Exact enumeration of self-avoiding walks on BCC and FCC lattices

TL;DR

This paper precisely enumerates self-avoiding walks on BCC and FCC lattices up to certain lengths, providing accurate estimates of growth constants and critical exponents, and compares these with previous results on cubic lattices.

Contribution

It introduces an exact enumeration method for BCC and FCC lattices and determines key parameters with high precision, extending the understanding of lattice models.

Findings

Enumerated walks up to length 28 (BCC) and 24 (FCC).

Provided accurate estimates for growth constants and amplitudes.

Results agree with previous studies but offer improved precision.

Abstract

Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents $\gamma$ and $\nu$ which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice $\mu=6.530520(20)$, $A=1.1785(40)$, and $D=1.0864(50)$, and for the FCC lattice $\mu=10.037075(20)$, $A=1.1736(24)$, and $D=1.0460(50)$.

Tables

Table 1. Table 1: Enumeration results for the number of three-dimensional self-avoiding walks Z N subscript 𝑍 𝑁 Z_{N} and the sum of their squared end-to-end distances P N subscript 𝑃 𝑁 P_{N} on the BCC lattice.

$N$	$Z_N$	$P_N$
1	8	24
2	56	384
3	392	4 248
4	2 648	40 704
5	17 960	358 008
6	120 056	2 987 232
7	804 824	23 999 880
8	5 351 720	187 661 376
9	35 652 680	1 436 494 872
10	236 291 096	10 816 140 768
11	1 568 049 560	80 339 567 112
12	10 368 669 992	590 168 152 512
13	68 626 647 608	4 294 543 350 696
14	453 032 542 040	31 003 097 851 872
15	2 992 783 648 424	222 268 142 153 784
16	19 731 335 857 592	1 583 984 756 900 544
17	130 161 040 083 608	11 228 345 566 400 136
18	857 282 278 813 256	79 223 666 339 548 320
19	5 648 892 048 530 888	556 634 161 952 309 400
20	37 175 039 569 217 672	3 896 382 415 388 139 840
21	244 738 250 638 121 768	27 181 650 674 871 447 672
22	1 609 522 963 822 562 936	189 042 890 267 974 827 744
23	10 588 362 063 533 857 304	1 311 064 323 033 684 408 072
24	69 595 035 470 413 829 144	9 069 398 712 299 296 227 648
25	457 555 628 726 692 288 712	62 590 336 418 536 387 660 248
26	3 005 966 051 800 541 943 464	431 019 462 253 450 273 360 416
27	19 752 610 526 081 274 414 584	2 962 188 249 772 759 155 770 280
28	129 713 248 317 927 812 262 200	20 319 964 852 485 237 389 626 176

Table 2. Table 2: Enumeration results for the number of three-dimensional self-avoiding walks Z N subscript 𝑍 𝑁 Z_{N} and the sum of their squared end-to-end distances P N subscript 𝑃 𝑁 P_{N} on the FCC lattice.

$N$	$Z_N$	$P_N$
1	12	24
2	132	576
3	1 404	9 816
4	14 700	144 288
5	152 532	1 951 560
6	1 573 716	25 021 536
7	16 172 148	309 080 808
8	165 697 044	3 714 659 040
9	1 693 773 924	43 714 781 448
10	17 281 929 564	505 948 384 608
11	176 064 704 412	5 777 220 825 912
12	1 791 455 071 068	65 234 797 723 584
13	18 208 650 297 396	729 724 191 726 408
14	184 907 370 618 612	8 097 639 351 530 304
15	1 876 240 018 679 868	89 239 258 469 121 912
16	19 024 942 249 966 812	977 545 487 795 069 952
17	192 794 447 005 403 916	10 651 662 728 070 257 016
18	1 952 681 556 794 601 732	115 520 552 778 504 791 136
19	19 767 824 914 170 222 996	1 247 619 751 507 795 906 248
20	200 031 316 330 580 106 948	13 423 705 093 594 869 393 216
21	2 023 330 401 919 804 218 996	143 942 374 595 787 212 970 696
22	20 458 835 772 261 851 432 748	1 538 749 219 442 520 114 999 744
23	206 801 586 042 610 941 719 148	16 403 200 314 230 418 676 555 512
24	2 089 765 228 215 904 826 153 292	174 411 223 302 510 038 302 309 440

Table 3. Table 3: Extrapolated coefficients of the various BCC series obtained from differential approximants. The confidence intervals are the standard deviations of the central 80% of estimates.

