J. Ambjorn et al- Effective sampling of random surfaces by baby universe surgery

8/3/2019 J. Ambjorn et al- Effective sampling of random surfaces by baby universe surgery

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E L S E V I E R

21 Apr i l 1994

P h y s i c s L e t t e r s B 3 2 5 ( 1 9 9 4 ) 3 3 7 - 3 4 6

PHYSICS LETTERS B

Effective sampling o f random surfaces by bab y universe surgeryJ . Am bjCr n

The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen ~ , DenmarkP . B i a / a s

Inst. of Comp. Science, Jagellonian University, ul. Nawojki 11, PL 30-072 Krakdw, PolandJ . J u r k i ewi czInst. of Phys., Jagellonian U niversity., ul. Reymo nta 4, PL- 30 05 9 Krak6w 16, Poland

Z . B u r d a 1 ,2 , B . P e t e r s s o nFakultiit fii r Physik, Universitdt Bielefeld, Pos(fach 10 O1 31, Bielefeld 335 01, Germ any

Rec e ived 25 January 1994 ; rev ised manuscr ip t rece ived 22 February 1994Edi to r : P .V. Land shof f

A b s t r a c tW e prop ose a new, very efficient algor i thm for sam pl ing of random surfaces in the M onte Car lo s imulat ions, based onso-cal led baby u niverse surgery, i .e . cut t ing and past ing of baby universes. I t drast ical ly reduce s s lowing dow n as com pare dto the s tandard local f l ip algor ithm, thereby al lowing simulat ions of large rando m surfaces coup led to mat ter f ields. A s anexam ple w e invest igate the eff iciency of the algor i thm for 2d s impl icial gravi ty in teracting wi th a one -com pon ent f ree scalarf ield . The radius o f gyrat ion is the s lowest m ode in the s tandard local f l ip /shi f t algori thm. Th e use of baby universe s urgerydecre ases the autoc orrelat ion t ime by three orde r of magni tude for a rando m surface of 0.5 105 tr iangles, whe re i t i s foundto be " l i n t -~ 150 "l- 31 swe eps.

1 . I n t r o d u c t i o nR a n d o m s u r f a c e s p l a y a n i m p o r t a n t r o l e i n m a n y

b r a n c h e s o f p h y s i c s . T h e y n a t u r a ll y a p p e a r i n c o n t e x to f m e m b r a n e s , s t r i n g th e o r y , 2 d g r a v it y , Q C D s t ri n g s ,d y n a m i c s o f N i e l s e n - O l e s e n v o rt e x , 3 d I s in g m o d e la n d m a n y o t h e r f ie l d s. T h e b a s i c c o n c e p t i n th e t h e o r yo f r a n d o m s u r f a c e s i s th e m e a s u r e o f i n t e g r a ti o n o v e rg e o m e t r i e s . T w o s u c c e s s f u l m e t h o d s o f f u n c t i o n a l in -1 A f e l l o w o f t h e A l e x a n d e r v o n H u m b o l d t F o u n d a ti o n .2 Perm anen t address : Ins t . o f Phys . , Jage l lon ian Univers i ty . , u l .R e y m o n t a 4 , P L - 3 0 0 5 9 , K r a k 6 w 1 6 , P o l a n d .

t e g r a t io n o v e r g e o m e t r i e s o f r a n d o m s u r f a c e s e x i s t:t h e c o n t i n u u m a p p r o a c h p r o p o s e d b y P o l y a k o v [ 1 ]w h i c h l e a d s t o t h e L i o u v i l l e t h e o r y , a n d t h e d i s c r e t ea p p r o a ch , b a s e d o n d y n a m i c a l t r ia n g u la t io n s [ 5 - 7 ] .T h e f o r m e r i s s o l v e d u s i n g c o n f o r m a l f i e ld t h e o r y [ 2 -4 ] . I t g iv e s p r e d i c t i o n s f o r t h e c r it i c a l e x p o n e n t y o ft h e e n t r o p y o f s u r f ac e s , e m b e d d e d i n d < 1 d i m e n -s i o n s, o r e q u i v a l e n t l y f o r c o n f o r m a l f i e ld s w i t h a c e n -t ra l c h a r g e c = d m i n i m a l l y c o u p l e d t o g r a v i ty . F o rd i m e n s i o n s d = c > 1 t h e e x p o n e n t 7 , a s w e l l a s th ec r i t i c a l e x p o n e n t f o r t h e m a t t e r f i e l d s e c t o r , g e t s a ni m a g i n a r y p a r t a n d t h e r e s u l t s h a v e n o d i r e c t p h y s i c a l

0 3 7 0 - 2 6 9 3 / 9 4 / $ 0 7 . 0 0 ( ~) 1 9 9 4 E l s e v i e r S c i e nc e B .V . A l l r i g h ts r e s e r v e dS S D I 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 0 2 8 3 - D


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338 J. Am bjorn et al. / Physics Letters B 325 (1994 ) 337-34 6m e a n i n g . I t h a s n o t y e t b e e n u n d e r s t o o d i f t h e b r e a k -d o w n a t c = d = 1 h a s i t ' s o r i g i n i n t h e m e t h o d i t -s e l f o r w h e t h e r i t re f l e ct s s o m e d r a s t ic c h a n g e i n t h es u r f a c e e n tr o p y , w h i c h c a n n o t b e d e s c r i b e d b y t h e e x -p o n e n t ~ [ 8 ] . T h e d i s c r e t e a p p r o a c h a l s o a l l o w s f o ra n a l y t i c a l s o l u t i o n s fo r c _ < 1 , u s i n g t h e e q u i v a l e n c ew i t h m a t r i x m o d e l s . I n a l l s o l v e d c a s e s t h e c r i t i c a l b e -h a v i o u r a g r e e s w i t h t h a t p r e d i c t e d b y t h e c o n t i n u u mc o n f o rm a l f ie l d t h e o ry . F o r c = d > 1 t h e d i s c r e t i z e dm o d e l s c a n n o t y e t b e s o l v e d a n a l y t i c a l l y , b u t t h e y a r ep e r f e c t l y w e l l d e f i n e d a n d o n e c a n s t u d y t h e m o d e l sb y n u m e r i c a l m e t h o d s .

