Universal scaling functions and quantities in percolation models

Physica A 266 (1999) 27–34

Universal scaling functions and quantities in percolation models Chin-Kun Hua; ∗ , Jau-Ann Chena , Chai-Yu Linb

a Institute

b Institute

of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan

Abstract We brie y review recent work on universal nite-size scaling functions (UFSSFs) and quantities in percolation models. The topics under discussion include: (a) UFSSFs for the existence probability (also called crossing probability) Ep , the percolation probability P, and the probability Wn of the appearance of n percolating clusters, (b) universal slope for average number of percolating clusters, (c) UFSSFs for a q-state bond-correlated percolation model corresponding to the q-state Potts model. We also brie y mention some very recent related c 1999 Elsevier Science B.V. All rights developments and discuss implications of our results. reserved. PACS: 05.50+q; 75.10-b Keywords: Percolation; Universality; Scaling function; Finite size

1. Introduction Universality and scaling are important concepts in the theory of critical phenomena [1] and percolation models [2] are ideal systems for studying critical phenomena. In this paper, we brie y review recent work on universal nite-size scaling functions (UFSSFs) and quantities in percolation models. Universality of critical exponents in critical systems is well known for a long time [1,2]. However, the universality of nite-size scaling functions (FSSFs) received much attention only in recent years. According to the theory of nite-size scaling [3], if the dependence of a physical quantity Q of a thermodynamic system on the parameter , which vanishes at the critical point  = 0, is of the form Q() ∼ ||a near the critical ∗

Corresponding author. E-mail: [email protected].

c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 5 7 0 - 6

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point, then for a nite system of linear dimension L, the corresponding quantity Q(L; ) is of the form Q(L; ) ≈ L−a= F( L1= ) ;

(1)

where  is the correlation length exponent, F(x) is the FSSF and x =L1= is the scaling variable. It is widely believed that systems within a given universality class usually have dierent FSSFs. In 1984, Privman and Fisher (PF) proposed UFSSFs and nonuniversal metric factors for static critical phenomena [4] for the temperature T near the critical temperature Tc and the external magnetic eld h near 0. Speci cally, they proposed that, near  = (T − Tc )=Tc = 0 and h = 0, the singular part of the free energy for a ferromagnetic system can be written as fs (; h; L) ≈ L−d Y (C1  L1= ; C2 hL( + )= ) ;

(2)

where d is the spatial dimensionality of the lattice, Y is a UFSSF, and C1 and C2 are adjustable nonuniversal metric factors [4]. From 1984 to 1994, the progress in research on UFSSF was very slow. In this paper, we brie y review recent work on UFSSFs for the existence probability [5,6] (also called crossing probability, see e.g. [7]) Ep , the percolation probability P, the probability Wn of the appearance of n percolating clusters, and universal slope for P∞ average number of percolation clusters, C, de ned by C = n=1 nWn . The considered models include bond and site lattice percolation, continuum percolation model, and a q-state bond-correlated percolation model (QBCPM) [8] coresponding to the q-state Potts model [9]. We also brie y mention some very recent developments and discuss our results.

2. UFSSFs of Ep and P for lattice percolation models In 1992, Hu proposed a histogram Monte Carlo simulation method (HMCSM) [5,6], which was then used by us to calculate the FSSFs for the Ep and the P of the percolation model and the QBCPM [8] corresponding to the QPM [10 –14]. Here, Ep is the probability that the system percolates and P is the probability that a given lattice site belongs to a percolating cluster. As L → ∞; Ep is 0 for p ¡ pc and is 1 for p ¿ pc ; if we write Ep ∼ (p − pc )a just above pc , then the critical exponent a of Ep is 0 [2]. On the other hand, the critical exponent of P is . According to Eq. (1), we may write Ep = F(x) and PL = = S(x) with x = (p − pc )L1= , where F(x) and S(x) are FSSFs. We found that Ep and P have very good nite-size scaling behavior and FSSFs depend sensitively on boundary conditions and aspect ratio of the lattice and spanning rules to de ne percolating clusters [10 –14]. Since the critical exponent of Ep is 0, Eq. (1) for Ep implies that Ep for all models in the same universality class must be equal at the critical point in order to have UFSSF. In 1992, Zi [15] found that Ep = 0:5 for site and bond percolation on large