$N$	$Z_N$	$P_N$	$P_N/Z_N$
29	8.51984378150(70)$\times {10}^{23}$	1.39148952051(11)$\times {10}^{26}$	163.323360851(42)
30	5.5928669767(12)$\times {10}^{24}$	9.5134610227(17)$\times {10}^{26}$	170.09989796(10)
31	3.6720987764(23)$\times {10}^{25}$	6.4944301898(72)$\times {10}^{27}$	176.85880953(40)
32	2.4097907972(39)$\times {10}^{26}$	4.4272318727(75)$\times {10}^{28}$	183.71851486(77)
33	1.5816583535(44)$\times {10}^{27}$	3.014025691(25)$\times {10}^{29}$	190.5611070(19)
34	1.037661297(10)$\times {10}^{28}$	2.049378203(42)$\times {10}^{30}$	197.4997221(33)
35	6.808628821(74)$\times {10}^{28}$	1.391831542(69)$\times {10}^{31}$	204.4217013(47)
36	4.46574383(26)$\times {10}^{29}$	9.44216466(95)$\times {10}^{31}$	211.435420(11)
37	2.929428561(97)$\times {10}^{30}$	6.3988380(13)$\times {10}^{32}$	218.432947(13)
38	1.9209657(36)$\times {10}^{31}$	4.3321295(17)$\times {10}^{33}$	225.518346(32)

Table 4. Table 4: Extrapolated coefficients of the various FCC series obtained from differential approximants. The confidence intervals are the standard deviations of the central 80% of estimates.

$N$	$Z_N$	$P_N$	$P_N/Z_N$
25	2.1111652709103(46)$\times {10}^{25}$	1.85010449211473(82)$\times {10}^{27}$	87.63428034806(26)
26	2.132245848773(38)$\times {10}^{26}$	1.9582778101818(72)$\times {10}^{28}$	91.8410891195(22)
27	2.15303362972(17)$\times {10}^{27}$	2.068615279889(35)$\times {10}^{29}$	96.079097491(11)
28	2.1735525326(10)$\times {10}^{28}$	2.18110187619(13)$\times {10}^{30}$	100.347327420(41)
29	2.1938240975(32)$\times {10}^{29}$	2.29572427539(38)$\times {10}^{31}$	104.64486552(13)
30	2.2138677922(93)$\times {10}^{30}$	2.41247069749(92)$\times {10}^{32}$	108.97085672(37)
31	2.233701285(63)$\times {10}^{31}$	2.5313307684(21)$\times {10}^{33}$	113.32449876(98)
32	2.25334058(14)$\times {10}^{32}$	2.6522953987(45)$\times {10}^{34}$	117.7050374(24)
33	2.2728013(51)$\times {10}^{33}$	2.7753566769(86)$\times {10}^{35}$	122.1117622(56)

Table 5. Table 5: Summary of our parameter estimates for γ 𝛾 \gamma and ν 𝜈 \nu , with comparison to values from the literature. Except where noted, the series estimates for γ 𝛾 \gamma and ν 𝜈 \nu from the literature come from the simple cubic lattice.

