Exact results for a correlated percolation model

Physica 119A (1983) 609-614

EXACT RESULTS

North-Holland

Publishing

FOR A CORRELATED Chin-Kun

Lash-Miller

Co.

Chemical Laboratories,

University

Received

PERCOLATION

MODEL

HU

of Toronto,

5 October

Toronfo, Ontario,

Canada MS.9 IA1

1982

The equivalence between a correlated percolation model and the Syozi model at T-+0 has been used to calculate the exact percocation probabilities (P) as a function of bare (PO) and renormalized @) occupation probability for such a percolation process on two-dimensional lattices. We also find that for both two and three-dimensional lattices P cc @,, -p0,J8 near critical point P,,~, where /I is the magnetic exponent of the corresponding spin model.

The percolation process is an interesting problem in statistical physics’-3). However, up to now, there has been no exact calculation of percolation pr.obabilities and the corresponding critical exponents for random percolation processes. It has been shown that the random bond percolation model corresponds to the q-state Potts model”) at q + 1 and a quenched dilute Ising model’). However, such correspondence does not help the exact solution of the random percolation problem, because there is also no exact solution for the q-state Potts model at q + 1 and the quenched dilute Ising model. Besides random percolation problem, the correlated percolation problems are also of interest in physics, because they might be relevant for understanding puzzling behaviour”) or phase transitions’) in interacting systems. In a previous paper”), it has been shown that a sublattice-dilute q-state Potts model on a decorated lattice (i.e. a case 1 lattice of ref. 10) at T+O, where q = 2 corresponds to the Syozi model”) or the annealed random bond model”) at T-0, is equivalent to a q-state bond-correlated percolation model (BCPM) which favours larger number of finite clusters. In this paper, we will calculate exact percolation probabilities for the 2-state BCPM on some two-dimensional lattices from the corresponding quantities of the Syozi model, which are known exactly13). The partition function of the Syozi model on a bond-decorated lattice can be written as13) Z(G, K, B, A) =

1 0,. 0, = + 1

n [l

+ exp(K(ai + Uj) + A)

(lj)

+ exp( - K(ai + 0,) + A)] X n exp(Boi) , where the first and the second product extend over all nearest-neighbor 0378-4371/83/000@-0000/$03.00 0

1983 North-Holland

(1)

(nn) bonds

CHIN-KUN HU

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and sites of the original undecorated lattice G, respectively, K = /iI is proportional to the nn coupling constant J, B = PH is proportional to the magnetic field H applied at sites of G, and A = /?p is proportional to the chemical potential p and responsible for controlling the concentration of decorated spins. Expanding the first product in (1) as summation over subgraphs of G’ E G, and keeping for the low temperature limit of Z(G, K, B, A) only the dominant terms where all spins in the same cluster have the same spin component, we have shown’O) that Z(G, K, B, A) for T-+0 (i.e. /I-+co) can be written as Z(G, K, B, A) = 1

exp((2K + A)e(G’))

G’cc

n (exp(Bn,) + exp( - Bn,)) . c

(2)

Here the product extends over all clusters c in G’, e(G’) is the total number of occupied bonds in G’ and n, is the total number of sites in cluster c. From (2), we can derive the spontaneous magnetization A4 and the average number of occupied bond p as T-rO. Suppose G has N sites and Nb bonds, then we have M(G,p,)

= ;y

D -’

c

ZZ(G’,p,)N*(G’)/N

(3)

~(G’,po)e(G’)lK,,

(4)

GGG

and P(G, po) =

JmmD-’ b-

c

GGG

respectively. Here

n

(G’, po) = p$““(

D=

c

1 _ po)Nb - @‘)~tG’)

,

~(G',P,),

(5) (6)

GiG

with p. given by pa = exp(2K

+ A)/(1 + exp(2K + A)),

(7)

N*(G’) of (3) is the total number of sites belonging to one of the extended clusters in G’, which extend from one side of G to opposite side of G and become infinite cluster as N-co. n,(G’) of (5) is the total number of finite clusters in G’. It is obvious that M(G, po) of (3) is the same as the percolation probability P(G, po) of the following 2-state BCPM on G: a) The bonds of G are occupied with a probability p,,. b) The overall probability of a subgraph G’ E G is enhanced by a factor 2 for each finite cluster in G’, including isolated sites which do not connect with any other sites. Because of the enhanced factor of(b), the average number of bonds at each edge

EXACT RESULTS FOR A CORRELATED PERCOLATION MODEL

of G is p(G,p,J

611

of (4) rather than p. of (7). For obvious reasons, we call p. and

p the bare and renormalized occupation probability, respectively. To calculate M(G, po) and p(G, po) of (3) and (4), we write the expression inside

the bracket of (1) as exp(KA + K’rr,~~). Here K;=iln(.l+exp(ZK+d)+exp(-2K+d))+fln(l+2exp(d)),

(8)

K’ = f ln(1 + exp(2K + A) + exp( T- 2K + A)) - f ln(1 + 2 exp(A)) .

