Bond percolation on simple cubic lattices with extended neighborhoods
BBond percolation on simple cubic lattices with extended neighborhoods
Zhipeng Xun ∗ and Robert M. Ziff † School of Materials and Physics, China University of Mining and Technology, Xuzhou 221116, China Department of Chemical Engineering and Center for the Study of Complex Systems,University of Michigan, Ann Arbor, Michigan 48109-2800, USA (Dated: July 7, 2020)We study bond percolation on the simple cubic (sc) lattice with various combinations of first,second, third, and fourth nearest-neighbors by Monte Carlo simulation. Using a single-clustergrowth algorithm, we find precise values of the bond thresholds. Correlations between percolationthresholds and lattice properties are discussed, and our results show that the percolation thresholdsof these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power law p c ∼ z − a , with exponent a = 1 . z , the threshold must approach the Bethe lattice result p c = 1 / ( z − p c ( z −
1) = 1 + 1 . z − / . PACS numbers: 64.60.ah, 89.75.Fb, 05.70.Fh
I. INTRODUCTION
Percolation is a fundamental model in statisticalphysics [1, 2]. It is used to describe a variety of nat-ural processes, such as liquids moving in porous media[3, 4], forest fire problems [5, 6] and epidemics [7]. It isalso a model for phase-transition phenomena. In perco-lation systems, sites or bonds on a lattice are occupiedwith probability p , and the value of p at which an infinitecluster (in an infinite system) first appears is known asthe percolation threshold p c .Many kinds of lattices, graphs, and networks have beeninvestigated to find the percolation thresholds and thecorresponding critical exponents. In two dimensions, ex-act values of percolation thresholds are known for severalclasses of lattices [8–16], but there are still many more lat-tices where thresholds cannot be found analytically, andin higher dimensions there are no exact solutions at all.Consequently, a main focus of investigation at present isstill based on approximation schemes or numerical simu-lations.Numerous algorithms and techniques have been devel-oped to find threshold numerically [2, 15, 17–28]. Manyrelated problems in percolation have also received atten-tion recently [29–38]—it remains a very active field.The study of three-dimensional lattices (the most com-mon ones being the simple cubic (sc), the face-centeredcubic (fcc), the body-centered cubic (bcc), and diamondlattices) is particularly important, due to their rele-vance for many natural processes. Much work in find-ing thresholds and critical exponents has been done inthree dimensions [20, 25, 26, 33, 39–44], and the valuesof percolation thresholds have been more and more ac-curate. Lorenz and Ziff [20] performed extensive MonteCarlo simulations to study bond percolation on three-dimensional lattices ( p c (sc) = 0 . p c (fcc) = ∗ [email protected] † rziff@umich.edu . p c (bcc) = 0 . p c (sc) = 0 . p c (fcc) = 0 . p c (bcc) = 0 . p c (sc) = 0 . p c (fcc) =0 . p c (bcc) = 0 . random numbers must be generated to achieve that levelof accuracy, and would require ≈ days of computa-tion on a single node. NN 2NN 3NN 4NN
FIG. 1. The neighborhoods considered here: nearest-neighbors (NN) (black with heavy bond, 6 vertices); 2NN(red with dashed bond, 12 vertices); 3NN (blue with no linksto the origin, 8 vertices); and 4NN (green with thin bond, 6vertices).
The problem of studying percolation on lattices with a r X i v : . [ c ond - m a t . d i s - nn ] J u l extended neighborhoods has received a great deal of at-tention in the last decades [43–47], with much work stim-ulated by the 2005 paper of Malarz and Galam [48]. Withextended neighbors, the coordination number z can bevaried over a wide range, so many types of systems canbe studied, and also there are applications where theseresults are useful [49]. Site percolation on lattices withextended neighborhoods corresponds to problems of ad-sorption of extended shapes on a lattice, such as k × k squares on a square lattice [50, 51]. Bond percolation re-lates to long-range links similar to small-world networks[52] and models of long-range percolation [53]. In twodimensions, having lattices with complex neighborhoodsmodels non-planar systems.For three-dimensional systems, some work has beendone for the sc lattice with extended neighborhoods[43, 44], although to relatively low precision and forsite percolation only. Precise percolation thresholdsare needed in order to study the critical behavior, in-cluding critical exponents, critical crossing probabili-ties, critical and excess cluster numbers, etc. There-fore, in this paper, we study bond percolation for sev-eral sc lattices with extended neighborhoods, includingcombinations of nearest-neighbors (NN), second nearest-neighbors (2NN), third nearest-neighbors (3NN), andfourth nearest-neighbors (4NN), as shown in Fig. 1. Weuse an effective single-cluster growth method similar tothat of Lorenz and Ziff [20] and what we have recentlyused to study percolation problems in four dimensions[54]. Thresholds for these systems were never studied forbond percolation, as far as we know, and thus we find allnew values. We find results to a precision of five or sixsignificant digits.With regard to the sc lattice with extended neighbor-hoods, crossing bonds exist in these kinds of structures.This bond percolation model with crossing bonds lives inan extended space of connectivities [55]. Here we showthat the single-cluster growth method we used in thispaper can be efficiently applied to these kinds of lattices.Another goal of this paper is to explore the relationbetween percolation threshold and coordination number.The value of percolation thresholds depends on kind ofpercolation (site or bond), lattice topology and assumedneighborhoods, etc. The study of how thresholds de-pend upon lattice structure, especially the coordinationnumber z , has also had a long history [56–61]. Havingthresholds of more lattices is useful for extending thosecorrelations.In the following sections, we present the underlyingtheory, and discuss the simulation process. Then wepresent and briefly discuss the results that we obtainedfrom our simulations. II. THEORY
