Critical Percolation Probabilities for the Next-Nearest-Neighboring Site Problems on Sierpinski Carpets

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1 Critial Perolation Probabilitie for the Next-Nearet-Neighboring Site Problem on Sierpinki Carpet H. B. Nie, B. M. Yu Department of Phyi, Huazhong Univerity of Siene and Tehnology, Wuhan , China K. L. Yao Center of Theoretial Phyi, CCAST (World Laboratory), Beijing , and Department of Phyi, Huazhong Univerity of Siene and Tehnology, Wuhan , China Abtrat In thi paper, we ompute the next-nearet-neighboring ite perolation (Connetion exit not only between nearet-neighboring ite, but alo between nextnearet-neighboring ite.) probabilitie P on the two-dimenional Sierpinki arpet, uing the tranlational-dilation method and Monte Carlo tehnique. We obtain a relation among P, fratal dimenionality D and onnetivity Q. For the 1.60 family of arpet with entral utout, ( 1 P ) /(1 P ) = ( D 1), where P = 0.41, the ritial perolation probability for the next-nearet-neighboring ite problem on quare lattie. A D reahe 2, P = P = 0. 41, whih i in agreement with the ritial perolation probability on 2-d quare lattie with next-nearetneighboring interation. I. Introdution In reent year, the tudy of fratal geometrie ha been reeiving great attention [1-12]. Gefen et al. [3-5] diued the elf-imilar lattie with a finite order of ramifiation R and with R =. In partiular, they applied the onnetivity Q, fratal dimenionality D to haraterize the Sierpinki arpet familie with R =, and found that direte-ymmetry pin model on thoe lattie undergo a phae tranition at T > 0. Gefen et al. [5] ontruted an approximate renormalization-group heme for the Iing model in the Sierpinki arpet family and found the dependene of T and the ritial exponent on the geometrial fator. Riera [7] analyed the ritial behaviour of the ferromagneti q-tate Pott model on Sierpinki arpet, and ued an approximate Migdal-Kadanoff real pae renormalization-group heme to find the 1

2 dependene of the ritial temperature and the ritial exponent on the number of tate q a well a on geometrial parameter. Yu and Yao [11] omputed the nearet-neighboring perolation probabilitie P on the two-dimenional Sierpinki arpet, and found a relation among P, fratal dimenionality D and onnetivity Q. Qu, Yao and Yu [12] ued the renormalization-group approah and Monte-Carlo imulation to treat the problem of perolation on two-dimenional quare lattie with next-nearet-neighboring interation. The ritial probability P and ritial exponent α, β, γ were obtained. In thi paper we ue the tranlational-dilation method (TDM) propoed Yu and Yao [11], and Monte Carlo tehnique to ompute P for the next-nearet-neighboring ite perolation problem (Connetion exit not only between nearet-neighboring ite, but alo between next-nearet-neighboring ite.) probabilitie on Sierpinki arpet familie. We find that P depend on the fratal dimenionality D and onnetivity Q. Epeially, we obtain a relation among P and D, and Q. We alo find that P in the family of arpet with entral utout i greater than that of the family with evenly attered utout. Compared with the nearet-neighboring perolation ' probabilitie P on the two-dimenional Sierpinki arpet [11], for the ame D and Q ' (ame b and l), we have P < P. Compared with the next-nearet-neighboring ite probabilitie P on the two-dimenional quare lattie [12], we have P > P. II. Method of Contrution for the Sierpinki Carpet Lattie For the onveniene of omputing P on the Sierpinki arpet family numerially, one need to tranfer the geometry problem for Sierpinki arpet lattie into a digital problem. In thi paper we ue the tranlational-dilation method (TMD) to ontrut a Sierpinki arpet lattie. Figure 1a (entral utout) and 1b (attered utout) how example of elf-imilar fratal lattie alled Sierpinki arpet, whih are formed by the TDM. The reurion formula for TDM are a follow: We firt aign an initial value for a quare lattie, for example M(I, J) = 1 (1) Where 1 I 2, 1 J 2, M(I, J) i an element of two-dimenional array, I and J repreent the row and olumn on a arpet lattie repetively, ee Fig. 3a. Then, the ite are tranlated toward the below aording to n 1 J M ( I, J ) = M ( I b, ), (2) where T b + 2 I ( T + 1) b + 1, 2