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square lattices with free boundary conditions (f bc) and Langlands, Pichet, Pouliot, and Saint-Aubin (LPPS) proposed that when aspect ratios for the square (sq), √ √ honeycomb (hc), and plane triangular (pt) lattices have the relative proportions 1 : 3 : 3=2, then site and bond percolation on such lattices have the same value of Ep at the critical point [7]. In 1995, we (HLC) [16,17] applied the HMCSM [5,6] to calculate Ep and P of site and bond percolation on nite sq, hc, and pt lattices, whose aspect ratios are √ √ approximately 1; 3, and 3=2, respectively [7]. Plotting Ep as a function of x = D1 (p − pc )L1= and D3 PL = as a function of x = D2 (p − pc )L1= , where D1 ; D2 and D3 are nonuniversal metric factors, we found that the six percolation models have very nice UFSSFs for Ep and P [16,17]. Within numerical uncertainties D1 = D2 and the nonuniversal metric factors for periodic boundary conditions (pbc) are consistent with those for fbc, although UFSSFs are quite dierent. We also found that the nonuniversal metric factors are independent of changes in aspect ratios holding the ratio between them constant [16,17]. These results indicate for each percolation model we need only two nonuniversal metric factors.

3. UFSSFs of Wn and universal slope for C It was proposed that the number of percolating clusters in the experimental sample max in of quantum Hall eect (QHE) at the critical point is useful for understanding xx the QHE [18]. Therefore, we studied the probability for the appearance of n percolating clusters, Wn , in percolation models. To mimic the Corbino disk often used in experimental studies of QHE, Hu [19] used the HMCSM to study bond percolation on L1 × L2 square lattices G with pbc in the horizontal L1 direction and fbc in the vertical L2 direction. A cluster which extends from the top row of G to the bottom row of G is a percolating cluster. A subgraph which contains at least one percolating cluster is a percolating subgraph and denoted by Gp0 . It should be noted that the de nition of Gp0 in [19,20] and this section is dierent from that of [10 –14,16,17] in which only the largest cluster is used to de ne Gp0 . A percolating subgraph which contains exactly n percolating clusters is P 0 0 denoted by Gn0 . Now we de ne: Wn = Gn0 ⊆ G pb(Gn ) (1 − p)E−b(Gn ) , where b(Gn0 ) is the number of occupied bonds in Gn0 ; E is the number of links in G, and the sum is over P∞ all subgraphs Gn0 of G. It is obvious that Ep = n=1 Wn and W0 = 1 − Ep . Using the HMCSM [5,6,19], Hu found that Wn as a function of z = (p − pc )L1= has very good scaling behavior. Hu also considered fbc in both horizontal and vertical directions and found that FSSFs for Wn depend sensitively on boundary conditions [19]. Using the HMCSM [5,6,19], Hu and Lin (HL) calculated Wn for bond and site percolation on sq, hc, and pt lattices with pbc in the horizontal direction and fbc in the √ vertical √ directions; the aspect ratios of sq, hc, and pt lattices are approximately 4; 4 3; 2 3, respectively. Using nonuniversal metric factors of [16], HL found that

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these percolation models have UFSSFs for Wn [20]. Similar results were found for three-dimensional lattices [21,22]. Using random deposition method and appropriate aspect ratio and nonuniversal metric factors [4], Hu and Wang have found that Ep and Wn for continuum percolation of soft disks and hard disks have the same UFSSFs as the lattice percolation [23,24]. The universality of Wn implies that the average number of percolating clusters, deP∞ ned by C = n=1 nWn , is also universal. Now we consider following four possible boundary conditions of the L1 × L2 lattice, where L1 is linear dimension in the horizontal direction and L2 is linear dimension in the vertical direction, BC1: periodic in L1 direction and free in L2 direction, BC2: free in both L1 and L2 directions, BC3: periodic in both L1 and L2 directions, BC4: free in L1 direction and periodic in L2 direction. In [19,20], HL calculated critical C for percolation on lattices with BC1 and BC2 for aspect ratio R between 0 and 10, where R = L1 =L2 . They found that for large R, critical C increases linearly with R with the same slope. In a recent letter, Zi et al. [25] calculated the number of clusters per lattice site, n, in bond and site percolation on two dimensional lattices with BC3 and the linear dimension L. They found that n = nc + b=N , where nc is n in the limit L → ∞; b is a constant and N is the number of lattice sites. They also found that b is universal and presented an argument that b is the number of percolating clusters so that the universality of b may be related to the universality of C found by Hu and Lin [20]. In a recent paper, Hu [26] used the HMCSM to calculate n and C for bond percolation on L1 × L2 sq lattices with BC1, BC2, BC3, and BC4. In BC3 and BC4, a cluster is percolating if each of L2 rows contains at least one site of that cluster [27]. Hu found that for four dierent boundary conditions, C increases linearly with R. However, we may have well-de ned slope b in n = nc + b=N only for BC3. Recently, Zi and Lorenz presented another argument for the universality of b [28].