I n f a c t , in t h e p a s t y e a r s m a n y a t t e m p t s h a v e b e e nu n d e r t a k e n t o s i m u l a t e r a n d o m l y t r i a n g u la t e d s u r f a c e sb y c o m p u t e r . I n t h e s t a n d a r d a p p r o a c h t h e c h a n g e i nt h e s u r f a c e g e o m e t r y i s o b t a i n e d w i t h t h e h e l p o f a l o -c a l m o v e ( c a l l e d f l i p ) , d e s c r i b e d fo r i n s t a n c e i n [ 7 ]a n d [ 9 ] . T h e b i g g e s t p r o b l e m c o n n e c t e d w i t h th e n u-m e r i c a l s t u d y o f t h e c r i t i c a l p ro p e r t i e s o f t r i a n g u l a t e ds y s t e m s , p a r t i c u l a r l y fo r d > 1 , i s t h e c r it i c a l s l o w -i n g d o w n , w h i c h r e s t ri c t s th e r a n g e o f s i m u l a t i o n s t ora t h e r s m a l l l a t t i c e s . A l r e a d y fo r l a t t i c e s o f a m o d e r -a t e s i z e i t i s d i f f i c u l t t o t h e rm a l i z e t h e s y s t e m a n d t h e ng e n e r a t e i n d e p e n d e n t s a m p l e s , b y t h e u s e o f t h is s t a n -d a r d a l g o r i t h m . T h e r e a s o n o f t h e s l o w i n g d o w n c a nb e t r a c e d t o t h e f a c t t h a t t h e l o c a l a l g o r i t h m , a l t h o u g he rg o d i c , i s n o t w e l l s u i t e d t o u p d a t e t h e t y p i c a l g e o m -e t r y k n o w n t o b e d o m i n a t e d b y t h e n o n l o c a l s t ru c t u re so f b a b y u n i v e r s e s o r b r a n c h e d p o l y m e r s , f o r w h i c h t h ee n t r o p y c o m i n g f r o m r e a r r a n g i n g w h o l e s u b - u n i v e r s e ss e e m s t o p l a y a n i m p o r t a n t r o l e f o r t h e e f f e c ti v e p ic -t u r e o f g e o m e t r i c a l f l u c tu a t io n s [ 1 0 - 1 9 ] .

S i n c e t h e s t a n d a r d a l g o r i t h m w a s p r o p o s e d [ 7 , 9 ] ,n o t m a n y i m p r o v e m e n t s h a v e b e e n m a d e . Q u a n t i t a -t i v e l y n e w a l g o r i t h m s w e re p ro p o s e d i n [ 1 0 ,1 1 ] . T h ea l g o r i t h m i n [ 1 0 ] g e n e ra t e s i n d e p e n d e n t t r i a n g u l a -t i o ns f o r d = 0 ( p u r e g r a v i t y ) m a k i n g u s e o f t h eD y s o n - S c h w i n g e r e q u a t i o n s w h i c h d e t e r m i n e r e l a -t i o n s b e t w e e n g ra p h s . T h e s e r e l a t i o n s a r e r e w r i t t e ni n M o n t e C a r l o l a n g u a g e a n d u s e d t o s a m p l e s u b -u n i v e r s e s i n a r e c u r s i v e w a y . R o u g h l y s p e a k i n g , t h em e t h o d i s b a s e d o n t h e t h e o r e t i c al i n p u t c o m i n g f r o me x a c t f o r m u l a s f o r t h e d i s t r i b u t i o n n ( A , l ) o f s u b -u n i v e r s e s w i t h a r e a A a n d p e r i m e t e r I . T h e r e l a t i o n sb e t w e e n n ( A , l ) a re k n o w n o n l y f o r p u r e g r a v i t y a n dt h e m e t h o d i s l i m i t e d t o t h i s c a s e . S i m i l a r r e m a r k sa re v a l i d fo r t h e a l g o r i t h m s u g g e s t e d i n [ 1 1 ] . I t i ss p e c i f ic to d = - 2 .

U s i n g " c l u s t e r " a l g o r i t h m s [ 1 2 ] c a n r e d u c e v e r ym u c h t h e c r it ic a l s l o w i n g d o w n i n t h e u p d a t i n g o f t h em a t t e r f i e l d s e c t o r . T h i s a l g o r i t h m s w e re s u c c e s fu l l yu s e d t o s t u d y s p i n s y s t e m s i n t e r a c ti n g w i t h t w o - a n df o u r - d i m e n s i o n a l s i m p l i c i al g r a v it y [ 1 3 , 1 4 ] . I n [ 1 5 ]a n a l g o r i t h m c a l l e d " v a l l e y s - t o - m o u n t a i n s r e f le c t i o n s "w a s p r o p o s e d t o r e d u c e t h e l a r g e c o r r e l a t i on t i m e i n t h eu p d a t i n g o f t h e c o n t i n u o u s m a t t e r f i e l d v a r ia b l e s . T h i sa l g o r i t h m w a s u s e d i n [ 1 6 ] t o s t u d y t h e c a s e d = 1 . A sw a s c l a i m e d t h e r e t h e a u t o c o r r e l at i o n t i m e s w e r e o f t h eo r d e r o f 3 0 0 s w e e p s f o r a s y s t e m w i t h 3 0 . 0 0 0 t r i a n g le s .I t i s n o t c o m p l e t e l y c l e ar t o u s h o w t h i s a n a ly s i s w a sm a d e . O u r e x p e r i e n c e w i t h " c l u s t e r " a l g o r i th m s s h o w st h a t e v e n a v e ry f a s t a l g o r i t h m i n t h e m a t t e r s e c t o rd o e s n o t h e l p t o i m p r o v e t h e u p d a t i n g o f g e o m e t r y a n di n e f f e c t w e w o u l d e x p e c t t h e a u t o c o r r e l a t i o n t i m e i nt h i s c a s e t o b e m u c h l o n g e r .

In th i s le t t e r w e p ro p o s e a n e w , v e ry g e n e ra l a n d e f -f i c i e n t u p d a t i n g s c h e m e . In t h i s a l g o r i t h m , a p a r t f ro mt h e s t a n d a rd s w e e p s o f t h e l a t t i c e , u s i n g a l o c a l f l i pm o v e , w e i n t r o d u c e a n e w t y p e o f m o v e , w h i c h w e c a l la b i g m o v e o r b a b y u n i v e r s e s u r g e r y . T h e b i g m o v ei s a g e n e r a li z a ti o n o f t h e A l e x a n d e r m o v e s [ 1 7 ] , d i s -c u s s e d r e c e n t l y i n t h e c o n t e x t o f h i g h e r d i m e n s i o n a ls i m p l i c i a l g r a v i t y [ 2 3 ] . T h e i d e a i s t o i n t ro d u c e l a rg ec h a n g e s i n t h e g e o m e t r y , t y p i c a l f o r s t r u c t u re s r e s e m -b l i ng b r a n c h e d p o l y m e r s , w h i c h a t t h e s a m e t i m e h a v el a rg e a c c e p t a n c e r a t e .

