropy growth of tile partition function in the scaling limit. The lunar natural ensemble in which to study this is tile canonical ensemble where die number of node., i...llowe,I to w,ry but tb.? to; elegy is I~eld fi.~:vd. Unfortunately, canonical ~:~semble c."lculations are very difficult and have never leaC to sat~-.faetory results [7]. We arc ther~?for~ motiva.e,I to attempt to calculate %,. in the trliercrc a :onical eusenlIJ|t ,~'rt(t tO d o this we use ideas of +;ross .'.nd Ilamber [20} W,, (Io5.. %tr by the behavior ,qf the microcanonical partition function describing the fluctuations of a surface of genus h with a fixed number of nodes N. ~ h ~ N-Zl+'~,t,l.l?NF 'lb i:l,'orpor~t,, rhanges in the number of triangles N it is n,rcessary to introduce a canonical partition function

S.M. Ca•recallet el. / NumericalStudy of Field TheoriesCoupled to 2d Qutmtu..nGravity Zh

-

-

tile surface n i (~) i = 1 , . . . D - 2. The induced metric h,, a is determined via the embedding

~--'~C-X,V~h N

The parametcr A is essentially a cosmological constant and must be tuned towards some critical value ,~c in order to recover continuum pUySles. |L IS stralgmiolwartl LO SI|OW that, close to Ae,

(A -

Z ~

A~) 2-~'''(n)

If the central charge of the matter fields is less th~n unity, then it has been shown [21,22] t h a t "/str is linear in genus -- "/,tr(h)

:

(1 - h ) ( 2 - 7 , t r ( 0 ) )

llowever, the central charge corresponding to the crumpling transition is not known precisely although the d a t a derived in [23] might suggest that it i~ k:-s than unity. So it wonld be extreme y interesting to try to measure %tr as a function of genus h, both to extract an important exponent arm to provide another check on the value of the ccntrai charge. A formula for "/str may be deriw d by examining the finite size scaling of the sqt~are of the intrinsic curvature density, r (~). Con rider th~ continuum fixed area (equtva~em, to ~xed .At} partition function with a term ft = afd2E,./~r 2 added to the usual action: Z ( A ) = j[ dp(g,X)6(F)exp(-Si,

,, - tt)

wh-re F ib the fixed area constraint r -- / a,2c ~v5

-

:

= a~x

.

oax

Under a global rescaling of the metric g ~ Ftg, Si,,~ is iuvariant whilst the new term H scales like [! "n . We make the additional assumption t h a t the measure dp is invariant. Consider now the partition function Z (A (1 4- Q)

g(A(i +O)= /dlt6 ( f J~- A(l +O)e -s If we now rescale 9 with f l = 1 + • we can rewrite this as t 1 Z ( A ( I + 0 ) = (-[-~-i J d # t F ) e - s e e ' f el'" Expanding the R.H.S to order • leads to Z(A(I 4-Q)

zC4)

=,+,(-l+~,f v~r:) 4-O(,q where the expectation values are evaluated with respect to the ensemble with area A. Using now tile definitLm of %t~ given earlier we car. derive tim result =.const - - - +.. A A Close to a r~it~-M point, the assmnption is t h a t the sum over discrete triangulations will generate the continuum partition function with A replaced by N and a

A

Si.~ = Sp + AS,.~

Sp

h~a

f d2~v~.q°'~h,,a

S r = f d2~x/hh"aVan. Van The term SK corresponds to the extrinsic curvature and involves the ~ t of uvit notmals to

i

\ai

/

where ai is the local coordination nmnber of node i. Notice t h a t the results should be insensitive to the value of a (which should prove to be a good check on the numerical results). %tr differs from ordinary exponents in t h a t its value varies with the topology of the surface.

