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J . Phys. A: Math. Gen. 19 (1986) L1169-1172. Printed in Gre at Britain

LE'ITER TO THE EDITOR

The efficient determination of the percolation threshold by afrontier-generating walk in a gradient

Ro bert M Ziff t and B SapovalSt Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan48109, USA$ Laboratoire d e Physique de la M atitre CondensCe, Ecole Polytechnique, 91128 Palaiseau,FranceReceived 11 Septem ber 1986Abstract. The frontier in gradient p ercolation is generated directly by a type of self-avoidingrando m w alk. The existence of the gradient permits one to generate an infinite walk on acomputer of finite memory. From this walk, the percolation threshold p c for a two-dimensional lattice can be determined with apparently maximum efficiency for a naiveMo nte Car lo calculation ( *N-"' ) . For a s qua re lattice, the value pc = 0.592 745 i .000002is found from a simulation of N = 2.6 x 10" total steps (occupied an d blocked perimetersites). The power of the method is verified on the KagomC site percolation case.

Recently, two new app roac hes for studying percolation clusters in two dimensions a ndfor efficiently finding the critical percolation probab ility p c have been independentlydeveloped. In one app roac h, Ziff er a1 [13 introduced a random walk which generatesthe perimeter of the percolation cluster directly, and this walk has been used to findp c by finding the point where internal a nd external perimeters ar e generated with equalprobability [2]. In the othe r app roac h, Sapoval et d [ 3 ] in troduced the idea of s tudyingpercolation in a system with a gradient in the occupation probability, which leads toan extended and controlled frontier between the percolating and non-percolatingregions and which also allows p c to be foun d (see below). These two approaches, bo thbased upon perimeters of percolation clusters, have led to very precise values of p c .Here we point out that these approaches can be combined to generate the frontierdirectly an d to find p c even more efficiently. Simply, if the perimeter-generating walkof [11 is carried ou t in a gradien t of p , then th e frontier of [3] will be exactly gene rated,without generating any of the occupied sites, in either region, that are not part of thefrontier. (For the algorithm of the walk, see [13.) In essence, th e walk is a very efficientmethod to generate the frontier alone. Th e value of p c may be fou nd from the frontier[4] by two methods. I n the first, p c is found as the average value of the probabilitysampled by both the occupied and blocked sites of the walk:

Pc = P ( F ) (1 )where j j is the average y value of all the sites in the walk and p ( y ) is the occupationprobability at y (the gradient is assumed to be in the - y direction). The second methodis to find p c is by the formula

Pc = No,,/ Ntotal (2)where No,, is the number of occupied sites and N,,,,, is the number of occupied plusblocked sites generated in the w alk.0305-4470/86/ 181169 + 04$02.50 @ 1986 The Institute of Physics L1169

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L1170 Letter to the EditorTh e net effect of the gradient on the walk is to m ake it move, on the average, in adirection perpendicular to the direction of the gradient. The strength of the gradientcontrols the width or the correlation length of the walk [3]. W hen the gradient is veryhigh, the walk is squeezed to a straight line, and (1) or (2) gives p c = 0 . 5 . As the

gradient is relaxed, the walk exp ands and the ap parent p c increases as more occupiedthan block ed sites are generate d. The average width CT of the walk is =0.51Vpl-o.57 3].We have carried out extensive simulations of this walk o n a two-dimensional squ arelattice fo r the purpose of determining p c . A lattice of 2048 x 2048 sites was used, witha gradient of 1/40 000 in the - y direction and the value p = 0.593 along the liney = 1024, so p varied linearly between ab ou t 0.567 an d 0.619. According to the analysisin [4], the finite-gradient correction to p c in this case is = 5 x which is smallerthan the statistical error tha t can be reached in o u r computations and so can be ignored.Also with this gradient CT = 215, so the edges of this lattice in the y direction shouldvery rarely be hit, and the information in the x direction can be forgotten after onepass of the box.To start the walk without any closures ( a completed perimeter), which would becomm on in such a low gradient, the x = 0 line was filled with o ccupied sites from y = 0to 1023 and blocked sites from y = 1024 to 2047, and the walk was started at x = 1,y = 1023. This boundary kept the walk at x > 0. After a while, the walk got startedand continued in the positive x direction, as shown in figure 1. The statistics of thefirst pass of th e lattice were discarded to eliminate a ny bias from th e sta rtup process.

Y

1 0 2 4I s t a r t )4

N o n -p e rc o l a t i n g r e g i o n

I P e r c o l a t i n g r e g i o n0 X 2047

Figure 1. The first pass of the walk on a 2048x2048 lattice. Th e gradient is in the - ydirection. I t can be seen that the walk hit the boundary x = 0 several times at the start.This exam ple show s rather large deviations in the position of the walk; more typical onestend t o stay closer to the centre ( y= 1024). This walk was stopped when x = 2047 was firstreached, and a total of 322 782 occupied plus blocked sites were generated . This figuremay be compared with the perimeter of a single large percolation cluster shown in [2].