A s w e w i l l s h o w , t h e a u t o c o r r e l a t i o n t i m e fo r t h en e w u p d a t e i s d r a m a t i c a l ly d e c r e a s e d c o m p a r e d t o t h es t a n d a rd , l o c a l a l g o r i t h m . F o r a l a t t i c e w i t h 0 . 5 1 05t r i a n g l e s w e fo u n d a n a u t o c o r r e l a t i o n t i m e o f 1 5 0 d: 3 1s w e e p s f o r t h e sl o w e s t m o d e . T h i s s h o u l d b e c o m p a r e dw i t h t h e c o r r e l a t i o n t i m e o b t a i n e d u s i n g t h e s t a n d a rda l g o r it h m , w h e r e w e a l r e a d y f o r a l a tt i c e w i t h 2 3 9 6t r ia n g l e s o b s e r v e d a n a u t o c o r r e l a ti o n t i m e o f 1 9 0 0 5 1 2 .

2. AlgorithmR e c e n t l y , t h e f r a c t a l s t ru c t u re o f 2 d g ra v i t y w a s d e -

s c r i b e d b y t h e d i s t r i b u t i o n o f s o -c a l l e d b a b y u n i v e r s e s[ 1 8 ] , w h i c h a r e s u b -u n i v e r s e s w i t h r e l a t i v e l y l a rg ea re a A , c o n n e c t e d w i t h t h e m o t h e r u n i v e r s e t h r o u g hl o o p s w i t h s m a l l p e r i m e t e r I . T h e i r d i s t r i b u t i o n d e t e r -m i n e s t h e l e a d i n g t e r m i n t h e e n t r o p y o f s u r f a c e s , af a c t w h i c h h a s a l r e a d y b e e n u s e d s u c c e s s fu l l y i n n u -


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J . Ambjc rn e t a l. / Phys ics Le t te rs B 325 (1994) 33 7-34 6 33 9

F ig . 1 . The b i g m o v e o f t h e s u r f a c e . T h e s i t u at i o n b e f o re a n d a f t e r t h e m o v e i s d i s p l a y e d .m e r i c a l s i m u l a t i o n s t o d e t e r m i n e y f o r s o m e q u a n t u mg r a v i t y m o d e l s [ 2 4 - 2 6 ] . T h e s e s u b - u n i v e r s e s w i l l b et h e m a i n o b j e c t i n o u r u p d a t i n g s c h e m e . M o r e p r e -c i s e ly , w e s h a l l c o n c e n t r a t e o n t h e m i n i m a l n e c k b a b yu n i v e r s e s , c a l l e d minbus.

L e t u s f i r s t d e s c r i b e t h e big move a l g o r i t h m f o r t h ec a s e o f p u r e g r a v i t y a n d f o r s u r f a c e s w i t h a s p h e r i c a lt o p o l o g y . In t h e f i r s t s t e p o f t h e a l g o r i t h m , w e f i n da m i n i m a l n e c k o n a s u r f a c e , i . e . a l o o p w h i c h h a st h r e e l i n k s a n d d i v i d e s t h e s u r f a c e i n t o t w o p a r ts . T h es m a l l e r p a r t w i l l b e c a l l e d minbu a n d i s t h e m a i n o b j e c t

i n o u r u p d a t i n g s c h e m e . T h e l a r g e r p a r t w i l l b e c a l l e dth e mother universe. The m i n i m a l n e c k i s a t r i a n g l e ,w h i c h d o e s n o t b e l o n g to t h e s u r f a c e ( s e e F i g . 1 ) . I f w ec u t t h e s u r f a c e a c ro s s t h i s n e c k a n d f i l l t h e t r i a n g u l a rh o l e s o n b o t h s i d e s o f t h e c u t , t h e s p h e re s p l i t s i n t ot w o s u r f a c e s , b o th w i t h a t o p o l o g y o f a s p h e r e . T h e s et w o s u r f a c e s c a n n o w b e g l u e d b a c k i n a d i f f e r e n tw a y . T o d o t h i s w e c h o s e r a n d o m l y o n e t r i a n g l e o ne a c h s u r f a c e a n d r e m o v e t h e s e t r i a n g le s , c h a n g i n g t h et w o s p h e r i c a l s u r f a c e s i n to d i s c s w i t h t r i a n g u l a r e d g e s .T h e s e e d g e s a r e g l u e d t o g e t h e r , f o r m i n g a m i n i m a l


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3 4 0 J . A m b j C r n e t a l . / P h y s i c s L e t t e r s B 3 2 5 ( 1 9 9 4 ) 3 3 7 - 3 4 6

neck of the new surface. This operation, which wedenote b a b y u n i ve r s e s u r g ery , can be performed in sixdifferent ways, depending on the way the vertices ofthe triangles are to be identified. This identification ischosen at random. It can be easily seen that such ab i g m o v e preserves the total area of the surface. Forthe pure gravity case the detailed balance condition isautomatically satisfied, because the move is reversibleand all the triangulations have the same weight. Thearea of the minbu involved in the move can be quitelarge. For surfaces with the polymer structure the movecan be visualized as cutting of a branch and gluing itback in a random way. Performing such a move withthe help of the local f l i p s would require very longcomputation time.

The move we use in this paper is a slightly sim-plified version of the one described above. The sim-plification lies in the fact that we choose a new trian-gle only on the bigger surface and keep the positionof the minimal neck on the m i n b u in the terminol-ogy introduced above unchanged. The b i g m o ves a r esupplemented with the standard sweeps of the lattice,making use of the local f l ips .

The concept of a b i g m o v e can be generalized tothe case of gravity coupled to matter fields. As an ex-ample consider a d-component free bosonic field on arandomly triangulated surface. In the discretized ap-proach we choose a version, where the field is locatedin the middle o f the triangles. The field configuration isrepresented by a set of real numbers x~, /z = 1 . . . d,where i labels the triangles of the surface. The actionof the field isS = 2 E ( x ~ - x~) 2, (1)

ij , z

where the sum runs over all pairs of the neighboringtriangles. In this case, the naive replacement of theposition of the minbu can lead to a large change ofthe action and very small acceptance rate. To over-come this problem, while performing the replacementone has to propose a new x-field configuration. In or-der to maximize the acceptance rate for the transitionone should minimize possible changes of the matterfield. It can be realized by keeping fixed most of thesquares of the relative differences between fields x~on the m i n b u and on the m o t h er u n i ve r s e, so that theonly change of the action comes from the change of

interactions between the x fields nearest to the loopson which the algorithm cuts and pastes a m i n b u .