Source111Abbreviations: MC $\equiv$ Monte Carlo, CB $\equiv$ conformal bootstrap, FT $\equiv$ field theory, MCRG $\equiv$ Monte Carlo renormalization group.	$\gamma$	$\nu$
This work	$1.15650\left(50\right)$	$0.58785\left(40\right)$
Clisby (2017) MC (2017)	1.15695300(95)
Clisby and Dünweg (2016) MC (2016)		0.58759700(40)
Shimada and Hikami (2016) CB (2016)	1.1588(25)	0.5877(12)
Schram et al. (2011) Series $N\le 36$ (2011)	1.15698(34)	0.58772(17)
Clisby (2010) MC (2010)		0.587597(7)
Clisby et al. (2007)222Using Eqs (74) and (75) with $0.516\le {\Delta }_1\le 0.54$. Series $N\le 30$ (2007)	1.1569(6)	0.58774(22)
Hsu and Grassberger (2004) MC (2004)	1.1573(2)
Prellberg (2001) MC (2001)		0.5874(2)
MacDonald et al. (2000) Series $N\le 26$ (2000)	1.1585	0.5875
Caracciolo et al. (1998) MC (1998)	1.1575(6)
Guida and Zinn-Justin (1998) FT $d=3$ (1998)	1.1596(20)	0.5882(11)
Guida and Zinn-Justin (1998) FT $ϵ$ bc (1998)	1.1571(30)	0.5878(11)
Butera and Comi (1997) Series $N\le 21$ (1997)	1.161(2)	0.592(2)
Butera and Comi (1997) Series $N\le 21$, biased (1997)	1.1594(8)	0.5878(6)
Butera and Comi (1997) BCC series $N\le 21$ (1997)	1.1612(8)	0.591(2)
Butera and Comi (1997) BCC series $N\le 21$, biased (1997)	1.1582(8)	0.5879(6)
Belohorec (1997) MCRG (1997)		0.58756(5)
Li et al. (1995) MC (1995)		0.5877(6)
MacDonald et al. (1992) Series $N\le 23$ (1992)	1.16193(10)
Guttmann (1989a) Series $N\le 21$ (1989)	1.161(2)	0.592(3)

Equations

$$Z_N$$

$$\frac{P_N}{Z_N}$$

$$Z_{2N}=Z_N^2+\sum _{S\ne \varnothing }(-1{)}^{|S|}{Z}_N^2(S),$$

$$P_N$$

$$\sum _{i=0}^K{Q}_i(z){\left(z\frac{d}{dz}\right)}^{i}F(z)=P(z).$$

$$\log Z_N$$

$$\log \frac{P_N}{Z_N}$$

Full text

I Introduction

The enumeration of self-avoiding walks (SAWs) on regular lattices is a classical combinatorial problem in statistical physics, with a long history, see e.g. Madras and Slade (1993); Janse van Rensburg (2015). Of the three-dimensional lattices, the simple cubic (SC) lattice has drawn the most effort, starting with a paper by Orr Orr (1947) from 1947, where the number of SAWs $Z_{N}$ was given for all $N$ up to $N_{\max}=6$; these results were obtained by hand. In 1959, Fisher and Sykes Fisher and Sykes (1959) used a computer to enumerate all SAWs up to $N_{\max}=9$; Sykes and collaborators extended this to 11 terms in 1961 Sykes (1961), 16 terms in 1963 Sykes (1963), and 19 terms in 1972 Sykes et al. (1972). In the following decade Guttmann Guttmann (1987) enumerated SAWs up to $N_{\max}=20$ in 1987, and extended this by one step in 1989 Guttmann (1989a). In 1992, MacDonald et al. MacDonald et al. (1992) reached $N_{\max}=23$, and in 2000 MacDonald et al. MacDonald et al. (2000) reached $N_{\max}=26$. In 2007, a combination of the lace expansion and the two-step method allowed for the enumeration of SAWs up to $N_{\max}=30$ steps Clisby et al. (2007). Recently, the length-doubling method Schram et al. (2011) was presented which allowed enumerations to be extended up to $N_{\max}=36$. To date, this is the record series for the SC lattice.

The body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are in principle equally as physically relevant as the SC lattice, but enumeration is hampered by the larger lattice coordination numbers, which detriments most enumeration methods severely. It is also very slightly more cumbersome to write computer programs to perform enumerations for these lattices. Consequentially, the SC lattice has served as the test-bed problem for new enumeration algorithms, and the literature on enumerations for the BCC and FCC lattices is far more sparse. For the BCC lattice, $Z_{N}$ was determined up to $N_{\max}=15$ in 1972 Sykes et al. (1972), and to $N_{\max}=16$ in 1989 Guttmann (1989a). The current record of $N_{\max}=21$ was obtained in 1997 by Butera and Comi Butera and Comi (1997) as the $N\rightarrow 0$ limit of the high temperature series for the susceptibility of the $N$-vector model. For the FCC lattice, enumerations up to $N_{\max}=12$ were performed in 1967 Martin et al. (1967), and the record of $N_{\max}=14$ was achieved way back in 1979 McKenzie (1979).