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Letter to the Editor L1171Every time the w alk hit a new higher value in the x direction, tha t colum n in the latticewas reset to be blank (unvisited) sites for all y. When the walk hit the end x = 2047,periodic boundary cond itions were used to w rap it back to x = 0. Thus the w alk couldbe continued indefinitely using a finite-memory computer.

A total of 2.6 x 10 occupied plus blocked sites were generated by carrying outsimultaneous runs on many Apollo workstation computers (with different randomnumber seed s) , using abo ut 5500 total hours of com puter time. (These are smallminicomputers, at least two o rders of magn itude slower tha n a high-speed m ainframeor supercomputer.) The bound aries in the y direction were hit only about 30 times,at which point the simulation s were stopped . Violation in the x direction (in whichthe walk makes i t back to discarded sites) was never found. Using either (1 ) or (2 ) ,we findp c= 0.592 745(2) (3)

where the number in parentheses represents the error in the last digit at the 68%confidence level (one sta nd ard deviation). The results of the other determ inations ofp c are shown in table 1. It can be seen that ou r value is consistent with previous resultsand is at least an order of magnitude more precise.Table 1. Determinations of pc for site percolation on a square lattice.

Value Method Reference Year0.48 Fifth-order series [6] 19600.55 Ninth-order series [7] 19610.581( 5) M C on 2000 sites [E] 19610.580( 8) MC on 782 [91 19630.59(1) Tenth-order series [IO] 19640.593(2) Nineteenth-order series [ 1 1 1 19760.595 MC on 1000~ [121 19760.591( ) Series analysis 1131 19760.592 7(3) MC on 4000 ~ 4 1 19780.593 l(6) M C on 500~ r151 19800.592 7(2) Transfer matrix [I61 19820.592 3(7) Series analysis ~171 19820.592 77(5) MC on 50000 [I81 19840.592 7( 1) MC on 160000 ~ 9 1 19850.592 4( 10) Transfer matrix P O I 19850.592 80( ) Gradient frontier [4] 19850.592 75(3) Perimeter walks PI 19860.592 73(6) Transfer matrix 1211 19860.592 45(2) Frontier walks This work 1986

The error quoted in (3) represents both the observed deviation from different runs,and the minimum deviation implied by the statistical uncertainty N - * . Because theobserved error is the statistical minimum , an d because no points other than those usedin the calculation were generated, we believe that this is the most efficient methodpossible for a naive Monte Carlo measurement of p c . We note that in previousdeterminations of p c , the given e rror is in general mu ch larger than the statistical limitN-l*, where N is the total num ber of sites visited ( or random num bers generated).We note also the simplicity of the program to generate these walks: the main part is

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L1172 Letter to the Editorabout 20 lines long, and there is no complicated bookkeeping or labelling, no clustersearching, no binning of distributions, and no delicate extrapolations. Furthermore,essentially only one run needs to be carried out; one does not have to try differentvalues of p as in most other methods.

As a check on the method a nd the rando m num ber generator ( a Tausworthe-typeshift generator), some runs were also carried out on a lattice with known p c . Thetriangular lattice, which is often used for this purpose because p c is exactly f, is notideal because of the perfect symmetry between t he occupied a nd blocked sites. Insteadwe used the K agomC lattice (th e matc hing lattice of bo nd percolation on the honeycomblattice), for which p,(site) is known exactly as 1- sin( 7 ~ / 8)= 0.652 7036 [ 5 ] . Generat-ing 1.2 x 10" steps on the 2048 x 2048 lattice with a gradient of 1/25 000, we foundNo,, / Ntotal0.652 704(9). Th e mean turn ed o ut to be correct well within one sta nda rddeviation.I n conclusio n, we find that applyin g the generating walk to the frontier in a gradientis a very efficient and well controlled way of generating those frontiers, and probablythe best Monte Car lo way to find p c for a two-dimensional lattice. We have found themost precise value of p c for a square lattice given to data. The striking agreementbetween p c found by this method and the other varied methods is good evidence thatthe general understanding of the percolation process in finite and infinite systems thatunderlies these studies is correct.References[ 13 Ziff R M, Cummings P T and Stell G 1984 J. fhys. A: Marh. Gen. 17 3009[2] Ziff R M 1986 fh ys . Rev. Lef t 56 54 5[3] Sapoval B, Rosso M a n d G o u y e t J F 1985 J . Physique Lerr. 46 L149[4] Rosso M, Gouyet J F and Sapoval B 1986 f h y s . Rev. B 32 6053[5] Sykes M F and Essam J W 1963 fhys. Rev. Lerr. 10 3[6] Elliott R J , H e a p B R, Morgan D J and R ushbrooke G S 1960 fh ys . Rev. Lett . 5 36 6[7] Domb C and Sykes M F 1961 fhys. Rev. 122 77[SI Frisch H L, Sonnenblick E, Vyssotsky V A and Hammers ley J M 1961 fhys. Rev. 124 1021[9] Dean P 1963 h o c . Ca mb . Phil. Soc. 59 397

G o u y e t J F, Rosso M, Sapoval B, Cassereau S and Couture B 1987 to be published

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