A quantity of central interest in percolation is the clus-ter size distribution n s ( p ), which is defined as the num- ber of clusters (per site) containing s occupied sites, as afunction of the occupation probability p . At the percola-tion threshold p c , n s is expected to behave as n s ∼ A s − τ (1 + B s − Ω + . . . ) , (1)where τ is the Fisher exponent, and Ω is the exponentfor the leading correction to scaling. Both τ and Ω areexpected to be universal—the same for all lattices of agiven dimensionality. In three dimensions, relatively ac-curate results for τ exist: 2 . . . . . . . A and B are constants that depend upon the system and arenon-universal.Note that even though we are considering bond perco-lation, we characterize the size of the cluster by the num-ber of sites it contains. This is in fact a common way todo it, and convenient for the growth method to generateclusters that we employ here, where we do not determinethe states of internal bonds. This is also natural in manytheoretical approaches such as the Temperley-Lieb cal-culation for percolation [66]. In any case, the number ofoccupied bonds of a cluster is proportional to the numberof occupied sites for large clusters, so either choice willyield the same scaling.The probability a site (vertex) belongs to a cluster withsize greater than or equal to s will then be P ≥ s = ∞ (cid:88) s (cid:48) = s s (cid:48) n s (cid:48) ∼ A s − τ (1 + B s − Ω + . . . ) , (2)where A = A / ( τ −
2) and B = ( τ − B / ( τ + Ω − p is away from p c , a scaling func-tion needs to be included. Then the behavior for large s (ignoring corrects to scaling here) can be represented as P ≥ s ∼ A s − τ f ( B ( p − p c ) s σ ) , (3)Here σ is another universal exponent, which is estimatedto be 0 . . . f ( x ) can be expanded as a Taylorseries, f ( B ( p − p c ) s σ ) ∼ C ( p − p c ) s σ + · · · . (4)where C = B f (cid:48) (0). We assume f (0) = 1, so that A = A . Combining Eqs. (3) and (4) leads to s τ − P ≥ s ∼ A + D ( p − p c ) s σ . (5)where D = A C .The theory mentioned above provides us two methodsto determine p c . The first way, we can plot s τ − P ≥ s vs s σ . Equation (5) predicts that s τ − P ≥ s will convergenceto a constant value at p c for large s , while it deviates froma constant value when p is away from p c . The second way,we can plot s τ − P ≥ s vs s − Ω . It can be seen from Eq. (2)that there will be a linear relationship between s τ − P ≥ s and s − Ω for large s , if we choose the correct value ofΩ, while for p (cid:54) = p c , where Eq. (2) does not apply, thebehavior will be nonlinear.We also consider a third method to study p c and τ . Itfollows from Eq. (2) that, at p c ,ln P ≥ s − ln P ≥ s ln 2 ∼ (2 − τ )(ln 2 s − ln s )ln 2 − B s − Ω (2 − Ω − ∼ (2 − τ ) + B s − Ω , (6)where (ln P ≥ s − ln P ≥ s ) / ln 2 is the local slope of a plotof ln P ≥ s vs ln s , and B = B (2 − Ω − / ln 2. Equation(6) implies that if we make of plot of the local slope vs s − Ω at p c , linear behavior will be seen for large s , andthe intercept s − Ω → − τ ). Again, if we are not at p c , the behaviorwill not be linear for large s . III. SIMULATION RESULTS
We carried out numerical simulations using the single-cluster growth algorithm. First, a site on the lattice ischosen as the seed. Under periodic boundary conditions,any site on the lattice can be chosen as the seed. Then,an individual cluster is grown at that seeded site. Togrow the clusters, we check all neighbors of a growthsite for unvisited sites, which we occupy with probability p , or leave unoccupied with probability 1 − p , and putthe newly occupied growth sites on a first-in, first-outqueue. To simulate bond percolation, we simply leave thesites in the unvisited state when we do not occupy them,i.e., when rnd < p , where rnd is a uniformly distributedrandom number in (0 , n , any site whose valueis less than n is considered unoccupied. When a site isoccupied in the growth of a new cluster, it is assignedthe value of the cluster number n . The procedure saves agreat deal of time because we can use a very large lattice,and do not have to clear out the lattice after each clusteris generated.Another advantage of the single-cluster growth methodis that it is very simple to record and analyze the results [20]. We attribute clusters of different sizes to differentbins. Clusters whose size (number of sites) fall in a rangeof (2 n , n +1 −
1) for n = 0 , , , . . . belong to the n -thbin. Clusters still growing when they reach the uppersize cutoff are counted in the last bin. Then, the onlything we need to record is the number of clusters in eachof the bins. Thus one does not need to study propertieslike the intersections of crossing probabilities for differentsize systems or create large output files of intermediatemicrocanonical results to find estimates of the thresh-old. The cutoff is 2 occupied sites for all the lattices inthis paper, meaning that the output files here are simplythe 17 values of the bins for each value of p . While themethod is not as efficient as the union-find method [69],which utilizes only one set of runs to simulate all valuesof p , it has the virtue that it is simple to analyze. If oneconcentrates the longest runs only to the values closestto p c (determined as one goes), the net disadvantage isnot that great.We have tested this method on the sc lattice, and find p c = 0 . τ = 2 . . L × L × L with L = 512, and with helical periodic bound-ary conditions. 10 independent samples were producedfor each lattice, representing several weeks of computertime each. Then the number of clusters greater than orequal to size s was found based on the data from oursimulations, and s τ − P ≥ s could be easily calculated.The plot of s τ − P ≥ s (with τ = 2 . s σ (with σ = 0 . p is shown in Fig. 2. For small clusters, there is asteep decline due to finite-size effects. For large clusters,the plot shows a linear region. The closer p is to p c , thelinear portions of the curve become more nearly horizon-tal. Then the value of p c can be deduced by plotting theslope of that linear part vs p , since by (5),d( s τ − P ≥ s )d( s σ ) ∼ D ( p − p c ) , (7)Finding the intercept where the derivative equals zeroyields p c . This is shown in the inset of Fig. 2. Thepredicted value of the percolation threshold, which is p c = 0 . x intercept in theinset plot.If we try different values of τ , we find the value of p c changes by a small amount. If instead we use τ =2 . τ , wefind p c = 0 . τ = 2 . p c = 0 . p c = 0 . τ .Fig. 3 shows the plot of s τ − P ≥ s (with τ = 2 . s − Ω (with Ω = 0 .
63) for the sc-NN+4NN lattice underdifferent values of p for large clusters. When p is verynear to p c , we can see better linear behavior, while thecurves show a deviation from linearity if p is away from p c . From this plot, we can conclude that 0 .