7g

SAL C~tterMt ot M. / NunwH~M ~:tu~(v of ["iehl Theories Coupled to 2d Quantum Gravity

T h e r e is a simple algorithm for building tria ulati(ms of surfaces with arbitrary topology. , ,w topology is defined by t , e genus (numb,,r of haydh~) or by the Enler ch,tracteristie "~=N-l+t

where 1 ~s ~he n a m b e r of links a v l I is the number of trlaugh~. For it triangulation, t and I arc related :It = 21 O n ' can place a bou~;d on tile nnmber ofti'~des netea.',ary to triangulate a surface of a given genus by using one more piece of information: the fret that there are a limited number of links that, can be drawn between a given number of nodes. t <_ ,v(~'

-

t h a t consistent results for % t , could be obtained over a finite region centred a b o u t the peak in the specilic heat. 'lb explore the first o f these questions we computed expectation values of the m e a n squared c u r v a t u r e on lattices from N = 72 to N = 324 fur couplings a in the range 0.2 to 2.0. T h e calculations revolved locating the peak in the specific heat for the largest lattice available and r u n n i n g at that coupling A for all the smaller lattices. Typically several million sweeps were .,~eded to obtain satisfactory error estimates on the largest lattice.

i)/~

The resul~iag bound o~ N is

,~

%

- 2.U

J *t : l . { )

For instau
5" ,,

112

'/.V F'ig.O. o j r 2 vs I/P.'. g = O

Fig. 9 illustrates tile results for spherical topology. We have plottea o f r 2 against I / N and att e m p t e d to fit the large N region by a straight line. Clearly for o ~. 1.0 the a s y m p t o t i c regime sets in for N > 144. F u r t h e r m o r e the limitiag gradi,u,ts (which yield e s t i m a t e s for "fttr - 2) are "q)proximately independent of a . O u r c u r r e n t best e s t i m a t e for the exponent ~'str is "r,,,

-" 2 =

-1,4(2)

X 2

-

9.3

77

S.M. Catterall el at. ] Numerical Study of Field Theories C o u p / e d to 2d Q u a n t u m Gra;'il.v

T h e finite size behaviour a p p a r e n t in our d a t a would clearly tavour the use of smaller o. IIowever the signal to noise ratio increases for small er and sets a practical limit on the smallest o we can use. ht practice we therefore adopted c~ = 0.2 for our runs on higher genus.

A !1 :

•** ~] m°. • ¢¢ ¢

ff : ( ,

Ac

*

IO

,

,

¢

,

,

J

,

,

I/N

¢

Fig. I 1. Grnus dependence of "V*tf

¢

t

ag

deed found a local observable (r~) w}fieh is sen-

¢

sitiv,' to tile global topology. T h i s is evidenced

o

oooe

o~tt

z/.~"0(a~

ooto

oo14

ool~

~0tfs oo:20

Fig. I0. Variation of Ac

'l'o explore the dependence on extrinsic curwt.ure couplhlg we ran at ,~ = 1.0 for A = A~ + bA = 1.05 + 0.05. T h e plot in fig. 10 i t lustrates this for the sphere. T h e qualitative behaviour is tile s a m e although the .-stimate for % t r - - 2 = --1.1(2) is s o m e w h a t difft.re~L from the value at A, (although compatible within errors). T h i s check then provides ,some reassurance t h a t our resu!t~ ~rc not Loo dependent on the location (,f s~me precise critical coupling. '10 address tile genus dependence of 7str we are carrying out. simulations on lattices with the topology of the one, two and three tori. T h e bulk of the~e results will be shown elsewhere [24]. T h e d a t a in fig 11 show ~ J ' r '~ versus I / N for the sphere and a genus ten surface. T h e first point to note is t h a t it appears that we have in-

by the estimates for 7str derived by least square fits to the d a t a at large N . T h e s e show a clear increase of'~str with genus 9. Clearly, the calculation described here is only exploratory, hut it is perhaps significant t h a t our e s t i m a t e for %tr OU the sphere a p p e a r s r a t h e r close to the branched p o l y m e r value of Vmtr "= 1/2. It can be shown t h a t this n u m b e r is essentially an upper b o u n d for the e x p o n e n t in any surface model which is not 'super.rigid'. Notice t h a t this does not imply triviality of the model &s the correspondieg mass gap exponent is almost certainly not mean-field.

