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Critical behaviour of the percolation probability for the bond problem on a three dimensional lattice
Volume 57A, number 3
PHYSICS LETTERS
14 June 1976
CRITICAL BEHAVIOUR OF THE PERCOLATION PROBABILITY FOR THE BOND PROBLEM ON A THREE DIMENSIONAL LA~FICE J.BLEASE, J.W. ESSAM and C.M. PLACE Department of Mathematics, Westfield College, University of London, London, NW3 7ST, UK Received 27 April 1976 The critical exponent /3 of the percolation probability for the bond problem on the FCC lattice is estimated by the Pad~approximant technique using a high density series expansion. It is found that /3 0.47 ±0.02.
The percolation probability P(,p), introduced in ref. [1] is the probability that a chosen site is connected to an infinite number of other sites, when each bond is present with probability p. Recent interest in the percolation probability stems from its connection with critical phenomena. For example, the magnetisation of the bond diluted Ising model tends to the percolation probability in the limitJ~’/CBT [2]. Further, the conductivity, a(p), of a random resistor network is given in ref. [3] by a(p) = D(p)P(p), where D(p) is the spin-wave stiffness coefficient of a dilute Heisenberg model. So far, the only published estimates of F(p) for a three-dimensional lattice have been obtained by Monte Carlo methods [4—6] The present calculation is based on the high density expansion P(p) 1 —q12 12q22 + l2q23 24q30 54q32 ,
P(p)
(p
as
—
p
-~
p~
(2)
and estimate p~and j3 by forming the Padé approximants to the series for (d/dq) logP(p). These estimates are displayed in a pole-residue plot (see fig. 1). The approximants which use 24 or less terms of the expansion may be misleading, since to this order in q the series for (d/dq) logPis the same as the expansion of dP/dq, which has a different type of singularity.
POSITION OF POLE IN p .7.125
.120 I
.115
.110
t35/2~.]
.
—
+ 204q33 —
+
—
126q34
—
8q36
276q40 + 768q41
—
608 q42
—
—
—
96q38 + 32q39 +
1304q45 —360q46 +480q47
1800q43
—
—
1056q48 —
—
+
l7240q57 —33720q58 +60120q59
—
l09676q60
+207072q61 —396792q62 +5039l6q63 —
565824q64 +783228q65 —737368q66
+426684q67,
[44/271 [38/271 139/271.138/28]
3066q44
+ 3360q49 5400q50 + l4280q51 21927q52 +23076q53 —41 138q54 +44604q56 17817q56 —
.6
134/261 [39/38] (43/211
W 0
(37/27] [34/25] 132/321 130/30] (32/33L[31/33[.131/341 •5 130/31]~~~_ (34/32] (36/27] [31/31k 138/26 (32/30] (33/2428/281 [29/29]’ 1 32/22]~/ 131./301 33133]1 / 36/291(35/28] ~4 33/211/ — 35/29 133/31] [32/211 133/30] [34/29] [42/211 [29/291(3~29]
(I) U_i
1
(1)
where q = 1 p, which we have obtained by the penmeter method [7]. We assume that
139/281 132/321 [42120]
132/281
.875
[3~28]
.880
.885
POSITION OF POLE IN q
.890
-
Fig. 1. Pole-residue plots for P(p)(o)andF(p)(.).
199
Volume 57A, number 3
PHYSICS LETTERS
The points on the pole-residue plot lie on a well defined curve, so that a good estimate of f3 could be ohtamed if p~were known. Since r~ is not known, we have used the estimate P~= 0.1190 ±0.0005 obtained in [8] from low density expansions. The pole-residue curve includes the central value and a linear approximation in its vicinity gives the estimate j3 = 0.474 24~p~±0.002 where Ap~ P~ 0.1190. Assuming that I~p~I ~ 0.0005 as above we obtain —
14 June 1976 Table 1
Model
P~
f.c.c. (bond) 0.1185 f.c.c. (site) 0.195
0.47 ±0.02.
(3)
Further estimates of p~and (3 may be obtained the probability (p) that a chosen bond, when present, belongs to an infinite cluster. We now show that P~p)has the same critical probability and exponent as P(p). Let I be the event that the origin belongs to an infinite cluster and C. the event that the origin is connected to its ith nearest neighbour (i = 1 z). By definition, P(p) = Pr{l} and pP(p) = Pr{IflC~} for all i. (4) from
Ref.
—
0.1195 0.201
0.49 —0.45 0.47 — 0.37
—
this work
f.c.c. b.c.c. (site) (site)
0.198 0.245
— —
0.202 0.251
0.40 —0.33 0.40 — 0.30
[91 [4,51
s.c. (site) (site)
0.3 10 0.311
— —
0.3 14 0.313
0.40 —0.37 — 0.35 0.41
[61
—
=
/3
_________________________________________
independent analysis of the low density series for the site problem on the FCC lattice [91gives a central value of p~of 0.197, which is consistent with a value of /3 in the upper half of the range in [10]. Similarly, extrapolation of the data in [5] to p~= 0.197 leads to a value of /3> 0.4. Correlation scaling was used in [8] to predict (3 = 0.39 ±0.07 for the FCC bond problem, which does not contain our central value. However, use of correlation scaling for the Ising model in three dimensions leads to similar discrepancies.