63 for the sc-NN+4NN lattice under the valuesof p = 0 . . . . . p to calculate τ . We determinethe value of τ falls in the interval of (2 . , . τ = 2 . . . p = 0 . . . p c .If we plotted points representing slopes from the lastsix bins, for example, we would have to use a quadraticto fit the data, as shown in Fig. 5. Here we are effec-tively assuming the next-order correction has exponent2Ω. However, the fit is not that good and the interceptdoes not agree with the value of τ found above, so we donot consider higher-order corrections further. In fact, wedo not report any of the plots of the local slopes (6) forthe other lattices.The simulation results for the other ten lattices we con-sidered are shown in the Supplementary Material in Figs.1-20. and the corresponding percolation thresholds aresummarized in Table I. We did not calculate the values of τ for all these lattices one by one; otherwise, the overallsimulation time would at least double. For all these plotswe assumed the values τ = 2 . σ = 0 . IV. DISCUSSION
In Table I, the lattices are arranged in the order of in-creasing coordination number z . As one would expect,the values of p c decrease with increasing z . For refer-ence, we have also added the site percolation thresholdsfound in Refs. [44] and [43]. Note that the ordering ofthe site thresholds is not always the same as for the bondthresholds, and the site thresholds are not all monotonicwith z as the bond thresholds are.In percolation research, there has been a long historyof studying correlations between percolation thresholdsand lattice properties [56, 58–60]. For example, in Ref. s (cid:1) s (cid:2) -2 P >= s p = 0 . 1 0 6 8 2 2 p = 0 . 1 0 6 8 2 3 p = 0 . 1 0 6 8 2 4 p = 0 . 1 0 6 8 2 5 p = 0 . 1 0 6 8 2 6 p = 0 . 1 0 6 8 2 7 p = 0 . 1 0 6 8 2 8 p = 0 . 1 0 6 8 2 9 p = 0 . 1 0 6 8 3 0 p = 0 . 1 0 6 8 3 1 p = 0 . 1 0 6 8 3 2 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. 2. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ =0 . p . The inset indicates the slope of the linear portions of thecurves shown in the main figure as a function of p , and thethreshold value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 1 0 6 8 2 2 p = 0 . 1 0 6 8 2 3 p = 0 . 1 0 6 8 2 4 p = 0 . 1 0 6 8 2 5 p = 0 . 1 0 6 8 2 6 p = 0 . 1 0 6 8 2 7 p = 0 . 1 0 6 8 2 8 p = 0 . 1 0 6 8 2 9 p = 0 . 1 0 6 8 3 0 p = 0 . 1 0 6 8 3 1 p = 0 . 1 0 6 8 3 2 FIG. 3. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+4NN lattice under different values of p . [43], Kurzawski and Malarz found that the site thresholdsfor several three-dimensional lattices can be fitted fairlywell by a simple power-law in z : p c ( z ) ∼ cz − a , (8)with a = 0 . z − − a rather than vs z − a . For bond percolation in four dimen-sions, we found a = 1 .
087 in Ref. [54] (where we calledthe exponent a as γ ). - 0 . 1 8 9 4- 0 . 1 8 9 2- 0 . 1 8 9 0- 0 . 1 8 8 8- 0 . 1 8 8 6 y = 0 . 1 7 9 8 8 x - 0 . 1 8 9 3 8 y = 0 . 1 3 8 5 3 x - 0 . 1 8 9 1 7 y = 0 . 1 0 7 6 1 x - 0 . 1 8 9 0 9 y = 0 . 0 6 8 5 7 x - 0 . 1 8 9 0 0 y = 0 . 0 0 5 2 7 x - 0 . 1 8 8 7 3 (ln P >=2 s -ln P >= s )/ln2 s - W p = 0 . 1 0 6 8 2 6 p = 0 . 1 0 6 8 2 6 3 p = 0 . 1 0 6 8 2 6 3 5 p = 0 . 1 0 6 8 2 6 4 p = 0 . 1 0 6 8 2 7 FIG. 4. Plot of local slope (ln P ≥ s − ln P ≥ s ) / ln 2 vs s − Ω with Ω = 0 .
63 for the sc-NN+4NN lattice under values of p =0 . . . . . (ln P >=2 s -ln P >= s )/ln2 s - W p = 0 . 1 0 6 8 2 6 3 p = 0 . 1 0 6 8 2 6 3 5 p = 0 . 1 0 6 8 2 6 4 y = - 4 0 . 2 2 6 5 8 x + 0 . 1 0 2 4 0 x - 0 . 1 8 8 9 6 y = - 4 2 . 1 1 4 2 9 x + 0 . 1 2 4 1 4 x - 0 . 1 8 9 0 2 y = - 4 3 . 6 2 1 3 5 x + 0 . 1 4 0 3 5 x - 0 . 1 8 9 0 7 FIG. 5. Plot of local slope (ln P ≥ s − ln P ≥ s ) / ln 2 vs s − Ω forthe sc-NN+4NN lattice under different values of p , consider-ing second-order finite-size corrections. Here we plot the log-log relation of p c vs z in Fig.6, along with the bond percolation thresholds of p c =0 . . . . z = 4), the sc ( z = 6), the bcc( z = 8), and the fcc ( z = 12) lattices, respectively. In Fig.6, we also make a comparison with site percolation forthe same lattices, using data from various sources [70]. Itcan be seen that bond percolation follows a much betterlinear behavior than site percolation, where there is morescatter in the plot. As z increases, the relative differencebetween site and bond thresholds grows, because in sitepercolation, a single occupied site automatically has theability to connect to the entire neighborhood at once,while for bond percolation only two sites are connected by TABLE I. Bond percolation thresholds determined herefor the simple cubic (sc) lattice with combinations ofnearest-neighbors (NN), second nearest-neighbors (2NN),third nearest-neighbors (3NN), and fourth nearest-neighbors(4NN). Also shown for reference are the site thresholds from a Ref. [44], b Ref. [43]lattice z p c (bond) p c (site)sc-NN+4NN 12 0.1068263(7) 0.15040(12) a sc-3NN+4NN 14 0.1012133(7) 0.20490(12) a sc-NN+3NN 14 0.0920213(7) 0.1420(1) b sc-NN+2NN 18 0.0752326(6) 0.1372(1) b sc-2NN+4NN 18 0.0751589(9) 0.15950(12) a sc-2NN+3NN 20 0.0629283(7) 0.1036(1) b sc-NN+3NN+4NN 20 0.0624379(9) 0.11920(12) a sc-NN+2NN+4NN 24 0.0533056(6) 0.11440(12) a sc-NN+2NN+3NN 26 0.0497080(10) 0.0976(1) b sc-2NN+3NN+4NN 26 0.0474609(9) 0.11330(12) a sc-NN+2NN+3NN+4NN 32 0.0392312(8) 0.10000(12) a l n z ln pc B o n d p e r c o l a t i o n S i t e p e r c o l a t i o n
FIG. 6. A log-log plot of percolation thresholds p c vs coor-dination number z (squares) for the diamond lattice, the sclattice, the bcc lattice, the fcc lattice, and the lattices simu-lated in this paper in the order of Table I, left to right. Theslope gives an exponent of a = 1 .