(9)

Thus

Z(G, K, B, A ) = exp(N,KZo(G, K’, B) ,

(10)

where Z,(G, K’, B) is the canonical partition function of the spin-$ Ising problem on G. Thus the spontaneous magnetization M(G, K, A) and the average number of occupied bond p(G, K, A) at any temperature are given by the equations M(G,K,A)=~~~~IV1~lnZ(G,K,B,A)=m(G,K’,f3=0), -t _

(11)

p(G, K, A) = irn, NC’ A In Z(G, K, B, A) b-’

(12) here z = 2N,/N, m(G. K’, B = 0) and u(G, K’, 0) are the spontaneous magnetization and internal energy of the spin-i Ising model on G with coupling constant K’, respectively, which are known exactly’4y15)for G being square (SQ), plane triangular (PT), and honeycomb (HC) lattices. From (7), (8) and (9), we have K’ = - i ln( 1 - p,),

=$K;=AK’=+p,,

(13)

in the limit T+O (i.e. fi-*co). With K’, (a/aA)Ki, (a/dA)K’ of (11) and (12) replaced by the corresponding left-hand-side expressions of (13), we arrive at the functions for M(G, po) (i.e. P(G, po) for the BCPM) and p(G, po), which are plotted in Fig. 1. Using the data of fig. 1, we also plot percolation probabilities of the 2-state BCPM as a function of renormalized occupation probabilities in Fig. 2, which is closer to percolation probability function of the random bond percolation problems. To avoid confusion, we use P and P’ to denote percolation probability as a function of p. and p, respectively. For the SQ lattice, it is easy to derive an analytic equation for p and P’. Since the BCPM favours subgraphs with larger number of finite clusters, it is obvious that P(G, po) in this paper is the lower bound of the percolation probability for the random bond percolation model with p. as bond occupation probability.

CHIN-KUN

612

HU

1.0 C t

RP 0.5 i

,

8’

I

,’ /’

Fig. 1. Percolation probability P (solid curve) and renormalized occupation probability p (dotted curve) as functions of bare occupation probability pOfor correlated percolation on (a) PT, (b) SQ, and (c) HC lattices.

EXACT

RESULTS

FOR

A CORRELATED

Fig. 2. Percolation probability P’ as a function of renorrnalized percolation on PT (a), SQ (0). and HC (0) lattices.

PERCOLATION

occupation

MODEL

probability

613

p for correlated

Now we turn to the problem of critical properties. Suppose m(G, K’, L = 0) of (11) can be expanded as: A (C)(K’ - Ki)fi + . . . near the critical coupling constant Ki, with the help of (13) it is easy to show that: P(G, PO)= C(G)@, -Po,Y + . . .

(14)

near the critical bare occupation probability pO.c, which equals 0.422649 . . . , 0.585786 . . . , and 0.732050. . . for PT, SQ, and HC lattices, respectively. C(G) = A(G)(2 - 2pJB and equals 1.3890. . . , 1.38649.. . , and 1.4274.. for G being PT, SQ, and HC lattices, respectively. Thus the critical exponent of P(G,A near P~.~ is the same as that of m(G, K’, L = 0) for both two and three-dimensional G. However, it has been shown by Essam and Garelick13) that the critical exponent of F’(G,p) near the critical concentration p, is modified by a factor (1 - LX’)for three-dimensional G and has a logarithmic correction for two dimensional G. From (7) and (9) we can show that, as T+O, pa = 1 - exp( - 2K’) = exp(2K + A)/(1 + exp(2K + A)) ,

(15)

and hence exp(2K’) - 1 = exp(2K + A),

(16)

where K’ = /?‘J’ and can be interpreted as the normalized nn coupling constant of the Ising model on G. From (16), (10) and (2) we can conclude that the Ising model at finite temperature also corresponds to the 2-state BCPM. From such connection, we can show’) that first and second order phase transitions in the Ising

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HU

model are due to percolation processes of the 2-state BCPM and have a unified picture for first and second order phase transitions and finite-size scaling at first order transitions16). The detail is given in ref. 9.

Acknowledgements

The author wishes to thank Professors J.P. Valleau and S.G. Whittington for useful conversations on the subject of this paper. He is currently supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

References 1) V.K.S. Shame and S. Kirkpatrick, Adv. Phys. 20 (1961) 325. 2) J.W. Essam, in: Phase Transitions and Critical Phenomena, vol. 2, C. Domb and M.S. Green, eds. (Academic Press, London, 1972). 3) J.W. Essam, Rep. Prog. Phys. 43 (1980) 833. 4) P.W. Kasteleyn and CM. Fortuin, J. Phys. Sot. Japan 26 (Suppl.) (1969) 11. 5) C.M. Fortuin and P.W. Kasteleyn, Physica 57 (1972) 536. 6) F.Y. Wu, J. Stat. Phys. 18 (1978) 115. 7) A.G. Dunh, J.W. Essam and J.M. Loveluck, J. Phys. C 8 (1975) 743. 8) H.E. Stanley, J. Phys. A 12 (1979) L329; H.E. Stanley and J. Teixeira, J. Chem. Phys. 73 (1980) 3404; H.E. Stanley et al., Physica 106A (1981) 260. 9) C.-K. Hu, University of Toronto preprint. 10) C.-K. Hu, Physica 116A (1982) 265. 11) I. Syozi and S. Miyazima, Progr. Theor. Phys. 36 (1966) 1083. 12) D.C. Rapaport, J. Phys. C5 (1972) 1830. 13) J.W. Essam and H. Garelick, Proc. Phys. Sot. 92 (1967) 136. 14) R.M.F. Houtappel, Physica 14 (1950) 425. 15) I. Syozi, in: Phase Transitions and Critical Phenomena, vol. 1, C. Domb and MS. Green, eds. (Academic Press, London, 1972), p. 269. 16) C.-K. Hu and P. Kleban, Bull. Am. Phys. Sot. 26 (1981) 241; P. Kleban and C.-K. Hu, Bull. Am. Phys. Sot. 27 (1982) 92, 325.