We propose to change all fields on the minbu byadding a constant shift A ~' to them:x ~ ~ x ~ + A~. (2)

When performing a b i g m o v e , a new field x~ has tobe created in the center of the new triangle T on thebigger surface, obtained in place of the minimal neck.At the same time the field xt in the center of the tri-angle t, which becomes a new minimal neck has todisappear. Altogether there are 2d numbers (d for A ~'and d for x~) to specify completely a transition be-tween the configurations, and therefore it is clear thatthe simple Metropolis question imposed on 2d ran-domly chosen fields, is ruled out, since one would getvery low acceptance. We found several possible solu-tions of how to update the field sector effectively. Themost efficient one, which we describe below, is a ver-sion of the heat-bath algorithm. We make the updatein two steps. First we ask, if the replacement of theminbu and triangle is accepted. We do not specify xrand AU but instead integrate over them. This gives usthe volume of the new available state space. To satisfya detailed balance condition the transition must be re-versible, and therefore this volume has to be comparedwith the analogous volume integrated over xt , Au forthe inverse transition. Denote the volumes for the con-figurations A, B, before and after transition by VA andVn. The detailed balance for this transition reads:V a p ( A ~ B ) = VB p ( B --+ A ) . (3)To compute the volumes, denote the fields around theminimal neck on the m i n b u by zl, z2, z3 and on thebigger surface by yl , y2, y3 (see Fig. 1 ). In the follow-ing the index/x will be omitted. The fields Yl, Y2, Y3interact with the new field xr in the center of the tri-angle, yielding the volume in the state space:I ( yx , y2 , y3 )

3/ d d x T e x p { - - 2 E ( y i - - X T ) 2 }

i= 1= ACexp {6((y2) _ ( y ) 2 ) } . (4 )

Let us denote the fields around the triangle t as xi.After gluing the fields x i interact with their counter-parts zi on the m i n b u . According to our prescription


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J . Amb j#rn e t a l . / Phys ics Le t te rs B 325 (1994) 337-3 46 341t h e w h o l e minbu c a n b e s h i f t e d b y A : Zi ~ z i "~- A.T h e s t a te v o l u m e i s o b t a in e d b y i n t e g r a t i n g o v e r A :J ( X l , X2, X3, El , Z2, Z3 )

3: f dda exp { - 2 Z (x' - z , +i=1= . M ' e x p { 6 ( ( ( x - z ) 2) - ( ( x - z ) ) 2 ) }= I(x l - z i ,x2 - - Z 2 , x 3 - - Z 3 ) . (5)

T h e d e t a i l e d b a l a n c e c o n d i t i o n c a n b e o b t a i n e d b yt h e c o m p a r i s o n o f t h i s m o v e w i t h t h e i n v e r s e m o v e . I th a s t h e f o r m :I ( y ) l ( x - z )p (A -- -+ B ) = # ( x ) I ( y - z ) p ( B - - + A )

( 6 )a n d i s s a t i s f i e d b y t h e p ro b a b i l i t y p (A --+ B) in thef o r m :

I ( y - z ) I I ( x - z ) }p ( A - * B) = m a x 1 , / R 3 - ) 7 ( - 5 / / 2 7 ) 7 ( T )e x p ( 6 ( ( y z } - ( y ) ( z } ) ) }

= m a x 1, e x p ( 6 ( ( x z } ( x } ( z } ) ) '(7)

3 . N u m e r i c a l s i m u l a t i o n sT o c h e c k t he p e r f o r m a n c e o f o u r a l g o r it h m w e p e r -

f o r m e d n u m e r i c a l s i m u l a t i o n s o f a t w o d i m e n s i o n a lr a n d o m s u r f a c e i n t e r a c t i n g w i t h a o n e - c o m p o n e n ts c a l a r f i e l d . T h i s s y s t e m c o r r e s p o n d s t o t h e b o u n d -a r y c a s e c = d = 1 a n d w e e x p e c t t h e a l g o r i t h m t op e r f o r m e v e n b e t t e r f o r c = d > 1 .

I n o u r s i m u l a t i o n s w e c o n c e n t r a t e d o n t h r e e t y p e so f o b s e r v a b l e s i m p o r t a n t f o r t h e e f f e c t i v e p i c t u r e o f af l u c t u a t i n g s u r f a c e : o b s e rv a b l e s r e f l e c t i n g

( a ) s h o r t r a n g e s t r u c tu r e o f t h e s u r f ac e ,( b ) a l o n g r an g e , g l o b a l g e o m e t r i c s t r u c t u r e a n d( c ) t h e m a t te r s e c to r , w h i c h t h r o u g h i t ' s c o u p l i n g

i n f l u e n c e s t h e in t e rn a l g e o m e t ry o f t h e s u r f a c e .T h e s e t h r e e t y p e s o f o b s e r v a b l e s a r e i n t h e s t a n d a r dl o c a l u p d a ti n g s c h e m e c h a r a c t e r i z e d b y t h r e e d i f f e r e n tt i m e s c a l e s o f e v o l u t io n , d e s c r i b e d b y t h e c o r r e s p o n d -i n g a u t o c o r r e l a t io n t i m e s . W e p r e s e n t h e r e r e s u l t s f o ro n l y f e w o b s e r v a b l e s w h i c h w e f o u n d r e p r e s e n t a t i v ef o r e a c h t y p e o f o bs e r v a b l e s . A s a e x a m p l e o f a t y p e( a ) o b s e r v a b l e w e c o n s i d e r th e a v e r a g e s q u a r e o f c u r-v a t u re w h i c h i s t h e d i s c r e t i z e d v e r s i o n o f :

< . ' > : l < . . ' l i < . . (11)w h e r e

(8 )i,lz i,j, l

I f t h e c u t / p a s t e m o v e i s a c c e p t e d , t h e n e x t s t ep i st o a s s i g n xr a n d A w i t h t h e a p p r o p r i a t e g a u s s / a n d i s-t r i b u t i o n s

O Z ~fddx T e x p ( - - 6 ( X r - ( y ) ) 2 ) ( 9 )f o r xr andoc f~dd A e x p ( - - 6 ( A + ( x - z } ) 2 ) ( 1 0 )

f o r A , w i th i V ' = ( q r / 6 ) 3 /2 , a s in ( 4 ) a n d ( 5 ) .N o t i c e , t h a t t h e t r an s i t io n p r o b a b i l i t y ( 7 ) h a s t o b em o d i f i e d i f t r ia n g l e s T a n d t h a v e a c o m m o n l i nk .T h e a l g o r i t h m c a n e a s i l y b e g e n e r a l i z e d t o s u r fa c e s

w i t h h i g h e r g e n u s . O t h e r m a t t e r f i e l d s c a n a l s o b ei n t r o d u c e d .