111 in Eq. (8), and the inter-cept ( z = 1) of the line is at ln p c = 0 . p c ≈ . z − . . Also shown on the plot are the site thresh-olds (provided by Refs. [20, 26, 43, 44]) for the same lattices,in which case the correlation of the thresholds with z is notnearly as good (circles). an added bond. By data fitting, we deduce a = 1 .
111 forbond percolation in three dimensions, and deviations ofthe thresholds from the line are within about 5% (except ≈
7% for the sc-NN+4NN lattice).For site percolation, one might expect a = 1 for com-pact neighborhoods and large z , because such neighbor-hoods can represent the overlap of extended objects. Forexample, consider the percolation of overlapping spheresin a continuum. Here the percolation threshold corre-sponds to a total volume fraction of adsorbed spheresequal to [71, 72], η c = 43 πr NV ≈ . r is the radius of the sphere, for N particles ad-sorbed in a system of volume V . Covering the space witha fine lattice, the system corresponds to site percolationwith extended neighbors up to radius 2 r about the cen-tral point, because two spheres of radius r whose centersare separated a distance 2 r apart will just touch. The ra-tio N/V corresponds to the site occupation threshold p c .The effective z is equal to the number of sites in a sphereof radius 2 r , z = (4 / π (2 r ) , for a simple cubic lattice.Then from Eq. (9) it follows that zp c / . p c = 2 . z (10)For the site thresholds available, this gives fairly accu-rate estimates; for example, for site percolation on thesc-NN+2NN+3NN lattice with z = 26, this predicts p c = 0 . η c = 0 . p c = 0 . z .) In any case,this analysis implies an exponent a equal to 1, for sitepercolation systems with compact neighborhoods. Thisargument does not seem to apply directly to bond perco-lation, although in general bond thresholds scale withsite thresholds, and it is known that bond thresholdsare always lower than site thresholds for a given lat-tice [74], so it is not surprising that the bond thresholdsshould follow similar 1 /z behavior. Of course, we are notconsidering just compact neighbors (like NN, NN+2NN,NN+2NN+3NN, NN+2NN+3NN+4NN) in our analysisin Fig. 6, but also more sparse ones, which may also affectthe apparent scaling of exponent a .For bond percolation, we have the bound that thethreshold must be greater than that of a Bethe latticewith coordination number z , namely p c = 1 / ( z − z , one would expect the Bethe result tohold asymptotically, because of the small chance thatthe bonds in a cluster will visit the same site. In Fig.7 we plot ( z − p c vs z − / , using additional thresholddata for larger z , and indeed find an intercept very closeto 1. The power − / z ,but have slightly different values of p c . For pairs of lat-tices with coordination number z = 18, 20 and 26, a far-ther distance between two neighborhood vertices seemsto lead to a smaller percolation threshold. For example,for z = 18, we have the two lattices sc-NN+2NN andsc-2NN+4NN, and the latter lattice, which has a lowerpercolation threshold, has 4NN vertices instead of the NNvertices of the first lattice. The exception to this trend ( z -1) p c z ) y = 1 . 2 2 4 x + 0 . 9 9 7 FIG. 7. A plot of ( z − p c vs. 1 / √ z for the compact latticessc-NN+2NN ( z = 18), +3NN ( z = 26), +4NN ( z = 32),+5NN ( z = 56), +6NN ( z = 80), +7NN ( z = 92), +8NN( z = 122), +9NN( z = 146) (right to left), using additionalthreshold data for larger z . This plot implies the behaviorshown in Eq. (11). is the sc-NN+3NN and sc-3NN+4NN lattices, both with z = 14, in which the latter lattice has a higher threshold.This behavior may be due to the special cluster structureof the later lattice. An example is shown in Fig. 8: forthe bond in red (grey) color, it is easy to form a loop,which has no contribution to percolation and, in fact, willbe forbidden in our growth process where we do not addbonds to previously occupied sites in the cluster. Withthe former lattice, however, loops cannot form from onlythree bonds, so it is easier for percolation to spread andthus the threshold is lower. In this case the threshold iscloser to the Bethe lattice prediction.Finally, we note that for the bcc and fcc lattices withcomplex neighborhoods, some thresholds follow from theresults of our paper here. For example, the bcc-NN+2NNlattice is equivalent to the sc-3NN+4NN lattice, and thefcc-NN+2NN lattice is equivalent to the sc-2NN+4NNlattices. In the same manner, the non-complex sc-2NNlattice is equivalent to the fcc lattice, and the sc-3NN isequivalent to the bcc lattice. V. CONCLUSIONS
In summary we have found precise estimates of thebond percolation threshold for eleven three-dimensionalsystems based upon a simple cubic lattice with multipleneighbor connections. Similar to what we have found re-cently in four dimensions, the thresholds decrease mono-tonically with the coordination number z , quite accu-rately according to a power law of p c ∼ z − a , with theexponent a = 1 .
111 here. This compares to the value
FIG. 8. An example of sc-3NN+4NN cluster. Suppose thebonds in black color are occupied at the n -th step, then theoccupation of the bond in red (grey) color will be forbiddenat the ( n + 1)-th step. a = 1 .