ISING M O D E L C O U P L E D T O G R A V I T Y

lutroduction and Model T h e previous discussions have focussed on the use of r a n d o m suff~ces as a discrete representation of certain hosonie string theories out of t h e

80

S.M. CatletMI et al / NumedcM Study of Field Theories Coupled to ~d Quantum C.ravity

critical dinlenshm

It i~ natural to t r y Io extend

an exact solution, :t serves t~ an important test-

Ihis Io the c';~, (if I'or~llionic strings, i.e. to add anticommnting d,'gr, ,.s of fre,,doni to the discretised world sllt','t. It is well known that the lsing ino&,l cm an arbitrary random graph is cqulvah'nt in the critical legion Io a MMorana fernfion [25]. Is'f. a~e thus molivated to consider tlw partition flnlctk)n el'an Islllg model coupled to (disercti"l quantlnn gravity

ing ground for our nunmfical hchniques. Specifically, how well do our Monte Carlo simulations,

Z : Z

Z

<'(J'//>

GN ¢,

s : .I ~<,,<,, + n E " ' (,JI ~;'¢ .,hall Le working on tile dlial graph to a triangulation -- a Ca graph Gr," and dealing with a ruicrocanonical en.~nible where the number of w.rliccs %' is hehl fixed. Ily allowing two loops hi a graph to Ilaxe ;it. lilo:~f, o111. lillk in ¢onnnoil W(! have (linlinat~.d all ,ivgeneracies associated with tadpoles ai,d :selL,.nerg2' insertions. We further coli[iiie :alr~eiv, s I.) graphs with the topology of thc Spill ,¢..,l'liri- o l r llrillw inters'st will be a niinlerical i;il'el.~llrl>ilh,lll ~[ ihe critical oXpOli,'nls, which an' illdellend( ill of gl'llllS+ this is llOt all hUlloriani rC~lhCiioli. A shnple MPiropolis algorithin is use(i to inll,lenlei!t changes in the k i n g sliin:~ alld local reconnectioll:; of *.he #3 graph. The model ha.¢ bt-en solved analytic,d y by e):ploiting its equivalence to a la%e N matrix niodel [26]. The system undergoes a third or(h'r I,liasl' lrali~i! ;'.)n (in zero liehl It ) Iwt'~een a low and high temperature (small J) i,tja.sc with a set of c:'itical exponents whicil differ markedly from their valu"s on a fixed ]alLice. It came as a surpri~, therefore, when a niinierleal study of this nlod,'l hy ( ;toss at al. [20] producl'd only the fixed laitlce i*Xllillll.l~t.~, ll,,,w for ill this c a ~ the anlhot~ ~i~d a Ilegg~, aPl)rOi,,'~ for siinulaiiag the gravity sector. Since this model is ol,e of the few which admit

which a t t e m p t to estimate the sunu.aL!on over gral)hs, produce exponents in agreement with the analytical prediction ? The results of such a study allow us to tune the algorithms and give us confidence in interpreting the numerical resuhs from ollwr models where analytic solution is not no¢~ihle Borrowhlg traditional ideas from statistical mechanics we will primarily be interested in studying the finite size behavi.~ar of various qua,lilies close to the infinite lattice critical coupling J¢ = iI In ( ~1os~ ' ) " A precise knowledge of J< eliminaU.'s a huge source of error in the analysis of flw finit:~ lattice data. One further difficulty arises in applying Hwse s t a n d a r d techniques; the nece~;sity of extracting a lillear scale L for tlie~c dynanlical sy.aenls. This we do by setting L~A'~ The exlionont d is an intrinsic llausdorff dinwnsion characterising the inlernal geometry of the. ~,lal,h. "['he [inite size scaling ansatz n.quires any thermodylianlic function Q, mer~;ured on a finite laliicc close to criticality, to be giwm by some scahng form [18]

QO,, N I

= N'~f

(¢N~')

The qnanlity ( = [I - J , / J I measures the deviaiioi, i~wa~ h u , . Ihc iiJhlite lattice critical point. The ar~ n~nent of the function f is essentially the ratio of the correlation length to the linear size of the sysl('lli. The expolicnts x and p are dehaed from th,. infinite lattice singularities o f Q and the correlalion lenglh ~. Q ( J ) ~ (a - J < ) - " F, (J) ~ (J - J.)-"