A t y p e ( b ) o b s e r v a b l e i s t h e g e o d e t i c d i s t a n c e dxyb e t w e e n t w o p o i n t s o n t h e s u r f a c e w i t h f i x e d l a b e l sx a n d y . A n o t h e r q u a n t i t y o f th i s t y p e i s t h e a v e r a g ei n t e rn a l s u r f a c e e x t e n s i o n fie. t h e d i s t a n c e a v e r a g e do v e r a l l p a i r s o f p o i n t sd = ( d x y } . ( 1 2 )T h e l a tt i ce i s i n v a r i an t w i t h r e s p e c t t o t h e p e r m u t a t i o n so f t h e p o i n t i n d ic e s , w h i c h a r e i n f a c t o n l y d u m m y a r -g u m e n t s . I t m e a n s t h a t d x y a n d d e s t i m a t e t h e s a m eq u a n ti t y. I n t h e u p d a t i n g p r o c e d u r e t h e e n d p o i n t s x , yp e r f o r m r a n d o m m o v e m e n t s o v e r t h e l a t t i c e a n d a f -t e r a l o n g m e a s u r e m e n t t i m e d x y s h o u l d e q u a l d , b u tw i t h m u c h b i g g e r er r o r. T h e r e a s o n w e s t u d y t h e mi n d e p e n d e n t l y i s t h a t w e a r e m a i n l y i n t e r e s te d i n t h ea l g o r i t h m d y n a m i c s , a n d t h e t w o s h o w d i f f e r e n t b e -h a v i o u r d u r in g t h e u p d a t i n g p r o c e d u r e . A s a t y p e ( c )o b s e r v a b l e w e c h o o s e t h e g y r a t i o n r ad i u s rr 2 = ( ( x - - .~ )2 ) . (1 3 )


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3 4 2 J . A m b j o r n e t a l . / P h y s i c s L e t t e r s B 3 2 5 ( 1 9 9 4 ) 3 3 7 - 3 4 6Table IThe integrated autoccorelation times for the local and global algorithms

N r "rint ( R 2 ) r int ( d xy ) ~" nt ( d ) 7"int ( r 2)local global local global local global local global

46 1.18(2) 1.66(3) 5.3(1 ) 2.61(7) 1.01(2) 1.20(2) 9.5(4 ) 3.26(9)76 1.60(3) 1.92(4) 7.9(3 ) 3.04(8) 1.87(4) 1.70(4) 2 4(2 ) 4.2(1 )116 1.97(4) 2.13(5) 11.5(5) 3.08(9) 2.85(8) 2.23(5) 38(3 ) 5.1(1 )156 2.12(5) 2.42(6) 14.7(8) 3.22(9) 4.2(1 ) 2.85(8) 63( 7) 5.7(2 )236 2 .36(6) 2 .57(7) 21(1) 3 .3(1) 5 .7(2) 4 .0(1) 86(11) 7 .3(2)316 2 .57(6) 3 .01(8) 26(2) 3 .4(1) 8 .1(3) 5 .1(1) 116(18) 8 .9(3)396 2 .63(7) 2 .91(8) 30(2) 3 .6(1) 11.4(5) 5 .7(2) 201(41) 8 .6(3)796 2 .81(8) 3 .5(1) 53(6) 4 .0(1) 24(2) 10.2(4) 430(126) 13.3(7)1 59 6 3 .04(6) 3 .53(7) 92(9) 4 .3(1) 4 0(3) 19.5(8) 1118(374) 20.4(9)2 3 9 6 2 .99 (3 ) 3 .7 (1 ) 97 (6 ) 4 .3 (2 ) 39 (2 ) 23 (2 ) 1900(512) 25 (2 )49996 4 .6(2) 9 .2(5) 241(65) 151(31)

T o c o m p a r e t h e a u t o c o r r e la t i o n ti m e s w e p e r f o r m e dr u n s u s i n g t w o t y p e s o f a l g o r i t h m . F o r b o t h t y p e s t h et i m e w a s m e a s u r e d i n sw eep s . Th e f i r s t a l g o r i t h m ,w h i c h w e c a l l th e l o ca l a l g o r i th m , m a d e u s e o n l y o f t h el o c a l m o v e s . T h e g e o m e t r y w a s u p d a t e d u s i n g t h e f l i pm o v e s . A s w e e p c o n s i s ts o f t h e n u m b e r o f a tt e m p t e df l i p s e q u a l t o t h e n u m b e r o f l a t t i c e l i n k s , fo l l o w e d b ya s h i f t , w h i c h i s t h e h e a t b a t h a n d o v e r r e l a x a t i o n u p -d a t e o f a l l t h e x ' s . I n p ra c t i c a l c a l c u l a t i o n s t h e o v e r -r e l a x a t i o n w a s u s e d w i t h a p r o b a b i li t y 5 0 % , w h i c hw e f o u n d t o m i n i m i z e t h e a u t o c o r r e l at i o n s f o r t h e l o -c a l a l g o r i t h m . T h e s e c o n d a l g o r i t h m , w h i c h w e c a l lth e g lo b a l a l g o r i th m , u s e d t h e l o c a l s w e e p s d e s c r i be da b o v e t o g e t h e r w i t h t h e g l o b a l s w e e p s c o n s i s ti n g o f t h eb ig mo ves , w h e r e t h e b ig mo ve w a s a t t e m p t e d a t e a c hm i n i m a l n e c k o f t h e s u r f a c e . F o r th e g lo b a l a l g o r i t h ma s w e e p m e a n s e i t h e r t h e l o c a l o r t h e g l o b a l s w e e p ,e a c h p e r f o r m e d a t r a n d o m w i t h e q u a l p r o b a b il i ty .

F o r e a c h o b s e r v a b l e w e m e a s u r e d t h e i n t eg r a t e d a u -t o c o r r e l a t i o n t i m e a n d f i t t e d i t t o t h e a s y m p t o t i c fo r -m u l aTi n t = c A z , ( 1 4 )t o e x t ra c t th e d y n a m i c a l e x p o n e n t z . I n ( 1 4 ) , A i s t h es u r f a c e a r e a , w h i c h i s e q u a l t o t h e n u m b e r o f t r i a n -g l e s . T h e f i t w a s m a d e u s i n g t h e s t an d a r d M I N U I Tp r o g r a m l ib r a r y [ 2 8 ] . T h e e x p o n e n t z c a n b e d i f f e r -e n t fo r d i f f e r e n t q u a n t i t i e s a n d t h e r e a l a u t o c o r r e l a t i o nt i m e c a n b e i d e n t i fi e d w i t h t h a t o f t h e s l o w e s t m o d e .I n o u r s i m u l a t i o n s w e c o v e r e d t h e r a n g e o f s i z e s u pt o 2 4 0 0 t r i a n g l e s fo r t h e loca l a l g o r i t h m a n d u p t o

0 .5 105 fo r the g lo b a l a l g o r i t h m . E f f e c t i v e s i m u l a -t i o n s w i t h l a t t i c e s o f t h a t s i z e w a s p o s s i b l e t h a n k s t oa v e ry l a rg e r e d u c t i o n o f t h e a u t o c o r r e l a t i o n t i m e s .