087 for 4d bond percolation [54], and the value0.790(26) for 3d site percolation found in Ref. [43]. How-ever, for large z , the threshold must be bounded by theBethe-lattice and site percolation results, and we find p c is given by p c = 1 z − (cid:16) . z − / (cid:17) (11) We also find that the correlation of thresholds with z for bond percolation is much better than it is for sitepercolation.In two, three, and higher dimensions, many percola-tion thresholds are still unknown, or known only to lowsignificance, for many lattices. Malarz and co-workers[43, 44, 46–48] have carried out several studies on lat-tices with various complex neighborhoods in two, threeand four dimensions. Their results have all concerned sitepercolation, and are generally given to only three signif-icant digits. Knowing these thresholds to higher preci-sion, and also knowing bond thresholds, may be usefulfor various applications and worthy of future study. Thesingle-cluster algorithm is an effective way of studyingthese in a straightforward and efficient manner. VI. ACKNOWLEDGMENTS
We are grateful to the Advanced Analysis and Com-putation Center of CUMT for the award of CPU hoursto accomplish this work. This work is supportedby the China Scholarship Council Project (Grant No.201806425025) and the National Natural Science Foun-dation of China (Grant No. 51704293). [1] S. R. Broadbent and J. M. Hammersley. Percolation pro-cesses: I. Crystals and mazes.
Mathematical Proceedingsof the Cambridge Philosophical Society , 53(3):629–641,1957.[2] Dietrich Stauffer and Amnon Aharony.
Introduction toPercolation Theory, 2nd ed.
CRC Press, 1994.[3] S. F. Bolandtaba and A. Skauge. Network modeling ofEOR processes: A combined invasion percolation anddynamic model for mobilization of trapped oil.
Transportin Porous Media , 89(3):357–382, 2011.[4] V. V. Mourzenko, J.-F. Thovert, and P. M. Adler. Per-meability of isotropic and anisotropic fracture networks,from the percolation threshold to very large densities.
Phys. Rev. E , 84:036307, 2011.[5] Christopher L. Henley. Statics of a “self-organized” per-colation model.
Phys. Rev. Lett. , 71:2741–2744, 1993.[6] Nara Guisoni, Ernesto S. Loscar, and Ezequiel V. Al-bano. Phase diagram and critical behavior of a forest-fire model in a gradient of immunity.
Phys. Rev. E ,83:011125, 2011.[7] Cristopher Moore and M. E. J. Newman. Epidemicsand percolation in small-world networks.
Phys. Rev. E ,61:5678–5682, 2000.[8] M. F. Sykes and J. W. Essam. Exact critical percolationprobabilities for site and bond problems in two dimen-sions.
J. Math. Phys. , 5(8):1117–1127, 1964.[9] Paul N. Suding and Robert M. Ziff. Site percola-tion thresholds for Archimedean lattices.
Phys. Rev. E , 60:275–283, 1999.[10] J. C. Wierman. A bond percolation critical probabilitydetermination based on the star-triangle transformation.
J. Phys. A: Math. Gen. , 17(7):1525–1530, 1984.[11] Christian R. Scullard. Exact site percolation thresholdsusing a site-to-bond transformation and the star-triangletransformation.
Phys. Rev. E , 73:016107, 2006.[12] Robert M. Ziff. Generalized cell–dual-cell transforma-tion and exact thresholds for percolation.
Phys. Rev. E ,73:016134, 2006.[13] F. Y. Wu. New critical frontiers for the Potts and per-colation models.
Phys. Rev. Lett. , 96:090602, 2006.[14] Robert M. Ziff and Christian R. Scullard. Exact bondpercolation thresholds in two dimensions.
J. Phys. A:Math. Gen. , 39(49):15083–15090, 2006.[15] R. M. Ziff and B. Sapoval. The efficient determination ofthe percolation threshold by a frontier-generating walk ina gradient.
J. Phys. A: Math. Gen , 19(18):L1169, 1986.[16] Ojan Khatib Damavandi and Robert M. Ziff. Percolationon hypergraphs with four-edges.
J. Phys. A: Math. Th. ,48(40):405004, 2015.[17] M. F. Sykes and J. W. Essam. Critical percolation prob-abilities by series methods.
Phys. Rev. , 133:A310–A315,1964.[18] J. Hoshen and R. Kopelman. Percolation and clusterdistribution. I. Cluster multiple labeling technique andcritical concentration algorithm.
Phys. Rev. B , 14:3438–3445, 1976. [19] Peter J. Reynolds, H. Eugene Stanley, and W. Klein.Large-cell Monte Carlo renormalization group for perco-lation.
Phys. Rev. B , 21:1223–1245, 1980.[20] Christian D. Lorenz and Robert M. Ziff. Precise determi-nation of the bond percolation thresholds and finite-sizescaling corrections for the SC, FCC, and BCC lattices.
Phys. Rev. E , 57:230–236, 1998.[21] M. E. J. Newman and Robert M. Ziff. Efficient MonteCarlo algorithm and high-precision results for percola-tion.
Phys. Rev. Lett. , 85:4104–4107, 2000.[22] B. Derrida and L. De Seze. Application of the phe-nomenological renormalization to percolation and latticeanimals in dimension 2.
J. Physique France , 43:475–483,1982.[23] Jesper Lykke Jacobsen. High-precision percolationthresholds and Potts-model critical manifolds from graphpolynomials.
J. Phys. A: Math. Th. , 47(13):135001, 2014.[24] Fumiko Yonezawa, Shoichi Sakamoto, and Motoo Hori.Percolation in two-dimensional lattices. I. A technique forthe estimation of thresholds.
Phys. Rev. B , 40:636–649,1989.[25] Junfeng Wang, Zongzheng Zhou, Wei Zhang, Timo-thy M. Garoni, and Youjin Deng. Bond and site per-colation in three dimensions.
Phys. Rev. E , 87:052107,2013.[26] Xiao Xu, Junfeng Wang, Jian-Ping Lv, and Youjin Deng.Simultaneous analysis of three-dimensional percolationmodels.
Frontiers of Physics , 9(1):113–119, 2014.[27] Stephan Mertens and Robert M. Ziff. Percolation in finitematching lattices.
Phys. Rev. E , 94:062152, 2016.[28] Cesar I. N. Sampaio Filho, Jos´e S. Andrade Jr., Hans J.Herrmann, and Andr´e A. Moreira. Elastic backbone de-fines a new transition in the percolation model.
Phys.Rev. Lett. , 120:175701, 2018.[29] Ivan Kryven, Robert M. Ziff, and Ginestra Bianconi.Renormalization group for link percolation on planar hy-perbolic manifolds.