Enumeration results derive their relevance from the ability to determine critical exponents, which, according to renormalization group theory, are believed to be shared between SAWs on various lattices and real-life polymers in solution de Gennes (1979). Two such exponents are the entropic exponent $\gamma$ and the size exponent $\nu$. Given the number $Z_{N}$ of SAWs of all lengths up to $N_{\max}$ and the sum $P_{N}$ of their squared end-to-end extensions, these two exponents can be extracted using the relations

[TABLE]

In these expressions, the growth constant $\mu$ and the amplitudes $A$ and $D$ are non-universal (model-dependent) quantities, while the leading correction-to-scaling exponent is a universal quantity with value $\Delta_{1}=0.528(8)$ Clisby and Dünweg (2016). Sub-leading corrections-to-scaling are absorbed into the $O(1/N)$ term. $\sigma$ is a lattice specific constant to ensure that our amplitude “$D$” is the same as in earlier work. $\sigma$ corrects for the fact that with our definition each step of the walk is of length $\sqrt{2}$ for the BCC lattice (leading to $\sigma=2$), and of length $\sqrt{3}$ for the FCC lattice (leading to $\sigma=3$). Note that for bipartite lattices, of which the SC and BCC lattices are examples, there is an additional alternating “anti-ferromagnetic” singularity, that is sub-leading but which still must be treated carefully as the odd-even oscillations tend to become amplified by series analysis techniques. Because of universality, the exponents are clearly more interesting from a physics perspective. However, accurate estimates for the growth constant and the amplitudes can also be very helpful for many kinds of computer simulations on lattice polymers.

In this paper, we used the length-doubling method Schram et al. (2011) to measure $Z_{N}$ and $P_{N}$ up to $N_{\max}=28$ and 24, on the BCC and FCC lattices, respectively. These lattices can be easily simulated as subsets of the SC lattice: the collection of sites in which $x$, $y$ and $z$ are either all even or all odd forms a BCC lattice, and the collection of sites $(x,y,z)$ constrained to even values for $x+y+z$ forms a FCC lattice. We then analysed these results to obtain estimates for the exponents $\gamma$ and $\nu$, the growth constant $\mu$, and the amplitudes $A$ and $D$. Our results for the two exponents $\gamma$ and $\nu$ agree with the values reported in literature which are obtained on the SC lattice, reinforcing the credibility of the literature values. Our results for the growth constant $\mu$ and the amplitudes $A$ and $D$ for the BCC and FCC lattices are the most accurate ones to date.

The manuscript is organized as follows. First, in Sec. II we present a short outline of the length-doubling method, and present the enumeration data. In Sec. III we describe the analysis method we use, before summarising our results and giving a brief conclusion in Sec. IV.

II Length-doubling method

We first present an intuitive description of the length-doubling method; a more formal description can be found in Schram et al. (2011). In the length-doubling method, the number $Z_{2N}$ of SAWs with a length of $2N$ steps, with the middle rooted in the origin, is obtained from the walks of length $N$, with one end rooted in the origin, and the number $Z_{N}(S)$ of times that a subset $S$ of sites is visited by such a walk of length $N$. The lowest-order estimate for $Z_{2N}$ is the number of combinations of two SAWs of length $N$, i.e. $Z_{N}^{2}$. This estimate is too large since it includes pairs of SAWs which overlap. The first correction to $Z_{2N}$ is the lowest-order estimate for the number of pairs of overlapping SAWs, which can be obtained from the number $Z_{N}(\{s\})$ of SAWs of length $N$ which pass through a single site $s$. The first correction is then to subtract $Z_{N}(\{s\})^{2}$, summed over all sites $s$. This first correction is too large, as it includes pairs of SAWs twice, if they intersect twice. The second correction corrects for this over-subtraction, by adding the number $Z_{N}(\{s,t\})$ of SAWs that pass through the pair of sites $\{s,t\}$. Continuing this process with groups of three sites, etc., the number $Z_{2N}$ of SAWs of length $2N$ can then be obtained by the length-doubling formula

[TABLE]

where $|S|$ denotes the number of sites in $S$.