The result follows since (IflC 1)CI and whenever I occurs at least one of the C1 must occur, therefore p~(p)~P(p)
=
Pr
z U i= 1
(IflC1) <~zpP(p).
(5)
The corresponding pole-residue curve is shown in fig. 1 from which we obtain /3 = 0.46030 ~Pc ±0.002. The discrepancy between the estimates of /3 could be reduced by taking 0.117 ~ ~ 0.118. However, the significance of this fact is not yet established. We have taken the estimate from the site-series since the bondseries has more leading zeros, and the error in (3) almost covers the range of /3 determined by the P-curve. The only direct estimates of 13 for three dimensional lattices that are available for comparison are those for the site problem (see table 1). It is currently believed (but not proved) that f3 is independent of the type of dilution and is a dimensional invariant; thus our estimate of /3 is apparently rather high. However, as the table shows, (3 is very sensitive to the choice of p~.An
200
We would like to thank Dr. M.F. Sykes and Miss M. Glen for making available their data for hp).
References [11 S.R. Broadbent
and J.M. Hammersley, Proc. Camb. Phil. Soc. 53(1957)629. [2] A.G. Dunn et al., J. Phys. C: Solid State Phys. 8 (1975) [3] BJ. Last, Ph.D. thesis, Univ. Birmingham, England (1971)~ [4] P. Dean and N.F. Bird, Mathematics Division Rept. Ma 61, Nat. Phys. Lab., Teddington, England (1966); Proc. Camb. Phil. Soc. 63 (1967) 477. [5] S. Kirkpatrick, Solid State Commun. 12 (1973) 1279. [6] S. Kirkpatrick, Phys. Rev. Lett. 36 (1976) 69.
[71 C. Domb,
Conf. on Fluctuation phenomena and sbchastic processes, Birkbeck College, London; Nature
(London) 184 (1959) 509. [8] 4219. A.G. Dunn et al., J Phys. C: Solid State Phys. 8 (1975) [91 M.A.A. Cox and J.W. Essam, to be published. [10] M.F. Sykes et al., to be published.
PHYSICS LETTERS
14 June 1976
CRITICAL BEHAVIOUR OF THE PERCOLATION PROBABILITY FOR THE BOND PROBLEM ON A THREE DIMENSIONAL LA~FICE J.BLEASE, J.W. ESSAM and C.M. PLACE Department of Mathematics, Westfield College, University of London, London, NW3 7ST, UK Received 27 April 1976 The critical exponent /3 of the percolation probability for the bond problem on the FCC lattice is estimated by the Pad~approximant technique using a high density series expansion. It is found that /3 0.47 ±0.02.
The percolation probability P(,p), introduced in ref. [1] is the probability that a chosen site is connected to an infinite number of other sites, when each bond is present with probability p. Recent interest in the percolation probability stems from its connection with critical phenomena. For example, the magnetisation of the bond diluted Ising model tends to the percolation probability in the limitJ~’/CBT [2]. Further, the conductivity, a(p), of a random resistor network is given in ref. [3] by a(p) = D(p)P(p), where D(p) is the spin-wave stiffness coefficient of a dilute Heisenberg model. So far, the only published estimates of F(p) for a three-dimensional lattice have been obtained by Monte Carlo methods [4—6] The present calculation is based on the high density expansion P(p) 1 —q12 12q22 + l2q23 24q30 54q32 ,
P(p)
(p
as
—
p
-~
p~
(2)
and estimate p~and j3 by forming the Padé approximants to the series for (d/dq) logP(p). These estimates are displayed in a pole-residue plot (see fig. 1). The approximants which use 24 or less terms of the expansion may be misleading, since to this order in q the series for (d/dq) logPis the same as the expansion of dP/dq, which has a different type of singularity.
POSITION OF POLE IN p .7.125
.120 I
.115
.110
t35/2~.]
.