Phys. Rev. E , 100:022306, 2019.[30] L. S. Ramirez, P. M. Centres, and A. J. Ramirez-Pastor.Percolation phase transition by removal of k -mers fromfully occupied lattices. Phys. Rev. E , 100:032105, 2019.[31] Oliver Gschwend and Hans J. Herrmann. Sequential dis-ruption of the shortest path in critical percolation.
Phys.Rev. E , 100:032121, 2019.[32] Zbigniew Koza. Critical p = 1 / Phys. Rev. E , 100:042115, 2019.[33] Stephan Mertens and Cristopher Moore. Percolationthresholds and Fisher exponents in hypercubic lattices.
Phys. Rev. E , 98:022120, 2018.[34] Stephan Mertens and Cristopher Moore. Series expansionof the percolation threshold on hypercubic lattices.
J.Phys. A: Math. Th. , 51(47):475001, 2018.[35] Christian R. Scullard and Jesper Lykke Jacobsen. Bondpercolation thresholds on Archimedean lattices from crit-ical polynomial roots.
Phys. Rev. Research , 2:012050,2020.[36] Yuri Yu. Tarasevich and Andrei V. Eserkepov. Percola-tion thresholds for discorectangles: Numerical estimationfor a range of aspect ratios.
Phys. Rev. E , 101:022108,2020.[37] Stephan Mertens and Cristopher Moore. Percolation isodd.
Phys. Rev. Lett. , 123:230605, 2019.[38] C. Appert-Rolland and H. J. Hilhorst. On the oddnessof percolation.
J. Phys. A: Math. Theor. , 53, 2020. [39] Steven C. van der Marck. Percolation thresholds of theduals of the face-centered-cubic, hexagonal-close-packed,and diamond lattices.
Phys. Rev. E , 55:6593–6597, 1997.[40] Steven C. van der Marck. Site percolation and randomwalks on d-dimensional kagom´e lattices.
J. Phys. A:Math. Gen. , 31(15):3449–3460, 1998.[41] Steven C. van der Marck. Calculation of percolationthresholds in high dimensions for FCC, BCC and dia-mond lattices.
Int. J. Mod. Phys. C , 9(4):529–540, 1998.[42] Stephan M. Dammer and Haye Hinrichsen. Spreadingwith immunization in high dimensions.
J. Stat. Mech.:Th. Exp. , 2004(7):P07011, 2004.[43] (cid:32)Lukasz Kurzawski and Krzysztof Malarz. Simple cubicrandom-site percolation thresholds for complex neigh-bourhoods.
Rep. Math. Phys. , 70(2):163 – 169, 2012.[44] Krzysztof Malarz. Simple cubic random-site percolationthresholds for neighborhoods containing fourth-nearestneighbors.
Phys. Rev. E , 91:043301, 2015.[45] I. V. Petrov, I. I. Stoynev, and F. V. Babalievskii. Cor-related two-component percolation.
J. Phys. A: Math.Gen. , 24(18):4421–4426, 1991.[46] M. Majewski and K. Malarz. Square lattice site percola-tion thresholds for complex neighbourhoods.
Acta Phys.Pol. B , 38:2191, 2007.[47] M. Kotwica, P. Gronek, and K. Malarz. Efficient spacevirtualization for the Hoshen-Kopelman algorithm.
Int.J. Mod. Phys. C , 30(8):1950055, 2019.[48] Krzysztof Malarz and Serge Galam. Square-lattice sitepercolation at increasing ranges of neighbor bonds.
Phys.Rev. E , 71:016125, 2005.[49] Ginestra Bianconi. Superconductor-insulator transitionin a network of 2d percolation clusters.
EPL (EurophysicsLetters) , 101(2):26003, 2013.[50] Zbigniew Koza, Grzegorz Kondrat, and KarolSuszczy´nski. Percolation of overlapping squares or cubeson a lattice.
J. Stat. Mech.: Th. Exp. , 2014(11):P11005,2014.[51] Zbigniew Koza and Jakub Po(cid:32)la. From discrete to con-tinuous percolation in dimensions 3 to 7.
J. Stat. Mech.:Th. Exp. , 2016(10):103206, 2016.[52] Jon M. Kleinberg. Navigation in a small world.
Nature ,406:845, 2000.[53] L. M. Sander, C. P. Warren, and I. M. Sokolov. Epi-demics, disorder, and percolation.
Physica A , 325(1):1 –8, 2003.[54] Zhipeng Xun and Robert M. Ziff. Precise bond percola-tion thresholds on several four-dimensional lattices.
Phys.Rev. Research , 2:013067, 2020.[55] Xiaomei Feng, Youjin Deng, and Henk W. J. Bl¨ote. Per-colation transitions in two dimensions.
Phys. Rev. E ,78:031136, 2008.[56] Harvey Scher and Richard Zallen. Critical density inpercolation processes.
J. Chem. Phys. , 53:3759, 1970.[57] Serge Galam and Alain Mauger. Universal formulasfor percolation thresholds.
Phys. Rev. E , 53:2177–2181,1996.[58] S. C. van der Marck. Percolation thresholds and universalformulas.
Phys. Rev. E , 55:1514–1517, 1997.[59] John C. Wierman. Accuracy of universal formulas forpercolation thresholds based on dimension and coordina-tion number.
Phys. Rev. E , 66:027105, 2002.[60] John C. Wierman and Dora Passen Naor. Criteria forevaluation of universal formulas for percolation thresh-olds.
Phys. Rev. E , 71:036143, 2005. [61] John C. Wierman. On bond percolation thresholdbounds for Archimedean lattices with degree three.
J.Phys. A: Math. Th. , 50(29):295001, 2017.[62] H. G. Ballesteros, L. A. Fern´andez, V. Mart´ın-Mayor,A. Mu˜noz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo.Measures of critical exponents in the four-dimensionalsite percolation.
Phys. Lett. B , 400:346–351, 1997.[63] Jean-Christophe Gimel, Taco Nicolai, and DominiqueDurand. Size distribution of percolating clusters on cu-bic lattices.
J. Phys. A: Math. Gen. , 33(43):7687–7697,2000.[64] Daniel Tiggemann. Simulation of percolation onMassively-Parallel computers.