The usefulness of this formula lies in the fact that the numbers $Z_{N}(S)$ can be obtained relatively efficiently:

• •

Generate each SAW of length $N$.

• •

Generate for each SAW each of the $2^{N}$ subsets $S$ of lattice sites, and increment the counter for each specific subset. Multiple counters for the same subset $S$ must be avoided; this can be achieved by sorting the sites within each subset in an unambiguous way.

• •

Finally, compute the sum of the squares of these counters, with a positive and negative sign for subsets with an even and odd number of sites, respectively, as in Eq. (3).

As there are $Z_{N}$ walks of length $N$, each visiting $2^{N}$ subsets of sites, the computational complexity is $\mathcal{O}(2^{N}Z_{N})\sim(2\mu)^{N}$ times some polynomial in $N$ which depends on implementation details. This compares favourably to generating all $Z_{2N}\sim\mu^{2N}$ walks of length $2N$, provided $\mu>2$. This is the case on the SC lattice, with $\mu=4.684$, and even more so for the BCC and FCC lattices, as we will show. The length-doubling method can also compute the squared end-to-end distance, summed over all SAW configurations; for details we refer to Schram et al. (2011). Details on the efficient implementation of this algorithm are presented in Schram et al. (2013).

The direct results of the length-doubling method, applied to SAWs on the BCC and FCC lattices, are presented in Tables 1 and 2, respectively. The BCC results for $N\leq 26$ and FCC results for $N\leq 22$ were obtained and verified by two independent computer programs: SAWdoubler 2.0, available from http://www.staff.science.uu.nl/~bisse101/SAW/, and Raoul Schram’s program. The BCC results presented for the largest problems $N=27,28$ were obtained by SAWdoubler 2.0 only, and the FCC results for $N=23,24$ were obtained by Raoul Schram’s program only. Thus the largest two problem instances for each lattice were not independently verified since these require a very large amount of computer time and memory. Still, based on our analysis we believe that the given values are correct.

III Analysis

We now proceed to analyse our series in order to extract estimates for various parameters. In addition to the expressions for $Z_{N}$ and $P_{N}/Z_{N}$ in Eqs (1) and (2), we also have

[TABLE]

As discussed earlier, we expect the critical exponents $\gamma$ and $\nu$ and the leading correction-to-scaling exponent $\Delta_{1}$ to be the same for self-avoiding walks on the SC, BCC, and FCC lattices. The amplitudes $A$ and $D$ are non-universal quantities, i.e. they are lattice dependent, while $\sigma=2$ for the BCC lattice and $\sigma=3$ for the FCC lattice. In the analysis below, we include a subscript to indicate the appropriate lattice.

The BCC lattice is bipartite, which introduces an additional competing correction which has a factor of $(-1)^{N}$, so causing odd-even oscillations. We reduce the influence of this additional sub-leading correction by separately treating the sequences for even and odd $N$. See Clisby et al. (2007) for more detailed discussion on this point for the asymptotic behaviour of $Z_{N}$ on the SC lattice, which is also bipartite.

We now describe the method of analysis we used, which involved two stages: extrapolation of the series via a recently introduced method involving differential approximants Guttmann (2016), and then direct fitting of the extended series with the asymptotic forms in Eqs (1), (2), and (4). We report our final estimates in Table 5 at the end of the section.

III.1 Extrapolation

Perhaps the most powerful general-purpose method for the analysis of series arising from lattice models in statistical mechanics is the method of differential approximants, described in Guttmann (1989b). The basic idea is to approximate the unknown generating function $F$ by the solution of an ordinary differential equation with polynomial coefficients. In particular if we know $r$ coefficients $f_{0},f_{1},\cdots,f_{r-1}$ of our generating function $F$, then we can determine polynomials $Q_{i}(z)$ and $P(z)$ which satisfy the following $K$th order differential equation order by order:

[TABLE]

The function determined by the resulting differential equation is our approximant. The power of the method derives from the fact that such ordinary differential equations accommodate the kinds of critical behaviour that are typically seen for models of interest.