S.M. Catle~all el at. / Numerical Study of Field Theories Coupled to 2d Quantum Gravity

Noti~:c that the ratio ~ can he measured by setting + = 0 i.e. J ~= Jc and analysing the Ndependence o! Q

Q(Jc,N)~I(O)N~ In order that we recover the infinite lattice result Q ~ ¢ - t as N - - oo, the scaling function f (z) must have the asymptotic behaviour f(=)

~

:--

81

up to 3 x 10z vector sweeps (a vector sweep in this coutext a t t e m p t s to update 20 independent lattice sites and 20 links). Errors are estimated by the usual b i , u i n g procedure where we demand a plateau in the 2 0 - 4 0 bin region and the susceptibility to agree with its binned average within one s t a n d a r d deviation. As we have argued, ratios of exponents can be extracted from the asymptotic N-dependence of observables at the critical cow piing Jc.

Thus a measurement of the limiting behaviour of the scaling function also allows a determination of critical exponents. We have considered two primary observables, the magnetisatlon M with exponent 3 and the snsceptibili*y X ~ith corresponding expc+.ent "7.

M--~?

i,

M~(J-J,.) ~ j./

-

(,w>-),

x

(s-s+)-~

Using the results of the previous discu~ion we have attempted to derive estimates for ratios of the criticalexponents ~ and b~d.To help reduce some of the tnnnelling effectson finitelatticesfor J > J~ we have replaced A4 by IMI. In addition we a~v~ studied the magnetic expo,i,:nt b defined tl~rough ~

=

t~'"9

\(s

_

s.)}

If we run at J = Jc then 6 can be extracted from

M ~tt ~t Result A. Finite SJze Stalin9 aud Critical Indices We have run on lattices with sizes K = 200 up to a +,~ximum of N = 3C00 sites, acc,nnulating

In N Fig. 12. Spin Susceptibility

We illustrate this in fig. 12 with a plot of In X versus In N at J = Jc. It. is clear t h a t the sealing regime oldy .sets ill at N = 1000, and a fit to tits last four d~ta points gives us the following estimate

7----= 0.6(I), X2 = 0.17 vd This is in good agreement with the exact result ~ = 2/3 and essentially excludes the fixed lattice number ~ = 7/8 at three standard deviations. Tile exponent ~ de.~cribing the eritkal behaviour of the magnetisaticn is, in F~inciple, more difficult to extract, the singular terms vanishing a.s N - - c o .

S.M. Catterall et M. / i!urnerlcM Study of Field Theories Coupled to 2d Quttt.lum Gravit3

with J = J¢ and the addition of a small ~,taguetic field H . At large m a g n e t i c field t | e sys-

_= +': ÷

+-..

.

.

.

.

.

. In .\"

Fig. 13 Mttgnetit,ation

tem in nnoved out, of the critical region and the magnetisation saturates. Conversely :i' the field t l is too small finite-size effects d ~ m i n a t e and again the magnetisation a p p r o a r h e s a plateau. Thus the scaling region corres!)onds to intermediate fieMs where In M should have a linear depewiance on In H. Fon N = 2tl0 we .see such a window for 0.007.5 < 1! < 0:915. A f t gives us an e s t i m a t e ~ = 0.14(2), with a X ~ = 0,5. In principle, this scaling window should broaden on larger lattices and indeed the d a t a for N = 500 illustrale this with the fitti,ag region extending from I! = 0.01 dowu to t! = 0.002. A linear fit. yields } = 0 1 6 ( 1 / , with a X~' = 1.0.