Int. J. Mod. Phys. C ,12(6):871–878, 2001.[65] H. G. Ballesteros, L. A. Fern´andez, V. Mart´ın-Mayor,A. Mu˜noz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo.Scaling corrections: site percolation and Ising model inthree dimensions.
J. Phys. A: Math. Gen. , 32(1):1–13,1999.[66] H. N. V. Temperley and E. H. Lieb. Relations be-tween the ’percolation’ and ’colouring’ problem and othergraph-theoretical problems associated with regular pla-nar lattices: Some exact results for the ’percolation’problem.
Proc. Roy. Soc. London. Ser. A , 322(1549):251–280, 1971.[67] J. A. Gracey. Four loop renormalization of φ theory insix dimensions. Phys. Rev. D , 92:025012, 2015.[68] P. L. Leath. Cluster size and boundary distribution nearpercolation threshold.
Phys. Rev. B , 14:5046–5055, 1976.[69] M. E. J. Newman and Robert M. Ziff. Fast Monte Carloalgorithm for site or bond percolation.
Phys. Rev. E ,64:016706, 2001.[70] Percolation Threshold Wikipedia page. https://en.wikipedia.org/wiki/Percolation threshold ,2020.[71] Christian D. Lorenz and Robert M. Ziff. Precise determi-nation of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algo-rithm.
J. Chem. Phys. , 114(8):3659–3661, 2001.[72] S. Torquato and Y. Jiao. Effect of dimensionality on thecontinuum percolation of overlapping hyperspheres andhypercubes. II. Simulation results and analyses.
J. Chem.Phys. , 137:074106, 2012.[73] E. Hyytia, J. Virtamo, P. Lassila, and J. Ott. Continuumpercolation threshold for permeable aligned cylinders andopportunistic networking.
IEEE Communications Let-ters , 16(7):1064–1067, 2012.[74] Geoffrey Grimmett.
Percolation, 2nd ed.
Springer-VerlagBerlin Heidelberg, 1999. Appendix: Supplementary Material
Supplementary Material for “Bond percolation on sim-ple cubic lattices with extended neighborhoods” [ZhipengXun and R. M. Ziff,
Phys. Rev. E s τ − P ≥ s vs s σ or s − Ω for the ten additional lattices not discussed in themain text: • sc-3NN+4NN, Figs. A.1 and A.2, • sc-NN+3NN, Figs. A.3 and A.4, • sc-NN+2NN, Figs. A.5 and A.6, • sc-2NN+4NN, Figs. A.7 and A.8, • sc-2NN+3NN, Figs. A.9 and A.10, • sc-NN+3NN+4NN, Figs. A.11 and A.12, • sc-NN+2NN+4NN, Figs. A.13 and A.14, • sc-NN+2NN+3NN, Figs. A.15 and A.16, • sc-2NN+3NN+4NN, Figs. A.17 and A.18, • sc-NN+2NN+3NN+4NN, Figs. A.19 and A.20,The plots for sc-NN+4NN lattice are discussed in themain text in Figs. 3 and 4. The resulting thresholds aresummarized in Table I of the text. We did not calculatethe apparent values of τ for all these lattices one by one;otherwise, the overall simulation time would have at leastdoubled. For all these plots we assumed the values τ =2 . σ = 0 . s (cid:2) -2 P >= s s (cid:1) p = 0 . 1 0 1 2 1 1 p = 0 . 1 0 1 2 1 2 p = 0 . 1 0 1 2 1 3 p = 0 . 1 0 1 2 1 4 p = 0 . 1 0 1 2 1 5 p = 0 . 1 0 1 2 1 6 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.1. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portions ofthe curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s (cid:1) -2 P >= s s - W p = 0 . 1 0 1 2 1 1 p = 0 . 1 0 1 2 1 2 p = 0 . 1 0 1 2 1 3 p = 0 . 1 0 1 2 1 4 p = 0 . 1 0 1 2 1 5 p = 0 . 1 0 1 2 1 6 FIG. A.2. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-3NN+4NN lattice under different valuesof p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 9 2 0 1 8 p = 0 . 0 9 2 0 1 9 p = 0 . 0 9 2 0 2 0 p = 0 . 0 9 2 0 2 1 p = 0 . 0 9 2 0 2 2 p = 0 . 0 9 2 0 2 3 p = 0 . 0 9 2 0 2 4 p = 0 . 0 9 2 0 2 5 p = 0 . 0 9 2 0 2 6 p = 0 . 0 9 2 0 2 7 d( s (cid:2) -2 P >= s )/d( s (cid:1) ) p FIG. A.3. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portions ofthe curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 9 2 0 1 8 p = 0 . 0 9 2 0 1 9 p = 0 . 0 9 2 0 2 0 p = 0 . 0 9 2 0 2 1 p = 0 . 0 9 2 0 2 2 p = 0 . 0 9 2 0 2 3 p = 0 . 0 9 2 0 2 4 p = 0 . 0 9 2 0 2 5 p = 0 . 0 9 2 0 2 6 p = 0 . 0 9 2 0 2 7 FIG. A.4. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+3NN lattice under different values of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 7 5 2 3 0 p = 0 . 0 7 5 2 3 1 p = 0 . 0 7 5 2 3 2 p = 0 . 0 7 5 2 3 3 p = 0 . 0 7 5 2 3 4 p = 0 . 0 7 5 2 3 5 p = 0 . 0 7 5 2 3 6 p = 0 . 0 7 5 2 3 7 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.5. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portions ofthe curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s (cid:1) -2 P >= s s - W p = 0 . 0 7 5 2 3 0 p = 0 . 0 7 5 2 3 1 p = 0 . 0 7 5 2 3 2 p = 0 . 0 7 5 2 3 3 p = 0 . 0 7 5 2 3 4 p = 0 . 0 7 5 2 3 5 p = 0 . 0 7 5 2 3 6 p = 0 . 0 7 5 2 3 7 FIG. A.6. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+2NN lattice under different values of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 7 5 1 5 6 p = 0 . 