Differential approximants are extremely effective at extracting information about critical exponents from the long series that have been obtained for two-dimensional lattice models, such as self-avoiding polygons Clisby and Jensen (2012) or walks on the square lattice Jensen (2016). However, differential approximants have been far less successful for the shorter series available for three-dimensional models such as SAWs on the simple cubic lattice Clisby et al. (2007); Schram et al. (2011). For short series, it seems that corrections-to-scaling due to confluent corrections are too strong at the orders that can be reached to be able to reliably determine critical exponents. (In fact, it is extremely easy to be misled by apparent convergence, while in fact estimates have not settled down to their asymptotic values.) The method that has proved most reliable is direct fitting of the asymptotic form Clisby et al. (2007), which we describe in the next sub-section.

However, we can do better than the usual method of performing direct fits of the original series, and adopt a promising new approach recently invented by Guttmann Guttmann (2016), which is a hybrid of the differential approximant and direct fitting techniques. The underlying idea is to exploit the fact that differential approximants can be used to extrapolate series with high accuracy even in circumstances when the resulting estimates for critical exponents are not particularly accurate, or even when the asymptotic behaviour is non-standard such as being of stretched exponential form. The extrapolations can be extremely useful in cases where corrections-to-scaling are large, as the few extra terms they provide may be the only evidence of a clear trend from the direct fits.

We have 28 exact terms for the BCC series, and 24 exact terms for the FCC series. We used second order inhomogeneous approximants to extrapolate the series for $Z_{N}$, $P_{N}$, and $P_{N}/Z_{N}$, where we allowed the multiplying polynomials to differ by degree at most 3. In each case we calculated trimmed mean values, eliminating the outlying top and bottom 10% of estimates, with the standard deviation of the remaining extrapolated coefficients providing a proxy for the confidence interval. Note that this is an assumption, and relies on the extrapolation procedure working well for our problem. In practice, this approach of inferring the confidence interval from the spread of estimates appears to be quite reliable in the cases for which it has been tested. We have also confirmed the reliability of the extrapolations by using the method to “predict” known coefficients from truncated series. We report our extended series in Tables 3 and 4.

III.2 Direct fits

We then fitted sequences of consecutive terms of the extrapolated series for $Z_{N}$ and $P_{N}/Z_{N}$ to the asymptotic forms given in Eqs (1) and (2), respectively. We found that fits of $P_{N}/Z_{N}$ were superior to fits of $P_{N}$ for estimates of $\nu$ and the parameter $D$, and hence we do not report fits of $P_{N}$ here.

To convert the fitting problem to a linear equation, we took the logarithm of the coefficients, which from Eqs (1) and (2) we expect to have the following asymptotic forms:

[TABLE]

We used the linear fitting routine “lm” in the statistical programming language R to perform the fits.

In all of the fits, we biased the exponent $\Delta_{1}$ of the leading correction-to-scaling term, performing the fits for three different choices of $\Delta_{1}=0.520,0.528,0.536$ which correspond to the best Monte Carlo estimate of $\Delta_{1}=0.528(8)$. We approximated the next-to-leading correction-to-scaling term with a term of order $1/N$, which we expect to behave as an effective term which takes into account three competing corrections with exponents $-2\Delta_{1},-1,-\Delta_{2}\approx-1$. For $\log Z_{N}$, we fitted $\log A$, $\log\mu$, $\gamma$, the amplitude $a$, and the amplitude of the $1/N$ effective term. For $\log(P_{N}/Z_{N})$, we fitted $\log D$, $\nu$, the amplitude $b$, and the amplitude of the $1/N$ term. For the BCC lattice, we minimised the impact of the odd-even oscillations by fitting even and odd subsequences separately. We included the extrapolated coefficients in our fits, repeating the calculation for the central estimates and for values which are one standard deviation above and below them.