,++l ~ N - 3

N,',ertheless our log-log I, Io¢ of M against N (:ig. 13) admits a reasonable liJ cr fit ou the 6 i,ojnts and gives

wl

=0.16(U,

~7

i

X"~ = 0 3 3

Wlnlst this number preci~,qy coipehto~+ with tho e~a-' result for a dynamical lat'Ace ~ = 1/6, fi, ring on s-bsets of the d a t a points yield numh,.rs d;7,cr;n¢ by several s t a n d a r d devlations. 1,tis is caused, at !ea.~t !2 part, by the onset of critical slowing down on the larger iaidre~ Therefore the eff'(ctive error is vrobably ta~ger the,. th:~t quoted a!,¢~ ,. l~¢,wcvcr, it is certainly w,ry dilficult to come up 'srith a fit which Is clo:,c to to the fixed (reguiar) lattice result of -P- = 1/16. Thus our data, at the very teast, exclude such a result and +~avour the dynamical : + + ' U [ ' ~ l i t • llutl, t.,~ dwse two results are in agree-

N = tO00

+ +

•

•

I

In It

Fig. 14. Ma~neti~atk,n ~ Fieki q his trend continaes at 5/ = 1000 wh,'.e our d a t a show (fig. 14), very co,vincingly, a linear r,,ginie extending to small nia&netic field. A fi~

r

neut with an e;,rller ~t.,ly [27] on triangulated ,attices. ~O llleaksute the exponent, 3 wc rail o11 three lattic~ sizes N -- 9.00, N = 500, and N = 1000

her,, giv,'s us our I,,:st e~timste for the exponent 1 -

= 0.18(2),

,¢~ = 0.09

] hic is ial excellent a g r e e m e n t with the

aria-

S.M. Catteran et al. / Numerical Study or F;eld Theories ( 'oupled to 2d Quantum Gravity

lyti~ :esult 1[/~ = 1/5 and the trend with N shows no sign o[ runniug tu the fixed lattice resuit lfli ---- t/15. Our results thus are g,merally in good agreement with the analytical results for m o d d s with dynamical counectivity and specifically rule out the fixed lattice exponents. This provides some reassurance t h a t the numerical techniques for summing over t r i ~ g u l a t i o n s are satisfactory The h c t that scaling sets in for N ~ 1000 is encouraging for the intcrpretation of results from such modcls ,~vith extrinsic curvature and is in s t a r k contrast to the case of pure two dimensional gra~il.y, where it is claimed [28] ~.hat far larger systems are needed to probe the continuum behavioar. B. Scaling Yunclions and Eq,~tfar of State

83

z = t N ~ for N = 500, N = 1000, N = 15fi0, assuming the exponents vd, 7 take the dynamical lattice values. The collapse of the d a t a onto a singh, curve is indicative of scaling. The slight deviations visible for tile N = 500 d a t a at small z are compatible with the our earlier conclusions - - t h a t the scaling regime for X sets in around N ~ 1000. The sca~ing function f 0e) is expected to behave ~ y m p t o t i c a l l y as f ',~.) ~ ~.-~

In p r i n d p ~ we c ~ . ~xtract an estimate for 3' by exat~fiuhlg ~hc gradient of the plot at large z. The linear fit shown in tile figure yields 7 = 1.4~1) witl~ a :~.2 = 3.0, which is somewhat different frou~a I I . analytic prediction 7 - 2. However, wil'a tile lattice sizes wc have considered, we are only able to go to = ~ 3.0, which is too small to really probe the limiti~tg behaviour. The num, dc,~l evide~ce points to an increase in 7 with lattice size, but dearly from this plot alone one cannot distinguisb the fixed (7 = 7/4) from dynautical exponent.

,1 > J,. i

7 ,. ill X

""

,.

j
Fig. 15. beating Plot for X 7-I To a t t e m r t a ~df-eonsisteat check on our resuits we have constructed finite-size sealing ;)lots for both '.he susceptibility and n~agnet]satiG~t. In fig. 1-~ we plot ! n ( X ~ - ~ ) against In(~) with

-a

-t5

i

-05

o lnx

Fig. 16. Scaling Plot fol M

s

S.M. (TaticrMI et al. / Numcricad Sicdy of Field Thec ri,.~ "'espied to 2d Quill ,uti'l "~ravity

A slt,lila, plot for fit," lil:~giwtisatioa hi /x[71l~) against hi (z) i~ shown in fig. 16. The two eurveo corrvspond to the cases J < J, (lower curve) and J > .L (upper curve). Again, as we have obs,.rved earlier, a scaling b e h a t i o u r for the nlag-