T h e m e a s u r e d a u t o c o r r e l a t i o n t i m e s ( i n s w e e p s )a r e p r e s e n t e d i n t h e T a b l e 1 b o t h fo r t h e l o c a l a n dt h e g l o b a l a l g o r i t h m . In t h e f i r s t c o l u m n w e s h o w t h es i z e d e p e n d e n c e o f t h e a u t o c o r r e l a t i o n t i m e Tin f o r(R 2 / . F o r t h i s q u a n t i t y Tin t S C al es v e ry s l o w l y w i t h t h el a t t ic e s i z e . T h e a v e ra g e c u rv a t u r e s q u a re i s t h e f a s t e s tm o d e i n t h e d y n a m i c s o f b o t h a l g o r i t h m s . F r o m t h et a b l e i t is s e e n t h a t fo r l o c a l q u a n t i t i e s t h e r e i s n o g a i ni n t h e a u t o c o r r e l a t i o n t i m e f r o m t h e b ig mo ves . Th e.flips d e c o r r e l a t e t h e l o c a l g e o m e t r i c a l q u a n t i t i e s f a s t e rthan big moves . In f a c t i n c l u d i n g t h e b ig mo ves m a k e st h e a u t o c o r r e l a t i o n t i m e s l i g h t l y l o n g e r . T h i s e f f e c t i sp r o b a b l y r e l a t e d t o t h e r e d u c t i o n o f t h e a u t o c o r r e l at i o nt i m e fo r t h e m a t t e r s e c t o r , d e s c r i b e d b e l o w . I t i s a l s oc l e a r t h a t i n b o t h a l g o r i t h m s t h e a u t o c o r r e l a t i o n s i nt h i s s e c t o r a r e s h o r t - r a n g e d a n d h a v e n o i n f l u e n c e o nt h e r e a l a u t o c o r r e l a t i o n t i m e .

I n t h e s e c o n d c o l u m n w e p r e s e n t t h e r e s u l t s f o r th ei n t e g ra t e d a u t o c o r r e l a t i o n t i m e fo r d xy. Th e g lo b a l a l -g o r i t h m , a s c o m p a r e d t o t h e s t a n d a r d loca l o n e , r e -d u c e s c o n s i d e r a b l y t h e a u t o c o r r e l a t i on t i m e o f t h i s o b -s e rv a b l e . I t i s c l e a r t h a t b y c u t t i n g a n d p a s t i n g a m i n b uo n e c h a n g e s t h e b r a n c h s t r uc t u r e o f t h e u n i v e r s e a n di n e f f e c t a d i s t a n c e b e t w e e n p o i n t s x a n d y , l y i n g o nd i f f e r e n t s u b -u n i v e r s e s , c a n c h a n g e v e ry m u c h . In t h i sw a y dxy g e t s e a s i l y d e c o r r e l a t e d . T h e f l i p s n e e d m u c hm o r e t i m e f o r t h i s. T h e r e d u c t i o n o f t h e a u t o c o r r e l a t io nt i m e i s r e f le c t e d i n th e e x p o n e n t z w h i c h w e f i n d t o b e


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J. Ambjcrn et al . / Physics Letters B 325 (1994) 337-34 6 3431 i i i i t i i

0 . 8

0 . 6

0 . 4

0 . 2

- 0 . 2 I i I I I I Ii 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0

Fig. 2. The normalized autocorrelation functions for gyration radius on the lattice with 239 6 points.z = 0 . 8 1 ( 6 ) f o r t h e lo c a l a l g o r i t h m , a n d z = 0 . 1 4 ( 2 )f o r t h e g l o b a l o n e . T o g e t s o m e i d e a a b o u t t h e p e r f o r -m a n c e o f b o t h a l g o r i t h m s f o r l a r g e r l a t t i c e s , w e t a k ea s a r e f e r e n c e p o i n t a l a t t ic e o f th e s i z e 0 . 5 . 1 0 5 , w h i c hw e s i m u l a t e d b y t h e n e w a l g o r i th m . I f o n e e x t r a p o l a t e sf o r t h e l o c a l a l g o r i t h m t h e r e s u l ts f r o m s m a l l e r l a tt i c es i z e s , o n e g e t s "lint o f t h e o rd e r 1 0 3 -1 0 4 , w h i c h i s tw o ,t h r e e o r d e r s o f m a g n i t u d e l a r g e r th a n 9 . 2 + .5 w h i c hw e g o t f r o m s i m u l a t i o n s w i t h t h e n e w a l g o r i t h m .

T h e r e l a t i v e l y s h o r t a u t o c o r r e l a t i o n t i m e o b s e r v e df o r d x y i s h o w e v e r n o t a g o o d e s t i m a t e o f t h e c o r r e l a -t i o n s f o r t h e l o n g - r a n g e o b s e r v a b l e s . S i n c e f i x i n g p o i n tl a b e l s i s n o t p h y s i c a l , a s w a s d i s c u s s e d a b o v e , i t c a nb e v i e w e d r a t h e r a s a m e a s u r e o f t h e m o b i l i t y o f t h ea l g o r i t h m . T h e m o r e r e a l i s t i c e s t i m a t e o f t h e s e c o r r e -l a t io n s c a n b e o b t a i n e d b y s t u d y i n g t h e o b s e r v a b l e d .I n f a c t n u m e r i c a l l y t h e t w o e s t i m a t e s a r e e q u a l w i t h i ne r r o rs . T h e t h i r d c o l u m n o f t h e T a b l e 1 s h o w s t h a t i nt h i s c a s e t h e a u t o c o r r e l a t i o n t i m e g r o w s f a s t e r w i t hv o l u m e t h a n f o r t h e o b s e r v a b l e d x y . T h e r e i s a g a i na l a r g e r e d u c t i o n o f t h e a u t o c o r r e l a t i o n t i m e f o r t h eg l o b a l a l g o r i t h m , t h e d y n a m i c a l e x p o n e n t s w e g e t a r er e s p e c t i v e l y z = 1 . 0 6 ( 3 ) a n d z = 0 . 7 6 ( 3 ) . T h e r e a-s o n o f t h e l a r g e i n c r e a s e o f z f o r t h e g l o b a l a lg o r i t h m ,c o m p a r e d t o t h a t fo r dxy c an b e a t t r i b u t e d t o t h e f a c t ,t h a t c u t t in g a n d p a s t i n g b r a n c h e d p a r t s o f t h e u n i v e r s ed o e s n o t c h a n g e i t s s iz e t o o m u c h . S o m e o f t h e c h a n g e sd o n e b y m o v i n g s u b - u n i v e r s e s a r e p r o b a b l y u n d o n e

b y n e x t m o v e s . W e h o p e h o w e v e r , t h a t t h i s e x p o n e n tc a n b e r e d u c e d b y p r o p e r l y a d j u s ti n g t h e r a t i o o f l oc a la n d g l o b a l u p d a t e s , s i m i l a r l y a s i s t h e c a s e fo r t h e l o -c a l o v e r r e l a x a ti o n i n th e s t a n d a r d a l g o r i t h m s , w h e r e zi s d r a s t ic a l l y r e d u c e d o n l y i f o v e r r e l a x a t i o n i s a p p l i e dw i t h a p r o p e r f r e q u e n c y . F r o m s i m u l a t i o n s o n t h e l a t -t i ce w i th a s ize 0 .5 105 we go t "lin = 241 . -4- 65. andi t i s t w o o r d e r s o f m a g n i t u d e l o w e r t h a n t h a t o b t a i n e df r o m e x t r a p o l a t in g t h e r e s u l t s f o r t h e l o c a l a l g o r i t h m .