This procedure gave us up to nine estimates for each sequence of coefficients (from the three choices of $\Delta_{1}$, and the three choices of extrapolated coefficient values). For the central parameter estimates we used the case where $\Delta_{1}=0.528$ (the central value) in combination with the central value of the extrapolated coefficients. We also calculated the maximum and minimum parameter estimates over the remaining 8 cases.

For the BCC lattice, we found that 5 of the extrapolated coefficients gave a spread which was only moderately greater than the spread arising from varying $\Delta_{1}$, effectively extending the series to 33 terms. For the FCC lattice, we found we could use 3 additional coefficients, extending the series to 27 terms.

For each of the parameter estimates, we plotted them against the expected relative magnitude of the first neglected correction-to-scaling term. This should result in approximately linear convergence as we approach the $N\rightarrow\infty$ limit which corresponds to approaching the $y$-axis from the right in the following figures. In Eqs (6) and (7) we expect that the next term, which is not included in the fits, is of $O(N^{-1-\Delta_{1}})$; given that $\Delta_{1}\approx 0.5$, we take the neglected term to be $O(N^{-3/2})$. The value of $N$ that is used in the plot is the maximum value of $N$ in the sequence of fitted coefficients, which we denote $N_{\max}$ in the plots.

We plot our fitted values in Figures 2–8. For ease of interpretation we converted estimates of $\log\mu$, $\log A$, and $\log D$ to estimates of $\mu$, $A$, and $D$. We note that the parameter estimates arising from the odd subsequence of the BCC series for $Z_{N}$ benefited dramatically from the extrapolated sequence. Examining estimates for $\gamma$ in Fig. 2, $\mu_{\rm bcc}$ in Fig. 4, and $A_{\rm bcc}$ in Fig. 6 we see in each case that the trend of the odd subsequence would be dramatically different were it not for the three additional odd terms in the extrapolated sequence. In other cases the additional coefficients are useful, and certainly make the trend for the estimates clearer, but are not as crucial.

Our final parameter estimates are plotted on the $y$-axes.

IV Summary and conclusion

We give our estimates for $\gamma$ and $\nu$ in Table 5, where we also include estimates coming from the literature. We observe that our estimates are consistent with the literature values, but that the recent Monte Carlo estimates of $\gamma$ and $\nu$, using the pivot algorithm, are far more accurate than the estimates from series. The estimates coming from our enumerations on the BCC and FCC lattices are not quite as precise as the estimates coming from the SC lattice only, but the fact that they are coming from two independent sources, with different systematic errors, makes these new estimates more robust.

In addition, our estimates of the non-universal quantities for the BCC lattice are $A_{\rm bcc}=1.1785(40)$, $D_{\rm bcc}=1.0864(50)$, and $\mu_{\rm bcc}=6.530520(20)$, which should be compared with earlier estimates of 6.5304(13) Guttmann (1989a) from 1989, and unbiased and biased estimates respectively of 6.53036(9) and 6.53048(12) Butera and Comi (1997) from 1997. Our estimates of the non-universal quantities for the FCC lattice are $A_{\rm fcc}=1.1736(24)$, $D_{\rm fcc}=1.0460(50)$, and $\mu_{\rm fcc}=10.037075(20)$, which should be compared with earlier estimates of 10.03655 McKenzie (1979) from 1979, and 10.0364(6) Guttmann (1987) from 1987 (where these estimates come from different analyses of the same $N\leq 14$ term series).

In conclusion, the length-doubling algorithm has resulted in significant extensions of the BCC and FCC series. The application of a recently invented series analysis technique Guttmann (2016), which combines series extrapolation from differential approximants with direct fitting of the extrapolated series, has given excellent estimates of the various critical parameters. In particular, estimates of the growth constants for the BCC and FCC lattices are far more accurate than the previous literature values.

Acknowledgements

This work was sponsored by NWO-Science for the use of supercomputer facilities under the project SH-349-15. Computations were carried out on the Cartesius supercomputer at SURFsara in Amsterdam. N.C. acknowledges support from the Australian Research Council under the Future Fellowship scheme (project number FT130100972) and Discovery scheme (project number DP140101110).