~L

netisation seems to set in on lattices with oniy N = S00 nodes (at lea.st in tile high t e m p e r a t u r e J < #, r,s;i,m). A linear fil of the* ,,t,per branch h'om .r ~ 1.0 troy ards aliowq a Im , lllt,illelil o f //i.e

i

;i" 2O

The N = 10fl0 d a t a yield # = 0 2:',(1 H w;,il a ~:~ -.: 9-8. Purthernlote tile fits Co," t'):, exlloilPnl yield ~y'~ielnatiraily large'i values a., 7 .,crea_~s. 'rht~% althoilgh wl, caniiot reach Jars, eliOUgll z where ill," tits beconie stable, the trend in oiil

data would favour the dynamical exponeut /~ z~ 1/2 rather than its llilich sn,allor value tJit th," ti×ed lattice sJ ~ 118. 'li~ ~lilllillalise, we sl'~ g o ) d i.vidence lop sealhis iii ihe Ihiii~-siz~ scaliliT, plots a . d ills to the I~ifgl' $' I'i'~llill' yi,'hl l e r l l i 'r eslilliat~'s lot Ill(! ,'x. t . m e . t s ~t aml "r which ~re compatihle with t l , exact resuhs. We have also att,mlptt:l Io detemliue die equat ion of state

, (,w) <-37 --- ,i, \ ,--/) Using a lattice with A' = 1090 node..', we determined the maglletisation ;il a scaling %glen corresl, olldhlg to small deviations away f r e t . the critical pohll, ~ = It07 - 0 12 and ~;i.all inagnet~fiehls II : O 0 0 2 - 0 , 0 0 5 . A lilot o f ~ . ~Lgaiqst the scah'd Illagli 'lisalioli 7M ( I l l 17) sholts that the data ,ollap.~es oilto a smooth curve which con~;I illltes a iioP,-pettu/balive lne~slllement of the filliet lOtl Pp.

Pil.

....4O ,-. *.,, /#/+ a, I / , i'~qll~tiOli

"

I ~O

el $liite

('. /u/+rna/ (;tometr!l Finally we have attempted i o extrael an estinh'llc for *.lit' correllltion length exllonent / / b y a • Ii,,, ;.......... i, .,lent o f ! k " c.3;,,,
w<. can :onslrnct ,~ ,nap of lhe intrinsic distance of all pOliils oil ally i,aHi,-ular graph from sortie rcfer,.iwe site. Each l a l t i : , liid~ is a.~,slgned unit lengll; and the distan¢+' bLt.ween two sites defined ~s the m i n h n a l w,~lk ou lhe gralib which cOillleels tilt" two sites. With this ilfformation we tht.ii forni an e s t i m a l e for th,, ¢orrelator G ( r ) by averaging over the en.%,ulhle of graphs and spin colJfiguratiolr~ ;~, follows

n(r) = ~-~J, (,l(i)- r) i

T h e w, ci ,r d ( i ) contains the disla.'lce ou the graph from site ," to the origin i0. T h e mass gap I.~:dcliw,d froin the a.symptotlc behaviovr of G (;(r~exp(-m(d)r),

~ --oo

S.M. Carler~ll et M. / Ntmwrlc, J .';t ~dr or" Fichl Theories C!oupled t o 2 d @ u n n t u m Grnvi~v

85

_o I

•

./= •

0 6 5 , V = 10(]0

.

N = 1000

.

•

o;

1

;L-

I

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

.

. . . . . . .

.....

r

:,

,;,,

,f

,;.