T h e r e a l p r o b l e m f o r t h e l o c a l a l g o r i t h m i s c r e a t e db y t h e s l o w e s t m o d e w h i c h i s t h e m a t t e r s e c to r . I n t h i ss e c t o r f o r th e s t a n d a r d l o c a l a l g o r i t h m w e f i nd t h e d y -n a m i c a l e x p o n e n t z = 1 . 4 + 0 . 1 f o r t h e g y ra t i o n r a -d i u s r 2 . T h e e x t r a p o l a t io n o f t h e r e s u l t s f r o m s m a l ll a tt i c es u p t o t h e s u r fa c e w i t h 0 . 5 . 1 0 5 t r i a ng l e s g i v e st h e n u m b e r o f 1 05 s w e e p s n e e d e d t o d e c o r r e l a t e c o n -f ig u ra ti on s . I n te r m s o f H P 7 2 0 C P U c o m p u t e r t i m e i tw o u l d m e a n t h a t o n e n e e d s r o u g h l y o n e w e e k t o p r o -d u c e t w o i n d e p e n d e n t c o n f i g u r a ti o n s . U s i n g o u r a l g o -r i t h m w e m a n a g e d t o r e d u c e t h i s t i m e t o 1 5 0 . + 3 1 .s w e e p s a n d t h e e x p o n e n t z = 0 . 5 0 + 0 . 0 3 . A l r e a d y f o rs m a l l v o l u m e s t h e j u m p i n p e r f o r m a n c e m a k e s a r e ald i f f e r e n c e b e t w e e n t h e t w o a l g o r i t h m s , w h i c h i s v i s u -a l iz e d i n t h e F i g . 2 , w h e r e t h e n o r m a l i z e d a u t o c o r r e -l a t i o n fu n c t i o n s fo r th e g y ra t i o n r a d i u s a r e d e p i c t e d ,s h o w i n g d r a s t i c c h a n g e o f t h e c o r r e l a t i o n r a n g e .

T h e r e s u l t s f o r t h e d y n a m i c a l e x p o n e n t s z o b t a i n e df r o m a f i t ~' in t = c A z f o r d i f f e r e n t o b s e r v a b l e s , a r e


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3 4 4 J . A m b j c r n e t a l. / P h y s i c s L e t t e r s B 3 2 5 ( 1 9 9 4 ) 3 3 7 - 3 4 6T a b l e 2T h e f i t s o f t h e v o l u m e d e p e n d e n c e o f t h e a u t o c c o r e l a t i o n t i m e s t o t h e f o r m u l a 7"in = cA z

1" int = c A z d xy d r 2

l o c a l g l o b a l l o c a l g l o b a l l o c a l g l o b a l

C 0 . 2 3 (6 ) 1 . 5 (2 ) 0 . 0 1 7 (3 ) 0 . 0 6 (1 ) 0 . 0 5 (3 ) 0 . 4 6 (9 )Z 0 . 8 1 (6 ) 0 . 1 4 (2 ) 1 . 0 6 (3 ) 0 . 7 6 (3 ) 1 . 4 (1 ) 0 . 5 0 (3 )x2/d .o.f. 0.11 2.18 2.66 1.40 1.17 1.12

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...

, , , , , , ,, I , , , , , , ,, I , , , ,1 0 0 1 0 0 0 1 0 0 0 0

Fig. 3. Th e autocorrelation ime vs, lattice size. The dashed line corresponds o the autocorrelation ime for r 2 in the loc al algorithm whichis the slowest mod e in this algorithm, the d ot-dashed line, the same but in the global algorithm, and the solid line, the autocorrelationtime of d. Notice, that for the lat t ice size of the ord er of 10 3 triangles, it bec omes the slow est mo de in the new algorithm.s u m m a r i z e d i n t h e T a b l e 2 . T h e e r r o r s q u o t e d a r e o b -t a in e d u s i n g t h e M I N U I T l i b ra r y p r o g ra m . C o m p a r i n gt h e s l o w e s t m o d e s f o r t h e a l g o r i t h m s o n e f i n d s t h a t th ed i f f e r e n c e i n e f f i c i e n c y i s g o v e r n e d b y t h e e x p o n e n td i f f e r e n c e ~tocat ,global = 0 . 6 ( 2 ) . N o t i c e th a t f o r"r 2 -- "~dl a r g e l a t t i c e s i n t h e global a l g o r i t h m , t h e l o n g e s t a u -t o c o r r e l a t i o n t i m e c o m e s f r o m th e o b s e r v a b l e d ( z =0 . 7 6 ( 3 ) ) i n p l a c e o f r E ( z = 0 . 5 0 ( 3 ) ) a s is s h ow ni n t h e F i g . 3 . A s m e n t i o n e d e a r l i e r , w e h o p e , h o w -e v e r, t h a t Z d c a n b e r e d u c e d b y a d j u s t i n g p r o p e r l y t h ef r e q u e n c y o f big moves, p r o b a b l y s c a l i n g i t w i t h t h el a t t i c e s i z e . W e p o s t p o n e t h i s d i s c u s s i o n t o a f u r t h e rs t u d y .

F r o m t h e p o i n t o f v i e w o f t h e s t r in g t h e o r y , t h e g y -r a t i o n r a d i u s g i v e s a n i n s i g h t i n t o h o w t h e s t r i n g g e -o m e t r y l o o k s l i k e i n th e t a r g e t s p a c e i n w h i c h a s t r in g

i s e m b e d d e d . F o r d = 1 , t h e t h e o r e t i c a l c a l c u l a t i o n sp r e d i c t f o r t h e b e h a v i o u r o f t h e g y r a t i o n r a d i u s t h ef o r m u l a [ 1 0 , 1 6 ] :( r 2) = a + b l o g A + c ( l o g A ) 2, (15)w h i c h w e v e r i f y h e r e n u m e r i c a l l y u s i n g t h e globala l g o r i t h m . I n t h e F i g . 4 , ( r 2 ) is p l o t t e d v s . l o g A , w h e r eA i s t h e n u m b e r o f tr i a n g le s o f t h e l a tt ic e . W e f i n d c =0 . 0 2 5 ( 2 ) , w h e r e t h e e r r o r is e s t im a t e d b y c o m p a r i n gt h e f it t o ( 1 5 ) w i t h t h e o n e , w h e r e t h e t e r m l o g A / Ai s i n c l u d e d . T h i s f i t c a n b e c o m p a r e d w i t h t h e o n eo b t a i n e d i n [ 1 6 ] . T h e n o r m a l i z a t i o n o f t h e x f i el dw e u s e h e r e i s d i ff e r e n t t h a n i n [ 1 6 ] . T h e i r r e s u l tc o r r e s p o n d s t o c = 0 . 0 2 0 ( 1 ) i n o u r n o r m a l i z a t io n . O u rv a l u e s e e m s t o b e a b o v e t h e o n e q u o t e d t h e r e w h i c hm a y b e d u e t o t h e l a r g e r s y s t e m w e u s e .