0 7 5 1 5 7 p = 0 . 0 7 5 1 5 8 p = 0 . 0 7 5 1 5 9 p = 0 . 0 7 5 1 6 0 p = 0 . 0 7 5 1 6 1 p = 0 . 0 7 5 1 6 2 p = 0 . 0 7 5 1 6 3 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.7. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portions ofthe curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 7 5 1 5 6 p = 0 . 0 7 5 1 5 7 p = 0 . 0 7 5 1 5 8 p = 0 . 0 7 5 1 5 9 p = 0 . 0 7 5 1 6 0 p = 0 . 0 7 5 1 6 1 p = 0 . 0 7 5 1 6 2 p = 0 . 0 7 5 1 6 3 FIG. A.8. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-2NN+4NN lattice under different valuesof p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 6 2 9 2 5 p = 0 . 0 6 2 9 2 6 p = 0 . 0 6 2 9 2 7 p = 0 . 0 6 2 9 2 8 p = 0 . 0 6 2 9 2 9 p = 0 . 0 6 2 9 3 0 p = 0 . 0 6 2 9 3 1 p = 0 . 0 6 2 9 3 2 p = 0 . 0 6 2 9 3 3 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.9. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portions ofthe curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 6 2 9 2 5 p = 0 . 0 6 2 9 2 6 p = 0 . 0 6 2 9 2 7 p = 0 . 0 6 2 9 2 8 p = 0 . 0 6 2 9 2 9 p = 0 . 0 6 2 9 3 0 p = 0 . 0 6 2 9 3 1 p = 0 . 0 6 2 9 3 2 p = 0 . 0 6 2 9 3 3 FIG. A.10. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-2NN+3NN lattice under different valuesof p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 6 2 4 3 5 p = 0 . 0 6 2 4 3 6 p = 0 . 0 6 2 4 3 7 p = 0 . 0 6 2 4 3 8 p = 0 . 0 6 2 4 3 9 p = 0 . 0 6 2 4 4 0 p = 0 . 0 6 2 4 4 1 p = 0 . 0 6 2 4 4 2 p = 0 . 0 6 2 4 4 3 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.11. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portionsof the curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s (cid:1) -2 P >= s s - W p = 0 . 0 6 2 4 3 5 p = 0 . 0 6 2 4 3 6 p = 0 . 0 6 2 4 3 7 p = 0 . 0 6 2 4 3 8 p = 0 . 0 6 2 4 3 9 p = 0 . 0 6 2 4 4 0 p = 0 . 0 6 2 4 4 1 p = 0 . 0 6 2 4 4 2 p = 0 . 0 6 2 4 4 3 FIG. A.12. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+3NN+4NN lattice under differentvalues of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 5 3 3 0 2 p = 0 . 0 5 3 3 0 3 p = 0 . 0 5 3 3 0 4 p = 0 . 0 5 3 3 0 5 p = 0 . 0 5 3 3 0 6 p = 0 . 0 5 3 3 0 7 p = 0 . 0 5 3 3 0 8 p = 0 . 0 5 3 3 0 9 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.13. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portionsof the curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 5 3 3 0 2 p = 0 . 0 5 3 3 0 3 p = 0 . 0 5 3 3 0 4 p = 0 . 0 5 3 3 0 5 p = 0 . 0 5 3 3 0 6 p = 0 . 0 5 3 3 0 7 p = 0 . 0 5 3 3 0 8 p = 0 . 0 5 3 3 0 9 FIG. A.14. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+2NN+4NN lattice under differentvalues of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 4 9 7 0 5 p = 0 . 0 4 9 7 0 6 p = 0 . 0 4 9 7 0 7 p = 0 . 0 4 9 7 0 8 p = 0 . 0 4 9 7 0 9 p = 0 . 0 4 9 7 1 0 p = 0 . 0 4 9 7 1 1 d( s (cid:2) -2 P >= s )/d( s (cid:1) ) p FIG. A.15. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portionsof the curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 4 9 7 0 5 p = 0 . 0 4 9 7 0 6 p = 0 . 0 4 9 7 0 7 p = 0 . 0 4 9 7 0 8 p = 0 . 0 4 9 7 0 9 p = 0 . 0 4 9 7 1 0 p = 0 . 0 4 9 7 1 1 FIG. A.16. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+2NN+3NN lattice under differentvalues of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 4 7 4 5 7 p = 0 . 0 4 7 4 5 8 p = 0 . 0 4 7 4 5 9 p = 0 . 0 4 7 4 6 0 p = 0 . 0 4 7 4 6 1 p = 0 . 0 4 7 4 6 2 p = 0 . 0 4 7 4 6 3 p = 0 . 0 4 7 4 6 4 d( s (cid:2) -2 P >= s )/d( s (cid:1) ) p FIG. A.17. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ = 0 . p . The inset indicates the slope of the linear portionsof the curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 4 7 4 5 7 p = 0 . 0 4 7 4 5 8 p = 0 . 0 4 7 4 5 9 p = 0 . 0 4 7 4 6 0 p = 0 . 0 4 7 4 6 1 p = 0 . 0 4 7 4 6 2 p = 0 . 0 4 7 4 6 3 p = 0 . 0 4 7 4 6 4 FIG. A.18. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-2NN+3NN+4NN lattice under differentvalues of p . s (cid:1) s (cid:2) -2 P >= s p = 0 . 0 3 9 2 2 8 p = 0 . 0 3 9 2 2 9 p = 0 . 0 3 9 2 3 0 p = 0 . 0 3 9 2 3 1 p = 0 . 0 3 9 2 3 2 p = 0 . 0 3 9 2 3 3 p = 0 . 0 3 9 2 3 4 p = 0 . 0 3 9 2 3 5 p d( s (cid:2) -2 P >= s )/d( s (cid:1) ) FIG. A.19. Plot of s τ − P ≥ s vs s σ with τ = 2 . σ =0 . p . The inset indicates the slope of the linear portionsof the curves shown in the main figure as a function of p , andthe center value of p c = 0 . x intercept. s - W s (cid:1) -2 P >= s p = 0 . 0 3 9 2 2 8 p = 0 . 0 3 9 2 2 9 p = 0 . 0 3 9 2 3 0 p = 0 . 0 3 9 2 3 1 p = 0 . 0 3 9 2 3 2 p = 0 . 0 3 9 2 3 3 p = 0 . 0 3 9 2 3 4 p = 0 . 0 3 9 2 3 5 FIG. A.20. Plot of s τ − P ≥ s vs s − Ω with τ = 2 . .
63 for the sc-NN+2NN+3NN+4NN lattice under dif-ferent values of pp