[

F i g . ! ~. C o r r e l a l i o n i " t m c t i e n J = 0.65

Fig. 19. M~s~ q a p vs J

A I~',,icvl ~raph of the log of ~.he '.orrelation function at J = 0.65 on ~ 1000 r o d e lattice is stlown m lig. 18. The linearity of the plot out to lattice distances of order 15 (v:here the signal is lost in the noise) indicaL~ the dominm:t contribution of a single state. The fitted masses arc plotted out as a functic,n of J in fig. 19. A lit of Otis d a t a to tire form a + b ( J c - J)~" yields l]~e estimat,'s a = 0.11)4(9), b --- 1.9(2), u = 1.15(8) with a x ~ = 1.6 Our d a t a i~ thus compatible with an exponent of u,,ity, .~i,nilat t~ ~lla, observed on a fixed lattice. Theory only provides us with the product ud = 3, so our d a t a would favour an intern:d Ila, sdorff dimensi,~n d ~ 7:, (n¢~liee that tb,ts i5 ~t purely intrinsic quantity, independenl of any 'external' llausdorff dimension associated with emhedding the string in some EueIideau space). Ih)wever it is possible, in principle, to measure d dir~,ctly I;y compLiAnt the nun~ber of lattice sites n.~ a flmction of drstance. Unlbrtunately we found that d was essentially independent of the lsing coupling J and close to its classical value d = 2. 'We interpret this as evideuce t h a t the lattices

in this sludy were simply t.Jo small to allow for the direct ineasurenlent of such non-local runetrlOn.~ Of tire geometry as the llausdorff dh ionsion (See, for example [2g]). Conversely. critical eXl)onenls for th~ lsing sector, being deternriued h~ the physics at small scales, m~y be extracted much n~ore cea(lily. S s t u l e a|'!l

In this section we have presented results of tqnite-size scaling studies of the lsing mo,!el on dynamical ~3-graphs !29]. The d a t a s~ow .~trong evidence for scaling on nloder~te latti,~e size~ (N ~ 1000), and yield estim~.~es for varrous critical exponents which ~re consistent ~, '~h the ~c-. ~nlts of inatrix rood/; calculations. ~)ur -results rule out tire fixco ~at.lce exponents which have I)~el, f~tvv red by a recent numerical study [20]. ~ Irave, in ,tddition, determined the equation of .~tate for the system and analysed the intrinsic correlations of Ising spins on the graph. T h e latter lead.~ to an estimate for the correlation exponeilt Av.

;.~,1. (%|ferMI et M. / NtmlerlcM Study of Pith/ "J'hc~uies (oupled to 2¢1 Quantum Gt wiry

8{;

Th,'se ~ot~elusions ar{' i m p o r t a n t for tile inh'rI)rela! [¢)1[ of o t h e r nanierical results concerning matt,,r fi,qds (perhaps ~H~ c > 1 ) coupled to two dinwnsoualquantumgra~ity It w o u h l b e l . t e r usting t<, extend this aualysis to models inrorp o r a | i a g fl~rtb,'r Ising s~)ecies, in a~} effort Io g o ¢ontiaut:asly through the so-called c = 1 t arrier. It would t~e Ifigldy instractivc to find "~ s t r o n g signal of ~,~v]e sort of palhological behaviour ,x~ the nmnber of Is|rig species is ihcr,'a.s+'d fronl two 1o Ihr,'e. Such a signal has heen conspicuously lacking it; the g a u s s | a n model., which haw~ been considered previotls/y m'e e.g. [2i

A r k no'c,'lodgctn<' nt s 'l'hts ~+'ork w~.s stJpport,'d by NSF granl I+IIY 87-111775. T h e nunteri,-al calculations werl pcrforlru.d using t h e resources of the' P i t t s b u r g h Superco;lll~l:h'r ('"lllr,- an,l the Amos (',.lllr,'.

E,.~arch

Also w(' acknowledge Na/io:tal Scil'lwe

Folindalioli sUi~l,orl through tlw Materials Hc~;,,arci! i,;d .,ralory ,tt the |l.iv,',~ity of Illiflois,

I'rbaila ('haml,,t~gt~. ~r.tlll NSI"-I)M|;..0-20;338. W~' thank A. Migdal for discussions a h . u t x,.ctoiisation and for directi:~g our at, te~itiol; to th,' u.'.' of [he dual lattice

refcrt~nta.s [11 .I ~t,t,j~,rn,H.l)urhuusandJ.FrOdlch, Ni-cI. Phys. ;S,~Y*7,433, 1985. f'Ti F. David, Nud Phys, H257, 5,13, in85. [:]] I)r B,,ulatov, V. K~.zmk{}v, l. Kr,shv all(I A. Mig, h,h '~,. I ],IL~ " "':, (;41, I!}g4L [.I} A, Hilb,i~',. ttnd F David, Nu,'[ ['hys, 8275, ,;I'7, 19gl;,