3 n T b 1, 1 J b 1, 1 n N, n i the erial number of tranlation, N i the tage of Sierpinki arpet (ee Fig. 3b, Fig. 2a-). T i a tranlational parameter. Equation (2) i alo adequate for the attered utout arpet. And then the known ite are tranlated toward the right by M ( I, J ) = M ( I, J b ), (3) n where 1 I ( b l) b / and ( b + l) b + 1 I b + 1, n 1 n b + 2 J b + 1; for the attered utout arpet, where ( T 1) b + 1 I T b + 1, T i an odd number le than or equal to b, n 1 n b + 2 J b + 1, ee Fig. 3. Finally, the ite are tranlated toward the below by n 1 J M ( I, J ) = M ( I b, ), (4) where ( b l) b / I ( b + l) b / 2 (if l = 1 and n = 1 thi inequality i 1 1 eliminated), b n + 2 J ( b l) b n / (if b- l = 2, thi inequality i n eliminated) a well a ( b + l) b / J b + 1; for the attered utout arpet, where ( T 1) b + 2 I T b (T i an even number le than b); ' ' T b + 1 J ( T + 1) b + 1(T i an even number le than b). If n = 1, the alulation of Eq. (4) i eliminated. If n i inreaed by 1, the dilation of the lattie i generated, then the tranlational alulation proee are repeated, and o on. Finally the requeted lattie for the Sierpinki arpet i formed (ee Fig. 2). For the utout of a arpet lattie, the omputer an automatially aign the zero to the utout ite (ee the irle in Fig. 3d). III. Site Oupation and Cluter Calulation on the Sierpinki Carpet Site on the Sierpinki Carpet are randomly oupied to row (l) and olumn (J). The probabilitie P i (i = 1, 2, 3, ) are eparately ompared with the peudo-random number y(0 ~ 1). If P i < y, the ite (I, J) i not oupied and aigned to be zero otherwie, a poitive integer i reaigned to the ite (I, J). If the ite (I, J) i oupied, nearet-neighboring ite (ee ite (I, J - 1) and (I - 1, J) in Fig. 4) and next-nearetneighboring ite (ee ite (I - 1, J - 1) and (I - 1, J + 1) in Fig. 4) already ompared with y are neearily invetigated to ee whether they are oupied or not, if o it belong to the luter ompoed of neighboring ite the erial number of the luter i aigned to the ite (I, J), the ize of the luter i then inreaed by unity; if not, the 3

4 ite (I, J) form a new luter (it ize i 1), and a new erial number whih i greater than the lat one by 1 i aigned to the new luter, and o on. Two luter oalee into a greater luter if two luter are joined by (at leat) a bond (inluding a bond whih join two next-nearet-neighboring ite) whih joining two ite that belong to different luter. If two luter oalee into a new greater luter, the erial number (e.g. no. 7) of ite of the latter luter i then hanged to be the erial number (e.g. no. 2) of ite of the former luter. The ize of the new luter i then equal to the um of ite of two luter, and finally zero i aigned to the latter luter ize. IV. Critial Perolation Probabilitie P and Diuion We again applied the modified eond moment 2 i S M = (5) i i ( S ) 2 i defined by Dean [9], where S i (i = 1, 2, 3, ) are the ize of luter and the ummation run over all the lattie luter. Then, P i taken to be that value of P at whih M / P i maximum, indiating an inreament of one tep. In the preent work we hooe P to be a ontant (e.g. P = 0.01) and et P i (i = 1, 2, 3, ), i alulated from Eq. (5). The ritial perolation probability for ite problem on Sierpinki arpet i Pi + Pi +1 P = (6) 2 when ( M 1 M ) P reahe the maximum. i+ i / From the omputation of the luter-ize ditribution the ritial perolation probabilitie an be obtained aording to Eq. (6). The reult for variou b, l of Sierpinki arpet are ummarized in Table 1 and 2, where the average are taken over 10 run. It an be een, from Table 1 and 2, that for the ame b the ritial perolation probabilitie P inreae with l, and P inreae with D dereaing and Q dereaing. From our numerial alulation we obtain the approximate relation, whih are hown in Fig 5, 6 and 7. For the family of arpet with entral utout: or 1.60 P = 1 (1 P )( D 1) (7) 4

5 and 1 P 1 P = ( D 1) P = 0.32Q 0.92Q (9) (8) k i 1 2 1/ 2 ei} with the remainder error (defined by er = {[ 1/( k 2)] = σ, here k = 22) and repetively (ee Table 1). Here P i the ritial perolation probability on the next-nearet-neighboring quare lattie, and P = For the family of arpet with evenly utout: with the remainder error (ee Table 2). 2 P = 0.36Q 0.88Q (10) From Eq. (7) and Fig. 5, we an find, a D reahe 2.0, for the family of arpet with entral utout, P = P =0. 41, whih i in agreement with the ritial perolation probability on 2-d quare lattie with next-nearet-neighboring interation [1]. From Eq. (9) and Fig. 7, we an find, a Q reahe 1.0 for the family of arpet with entral utout, P = P =0. 41, whih i in agreement with the ritial perolation probability on 2-d quare lattie with next-nearet-neighboring interation [1]. From Eq. (10) and Fig. 7, we an find, for the family of arpet with evenly attered utout, a Q reahe 1.0, P =0.40± 0. 01, whih i alo in agreement with the ritial perolation probability on 2-d quare lattie with next-nearet-neighboring interation [1]. From Table 1-2 and Fig 5-8, one eaily find that for the ame D or Q (ame b and l), P in the family of arpet with entral utout i greater than thoe of the family with evenly attered utout. Thi i expeted, ine they belong to two different univerality lae [11]. ' Compared with the nearet-neighboring perolation probabilitie P on the two-dimenional Sierpinki arpet [11], for the ame D or Q (ame b and l), we have ' P P <. Thi i reaonable, ine for the next-nearet-neighboring ite perolation, onnetion exit not only between nearet-neighboring ite, but alo between nextnearet-neighboring ite, thu P beome maller. Compared with the ite perolation probabilitie quare lattie [12], we have P on the two-dimenional P > P. Thi i alo reaonable, ine for the Sierpinki 5