—
+ 204q33 —
+
—
126q34
—
8q36
276q40 + 768q41
—
608 q42
—
—
—
96q38 + 32q39 +
1304q45 —360q46 +480q47
1800q43
—
—
1056q48 —
—
+
l7240q57 —33720q58 +60120q59
—
l09676q60
+207072q61 —396792q62 +5039l6q63 —
565824q64 +783228q65 —737368q66
+426684q67,
[44/271 [38/271 139/271.138/28]
3066q44
+ 3360q49 5400q50 + l4280q51 21927q52 +23076q53 —41 138q54 +44604q56 17817q56 —
.6
134/261 [39/38] (43/211
W 0
(37/27] [34/25] 132/321 130/30] (32/33L[31/33[.131/341 •5 130/31]~~~_ (34/32] (36/27] [31/31k 138/26 (32/30] (33/2428/281 [29/29]’ 1 32/22]~/ 131./301 33133]1 / 36/291(35/28] ~4 33/211/ — 35/29 133/31] [32/211 133/30] [34/29] [42/211 [29/291(3~29]
(I) U_i
1
(1)
where q = 1 p, which we have obtained by the penmeter method [7]. We assume that
139/281 132/321 [42120]
132/281
.875
[3~28]
.880
.885
POSITION OF POLE IN q
.890
-
Fig. 1. Pole-residue plots for P(p)(o)andF(p)(.).
199
Volume 57A, number 3
PHYSICS LETTERS
The points on the pole-residue plot lie on a well defined curve, so that a good estimate of f3 could be ohtamed if p~were known. Since r~ is not known, we have used the estimate P~= 0.1190 ±0.0005 obtained in [8] from low density expansions. The pole-residue curve includes the central value and a linear approximation in its vicinity gives the estimate j3 = 0.474 24~p~±0.002 where Ap~ P~ 0.1190. Assuming that I~p~I ~ 0.0005 as above we obtain —
14 June 1976 Table 1
Model
P~
f.c.c. (bond) 0.1185 f.c.c. (site) 0.195
0.47 ±0.02.
(3)
Further estimates of p~and (3 may be obtained the probability (p) that a chosen bond, when present, belongs to an infinite cluster. We now show that P~p)has the same critical probability and exponent as P(p). Let I be the event that the origin belongs to an infinite cluster and C. the event that the origin is connected to its ith nearest neighbour (i = 1 z). By definition, P(p) = Pr{l} and pP(p) = Pr{IflC~} for all i. (4) from
Ref.
—
0.1195 0.201
0.49 —0.45 0.47 — 0.37
—
this work
f.c.c. b.c.c. (site) (site)
0.198 0.245
— —
0.202 0.251
0.40 —0.33 0.40 — 0.30
[91 [4,51
s.c. (site) (site)
0.3 10 0.311
— —
0.3 14 0.313
0.40 —0.37 — 0.35 0.41
[61
—
=
/3
_________________________________________
independent analysis of the low density series for the site problem on the FCC lattice [91gives a central value of p~of 0.197, which is consistent with a value of /3 in the upper half of the range in [10]. Similarly, extrapolation of the data in [5] to p~= 0.197 leads to a value of /3> 0.4. Correlation scaling was used in [8] to predict (3 = 0.39 ±0.07 for the FCC bond problem, which does not contain our central value. However, use of correlation scaling for the Ising model in three dimensions leads to similar discrepancies.
The result follows since (IflC 1)CI and whenever I occurs at least one of the C1 must occur, therefore p~(p)~P(p)
=
Pr
z U i= 1
(IflC1) <~zpP(p).
(5)
The corresponding pole-residue curve is shown in fig. 1 from which we obtain /3 = 0.46030 ~Pc ±0.002. The discrepancy between the estimates of /3 could be reduced by taking 0.117 ~ ~ 0.118. However, the significance of this fact is not yet established. We have taken the estimate from the site-series since the bondseries has more leading zeros, and the error in (3) almost covers the range of /3 determined by the P-curve. The only direct estimates of 13 for three dimensional lattices that are available for comparison are those for the site problem (see table 1). It is currently believed (but not proved) that f3 is independent of the type of dilution and is a dimensional invariant; thus our estimate of /3 is apparently rather high. However, as the table shows, (3 is very sensitive to the choice of p~.An
200
We would like to thank Dr. M.F. Sykes and Miss M. Glen for making available their data for hp).
References [11 S.R. Broadbent
and J.M. Hammersley, Proc. Camb. Phil. Soc. 53(1957)629. [2] A.G. Dunn et al., J. Phys. C: Solid State Phys. 8 (1975) [3] BJ. Last, Ph.D. thesis, Univ. Birmingham, England (1971)~ [4] P. Dean and N.F. Bird, Mathematics Division Rept. Ma 61, Nat. Phys. Lab., Teddington, England (1966); Proc. Camb. Phil. Soc. 63 (1967) 477. [5] S. Kirkpatrick, Solid State Commun. 12 (1973) 1279. [6] S. Kirkpatrick, Phys. Rev. Lett. 36 (1976) 69.
[71 C. Domb,
Conf. on Fluctuation phenomena and sbchastic processes, Birkbeck College, London; Nature
(London) 184 (1959) 509. [8] 4219. A.G. Dunn et al., J Phys. C: Solid State Phys. 8 (1975) [91 M.A.A. Cox and J.W. Essam, to be published. [10] M.F. Sykes et al., to be published.