[5] ?. A,nI,jorn, H I)urhuus, a.,| J. I"rohlid,. Nu, I. P!,ys. H275, 161, 198~L ['3] J. Juikiewic~, A. Kr'zywicki and B. Peterss~L Ph}s. Left. 1"1177,89, 1986.

[7] F. David, J. Jurkiewicz, A. KrLywlckl trod H . Pelersmm, Nu¢l. Phyl. B290, 218, 1987'. [,~] J. Ambjorn, Ph De' ForY.ratld, P. KoukJou and D, |)etrltls, Phys. bert. BI97. 5,18, 1987. [!~] J. Ambjorn and B. Durhuus, Phys. Lett. BI88, 253, 1987. [10] A. l'olyakov, Nucl. Phys. B26~, 106, 1986. [I i| I). W[esmann, Nu -I Phys 1:1323.330, 1989. [12] S. Catterall, Phya. Let|. B220, 207. 1989. [131 C. HailliP, D. Johnston. and FL Williams, N u t . Phys. B.~5, 469, IDa. [I,1] IL Rcld~en and J. Kogut, Nud. Phys. 1:3348, 543 1091. [15] C. Baillie, S. ~'atterall, D, Johns,c.n ~ld R. Wi||iams, Nucl. Phys. B348, 543, 1991. [16] S. Catterall. Phys. le~t, H243, 358, 10130. [!7] J. ambjorn, .I. Jurkiewicz. S, Varsted, A. Irback, B. P~tersson, 'Critical Properties of tae O ~ ' n a m i vzJ J{al.l,~t. Surf~'e wlth Extrinsic CJ~rvattwe' NklI|11'~-91- 1,I. |Ill| |'~. [)*~ll~a.y, K. Moll, (;. Chester and M. WJaller, Phys. |[ev. HI2, 5025, 1975. [1!)} S. (!all,,radl, |). |£|s~nslein, J. S.ogut, R. 11enken, "Cluniplhlg on Dynmnical ~'a-Gr;Lphs', ILL-{ rH)91-15, r,¢cPpted for publication Nucl. P,,ys. B. [20] M, (;r,~s and ]l. Ihunhcr, 'Critical Propertles of 2d Simpli,'iM (~i.%l|tillln Gravily', UC|-90-33. [,~1] V. I'~ni~hnik, A. Polyakov and A. 2amolodchikov, M,.I. Phys, l.ett. A3, all9, 1988. [22] I. I)istl,,t aald H. I'[awaJ, Nucl. Phys. R3~I, 509, 198~). [231 It. I{,.nk,.n ~uld I. K,)gut, Nud. Ph','s. B348, 580, P.JS~I. [21] S. L'~dl,'rMI, l{ |lenken, aaltI J. I(t~ga:t, in prepara~ ||tilt. [251 ,~1. H,.r~ha~l~ky asul A. Mig(lal, Phys. bcte. P,74. 3~3, 1984; 126] D. Boulatov and V. Kazakov, Nuct. Phys. B186, 379, 1~187;Z. ]'lurda ~nd J. Jurkiewlcz, Act,s Phy~ica P.h.fi~a H20, 9-19, If~9. [271 .I. )ulki,'wi~"z, A, Krzywickl, FI, Petel~ol, and H. S,uh'd.'rg, I'hy~. Let|. H213, 511, 1988. {~] M. Agi~hleln. 1.. Jacob,, A. Migdal, J Richardson, ~11"1 lU',';~rint c ' r P 1840, ,M~th IreS. [29] S. Calte,~ll, J. Kogut, FL I)enkc;~, 'Scaling Behavi,;tir {~fthe Is|rig Model Coupled to 24 Quantum C,avily', ILl- {'r|1}-91-19.