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J. Ambjcr n et al. / Physics Letters B 325 (1994) 337-3 46 345. . . . . i . . . . . . . . , . . . . . . . . i . . . .

2 . 5

2

1 . 5

1

0 . 5

0 . . . . . t . . . . . . . . i . . . . . . . . i

1oo 1ooo 10000

Fig. 4. Gyra tion radius as a function of the lattice size fitted to the formula 2 = a + blogA + c(logA) 2. The error bars are smaller thanthe symbols used for the data points.4 . D i s cu s s io n

T h e a l g o r i t h m p r e s e n t e d a b o v e c a n b e u s e d t o s t u d yt h e m o d e l o f 2 d g r a v i t y in t e r a c t in g w i t h t h e G a u s s i a no r s p i n - l i k e f i e l d s fo r c a s e s w h e n c ~> 1 . T h i s p ro b l e mi s p r e s e n t l y b e i n g i n v e s t i g a t e d . A n o t h e r p o s s i b l e a p -p l i c a t i o n is i n n u m e r i c a l s i m u l a t i o ns o f s t r i n g m o d e l sw i t h e x t r i n s i c c u rv a t u re t e rm s i n t h e a c t i o n [ 2 1 ] . F o rs u c h m o d e l s t h e a u t o c o r r e l a t i o n t i m e f o r o b s e r v a b l e sl i k e t h e g y ra t i o n r a d i u s , w h i c h a r e d e f i n e d i n t a rg e ts p a c e , b e c o m e s e n o r m o u s . E v e n f o r l a tt i c e s o f t h e s i z e1 22 i t i s o f t h e o r d e r o f 1 04 [ 2 0 ] . T h i s i s o f c o u r s e as e r i o u s b a r r i e r f o r g o i n g t o l a r g e r v o l u m e s , a n d e v e nf o r v o l u m e s s i m u l a t e d s o f a r , i t s e e m s t o b e a s o u r c eo f d e b a t e c o n c e r n i n g t h e i n t e r p r e t a ti o n o f t h e r e s u lt s ,a s f o r e x a m p l e t h e o r d e r o f th e p h a s e t r a n s i ti o n [ 2 2 ] .W e h o p e t h a t t h e n e w a l g o r i t h m w i l l b e e f f i c ie n t a ls oi n t h e s t u d y o f h ig h e r d i m e n s i o n a l g r a v i t y [ 2 3 , 2 7 ] ,w h e r e a c o l d p h a s e i s k n o w n t o b e d o m i n a t e d b y e l o n -g a t e d b r a n c h i n g s t r u c tu r e w h i c h s l o w s d o w n t h e s t a n -d a r d a l g o r i t h m b a s e d o n l o c a l d e c o m p o s i t i o n s o f t h es i m p l i c i a l m a n i f o l d .

A s c o m p a r e d t o " c l u s t e r " a l g o r i t h m s [ 1 3 , 1 5 , 1 6 ] ,t h e b i g m o v e a l g o r i th m p e r f o r m s l a r g e c h a n g es o f g e -o m e t r y a n d n o t o n l y l a r g e c h a n g e s o f t h e m a t t e r f ie l ds .T h e a l g o r i t h m o w n s i t s e f fi c i e nc y t o t h e f a c t t h a t it u p -d a t e s d i r e c t l y t h e d e g r e e s o f f r e e d o m w h i c h s e e m t o

b e i m p o r t a n t f o r t h e e f f e c t i v e p i c t u r e o f t y p i c a l w o r l d -s h e e t g e o m e t r y . I n c i d e n t a l l y , w h e n p e r f o r m e d o n as i n g u l a r s p o t o f a l a t t i c e , a f l i p m o v e c a n d r a s t i c a l l yc h a n g e t h e m i n b u s t r u c t u r e b y s p l i t ti n g o r g l ui n g s o m eo f t h e m t o g e th e r . I n g e n e r a l o n e c a n t h i n k o f t h e f l i p a st h e m o v e w h i c h i s r e s p o n s i b l e f o r u p d a t i n g l o c al f l u c -t u a ti o n s o n t h e s u r f a c e , l i k e f o r e x a m p l e t h e c u r v a t u r ef l u c t u a ti o n s . In t u rn , c u t t i n g a n d p a s t i n g m i n b u s i s r e -s p o n s i b l e f o r u p d a t in g t h e g l o b a l b r a n c h i n g s t r u c t u reb y c o n t r o l l i n g t h e p a r t o f t h e e n t r o p y w h i c h r e s u l t sf r o m b a b y u n i v e r s e s u r g e r y . A s s h o w n r e c e n t l y , t h i st y p e o f d e g r e e s o f f r e e d o m c a n d r a s t i c a ll y c h a n g e t h et o ta l e f f e c t iv e e n t r o p y o f t h e m o d e l [ 1 9 ] , a n d c a n b ev e r y i m p o r t a n t i n t h e e f f e c t i v e p i c t u r e o f t h e s u r f a c e .W e h o p e t h a t t h e n e w a l g o r i t h m w i l l p l a y t h e s a m er o l e f o r t h e r a n d o m s u r f a c e s a s t h e c l u s t e r a l g o r i t h mf o r s p i n s y s t e m s o r r e c u r s i v e s a m p l i n g i n c = 0 q u a n -t u m g ra v i t y .

5 . A c k n o w l e d g e m e n tO n e o f us ( Z . B . ) w o u l d l i k e t o t h a n k A l e x a n d e r v o nH u m b o l d t F o u n d a t i o n f o r t h e f e l l o w s h i p . W e t h a n k

H L R Z - J t i l i c h f o r t h e c o m p u t e r t i m e . T h e w o r k w a sp a r t i a l l y s u p p o r t e d b y K B N g r a n t s : 2 P 3 0 2 1 6 9 0 4 a n d2 P 3 0 2 0 4 7 0 5 .


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MIN UIT F unc t ion Min imiza t ion and E r ro r Ana lys i s , 1988.

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