4. UFSSFs of Ep for a q-state bond-correlated percolation model Based on the subgraph expansion of Ising-type models in external elds, Hu has shown that phase transitions of many Ising-type models can be described as geometric percolation transitions [8]. In particular, Hu has shown that phase transitions of the QPM on G are percolation transitions of a QBCPM [8] on G, in which each NN bond of G is occupied with a probability p, where p = 1 − exp(−J=kB T ) with J being the ferromagnetic Potts coupling constant. The sites connected by occupied bonds are in the same cluster and a cluster may have any one of q dierent directions. There are 2E dierent bond con gurations G 0 , also called “subgraphs”, of G. A subgraph G 0 of b(G 0 ) occupied bonds and n(G 0 ) clusters will appear with the probability weight: 0 0 0 (G 0 ; p; q)=pb(G ) (1−p)E−b(G ) q n(G ) . The spontaneous magnetization and the magnetic susceptibility of the QPM are related to the percolation probability P and the mean cluster size of the QBCPM, respectively [8]. The probability for the appearance of

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Fig. 1. (a) Ep for the QBCPM as a function of x = (1 − T=Tc )L1= . The scaling function is F(q; x). For the sq lattice, the results for a model with both NN and NNN couplings is also shown. Here q = 2. (b) Wn for the QBCPM as a function of x = (1 − T=Tc )L1= . The scaling function for Wn is Un . Here q = 1.

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percolating clusters, Ep , of the QBCPM is de ned by P 0 G 0 ⊆ G (Gp ; p; q) : Ep (G; p; q) = P p 0 G 0 ⊆ G (G ; p; q)

(3)

Here the sum in the denominator is over all subgraphs G 0 of G and the sum in the numerator is restricted to all percolating subgraphs Gp0 of G. We use a cluster Monte Carlo simulation method [29] to simulate the QBCPM and QPM on 64×64 sq lattices, 97×56 hc lattices, and 52×60 pt lattices. The aspect ratios of these lattices approximately match the LPPS relative proportions. Typical calculated results of Ep as a function of the scaling variable x with x = tL1= (t = (T − Tc )=T ) is presented in Fig. 1a, which shows that Ep for the QBCPM and QPM on sq, hc, and pt lattices have UFSSF near x = 0 without nonuniversal metric factors. For the sq lattice, we also consider a model with both NN and NNN coupling and nd that the calculated Ep as a function of x have the same FSSF as the model without NNN coupling. Since the QPM for q = 1 corresponds to the bond random percolation model (BRPM), we also plot data of Wn in [20] for bond percolation on sq, hc, and pt lattices as a function of x = tL1= (t = (T − Tc )=T ) and obtain Fig. 1b, which shows that we have UFSSF for Wn without using nonuniversal metric factors. These results are analogous to setting C1 = 1 in PFs theory (see Eq. (2) in this paper) for all lattices of a given dimension. We also nd similar results for three-dimensional lattices. It should be noted that if we use the scaling variable z = (p − pc )L1= or z 0 = (p=pc − 1)L1= as the horizontal axis, then we need to use metric factor for each lattice. This suggests that t = 1 − T=Tc is a fundamental variable for describing critical phenomena near the critical point even for BRPM.

5. Some recent developments Recently, Okabe et al. [30] calculated FSSFs of Binder parameter g and magnetization distribution function p(m) for the Ising model on L1 × L2 square lattices with pbc in the horizontal L1 direction and tilted boundary conditions with tilt parameter c in the vertical L2 direction. For appropriate sets of (R; c) with R = L1 =L2 , the FSSFs of g and p(m) are universal and in such cases R=(c2 R2 + 1) is an invariant. For percolation on lattices with xed R, FSSF of the existence probability does not change as c increases from 0. Very recently, Hu et al. [31] used a cluster Monte Carlo method to calculate the number of clusters per site, n, at the critical point of the QBCPM on L0 × L sq lattices. They proposed that n as a function of 1=L for q¿2 and xed L0 =L has an energy-like singularity. For q = 2, i.e. the Ising model, they found that the data can be well represented by n = nc − c=L + b=L2 + · · · ; where b can be calculated exactly from conformal eld theory (CFT), c ¿ 0 and can be calculated exactly from formula for the internal energy of the Ising model.

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6. Discussion For random percolation, Cardy [32,33] used CFT to obtain exact formula any R and Wn for large R; the former agrees very well with the LPPS’s It is valuable to extend such analytic and numerical calculations of Ep and QBCPM. Fig. 1 of this paper represents eort in this direction. Fig. 1 suggests that t = 1 − T=Tc is more fundamental than p − pc or A physical interpretation of T in the BRPM is needed.

for Ep for result [7]. Wn to the p=pc − 1.

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