6 arpet ite perolation, the onnetivity i maller than that of the quare lattie ite perolation, thu P i larger than P. For the family of arpet with entral utout, and for the nearet-neighboring ite perolation, Yu and Yao [11] obtained, 1 P 2.35 = ( D 1), (11) 1 P where P = i the ritial perolation probability on the nearet-neighboring quare lattie. For the next-nearet-neighboring ite perolation on the Sierpinki arpet, we obtain 1 P 1.60 = ( D 1), (12) 1 P where P =0. 41 i the ritial perolation probability on the next-nearet-neighboring quare lattie. It i intereting that the exponent 1.60 for the next-nearet-neighboring ite perolation on the Sierpinki arpet i different from the exponent 2.35 for the nearet-neighboring ite perolation on the Sierpinki arpet. Until now, no knowledge about the phyial origin of exponent 1.60 and 2.35 i reported. It i an open quetion and need further tudy. V. Conluion In thi paper, we ompute the next-nearet-neighboring ite perolation probabilitie P on the two-dimenional Sierpinki arpet, uing, uing the tranlational-dilation method and Monte Carlo tehnique. We obtain a relation among P, fratal dimenionality D and onnetivity Q. For the family of arpet with entral 1.60 utout, ( 1 P ) /(1 P ) = ( D 1), where P i the ritial perolation probability for the next-nearet-neighboring quare lattie. A D reahe 2, P = P = 0.41, whih i in agreement with the ritial perolation probability on 2- d quare lattie with next-nearet-neighboring interation [1]. Interetingly, we notied that the exponent 1.60 for the next-nearetneighboring ite perolation on the Sierpinki arpet i different from the exponent 2.35 for the nearet-neighboring ite perolation on the Sierpinki arpet. Until now, no knowledge about the phyial origin of exponent 1.60 and 2.35 i reported. It i an open quetion and need further tudy. 6

7 Referene [1] R. Zallen, The Phyi of Amorphou Solid, Wiley-Interiene Publiation (1983). [2] B. B. Mandelbrot, The fratal Geometry of Nature, San Franio, Freeman (1982). [3] Y. Gefen, A. Aharony and B. B. Mandelbrot, J. Phy. A: Math. Gen. 16 (1983) [4] Y. Gefen, A. Aharony, Y. Shapir and B. B. Mandelbrot, J. Phy. A: Math. Gen. 17 (1984) 435. [5] Y. Gefen, A. Aharony and B. B. Mandelbrot, J. Phy. A: Math. Gen. 17 (1984) [6] Y. Gefen, B. B. Mandelbrot and A. Aharony, Phy. Rev. Lett. 45 (1980) 855. [7] R. Riea and C. M. Chave, Z. Phy. B: Condened Matter 62 (1986) 387. [8] B. Lin and Z. R. Yang, J. Phy. A: Math. Gen. 19 (1986) L49. [9] L. P. Kadanoff, Ann. Phy. (NY) 100 (1976) 359. [10] P. Dean, Pro. Cambridge Philo So. 59 (1963) 397. [11] B. M. Yu and K. L. Yao, Z. Phy. B: Condened Matter 70 (1989) 209. [12] S. H. Qu, K. L. Yao and B. M. Yu, Ata Phy. Sinia 40 (1991)

8 Table 1. Critial perolation probabilitie P for the next-nearet-neighboring ite problem on Sierpinki arpet with entral utout ( l l). Here σ e i an error between experiment value and predited value from Eq. (7) or Eq. (9), σ i the tandard error. b l N D Q Experiment Value Predited value from Eq. (7) Predited value from Eq. (9) P ± σ P σ e P σ e ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 2. Critial perolation probabilitie P for the next-nearet-neighboring ite problem on Sierpinki arpet with evenly attered utout ( l l). Here σ e i an error between experiment value and predited value from Eq. (10), σ i the tandard error. b l N D Q Experiment Value Predited value from Eq. (10) P ± σ P σ e ± ± ± ± ± ± ±

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