Bond percolation between k separated points on a square lattice
BBond percolation between k separated points on a square lattice S. S. Manna
Satyendra Nath Bose National Centre for Basic Sciences,Block-JD, Sector-III, Salt Lake, Kolkata-700106, India
Robert M. Ziff ∗ Center for the Study of Complex Systems and Department of Chemical Engineering,University of Michigan, Ann Arbor, Michigan 48109-2136, USA
We consider a percolation process in which k points separated by a distance proportional to systemsize L simultaneously connect together ( k > k = 1), through adjacent connected points of a single cluster. These processes yieldnew thresholds p ck defined as the average value of p at which the desired connections first occur.These thresholds are not sharp as the distribution of values of p ck for individual samples remainsbroad in the limit of L → ∞ . We study p ck for bond percolation on the square lattice, and find that p ck are above the normal percolation threshold p c = 1 / p ck can be related to integrals over powers of the function P ∞ ( p ) equal to the probability apoint is connected to the infinite cluster; we find numerically from both direct simulations and frommeasurements of P ∞ ( p ) on L × L systems that, for L → ∞ , p c = 0 . p c = 0 . p c = 0 . p c = 0 . . The percolation thresholds p ck remain the same, even whenthe k points are randomly selected within the lattice. We show that the finite-size corrections scaleas L − /ν k where ν k = ν/ ( kβ + 1), with β = 5 /
36 and ν = 4 / ν = 48 / ν = 24 / ν = 16 / ν = 6 /
7, etc. We also studythree-point correlations in the system, and show how for p > p c , the correlation ratio goes to 1 (nonet correlation) as L → ∞ , while at p c it reaches the known value of 1.022. I. INTRODUCTION
Percolation is the study of long-range connectivenessin systems such as graphs or lattices in which the sites orbonds are randomly occupied with probability p . Thereis a well-defined threshold p c at which the average size ofa cluster first becomes infinite. The threshold can alsobe defined by considering finite systems (say an L × L square), and studying the probability that a single clusterconnects or spans two opposite sides. The average valueof p at which spanning first occurs yields an estimate for p c ( L ), and using finite-size scaling one can predict thevalue of p c for L → ∞ . In this case, the threshold is sharpas L → ∞ . For a square lattice with bond percolation,for example, one has p c = 1 / k -mers undergoing random se-quential adsorption [7], percolation disassortativity onrandom networks [8], percolation for random sequentialadsorption with relaxation [9], percolation over a rangeof interactions [10], percolation in high dimensions andon a random graph [11], percolation on hypercubic lat-tices in high dimensions [12, 13], percolation of the elasticbackbone [14], universality in explosive percolation [15], ∗ rziff@umich.edu crossing probabilities for polygons [16], rigorous boundsfor percolation thresholds [17], percolation on randomjammed sphere packings [18], and percolation on hyper-bolic manifolds [19]. Clearly, percolation remains a veryactive field.For the ordinary percolation problem in d dimensions,the connectivity is usually considered between the pairof opposite ( d −
1) dimensional hypersurfaces. Natu-rally, the question arises, what happens if the connec-tivity is considered between the ( d − d − d −
2) dimensional hypersurfaces in d = 2.More specifically, we study the percolation problem be-tween the k widely separated points (dimension 0) on thetwo-dimensional square lattice, or between a single pointand the boundary of the system.The first threshold we consider is defined as the aver-age value of p at which a point in the center of a squaresystem first connects to any point on the boundary. Thisdefines the threshold p c . The other thresholds are de-fined as the average value of p at which k points sep-arated far apart in a periodic system all first connect;we call those thresholds p ck . These thresholds are allgreater than p c , indicating that we are in the supercriti-cal regime of percolation where there is a percolating netthroughout the system. Being in a supercritical state isexpected since connecting a large cluster to a specific sin-gle point at the normal critical point p c occurs with lowprobability (unlike connecting to a boundary, for exam-ple, which can occur through many paths and is mucheasier). Connecting to a boundary is a universal prop- a r X i v : . [ c ond - m a t . d i s - nn ] J u l erty that survives at the critical point when the latticespacing goes to zero, while in that limit the probabilityof connecting to a single point goes to zero. When goingto the supercritical regime, the probability of connectingto a point can be raised to a significant value, and this al-lows different points to connect together simultaneouslywith a sufficient probability to be observed.We carried out computer simulations to find the valuesof p ck directly for k = 1 , , p ck to P ∞ , the percolation functionthat gives the probability a given point belongs to theinfinite cluster, or the largest cluster for a finite system.By directly simulating P ∞ for this system, we are ableto verify numerically that the relation to p ck is valid.The analysis also shows that, unlike in the case of theusual percolation threshold, the distribution of p ck forindividual systems is broad and does not become sharpas the system size goes to infinity. That is, there arelarge fluctuations in the states of these systems definedby these percolation criteria.In Fig. 1 we show pictures of simulations of a 64 × p c = 4415 / ≈ . p = 4096 / / p c , at which point no connection exists betweenthe two anchor points for this system. The value of p c for this sample is close to the average value p c = 0 . p c , there is one overwhelming “infinite” clusterthroughout the system, and finite clusters are very small.This behavior illustrates the idea behind our conjecturethat in the supercritical region, the probability that k points are connected together is equal to [ P ∞ ( p )] k .In Fig. 2 we show a very rare case where the connectionbetween the anchor points occurred at a value substan-tially below p = 1 /
2; for large systems such cases appearwith very low probability.We also studied a ratio involving three-point corre-lations and two-point correlations, and show how thatvaries with the separation of the points compared withthe size of the system. This ratio has been studied pre-viously at the critical point only [20, 21]; here we studyit for all p .In section II we develop our theory for p ck , includingthe scaling of the estimates. In section III we describeour simulation methods, and in section IV we give theresults of our simulations. In section V we consider theproblem of the three-point correlation ratio. In sectionVI we discuss our results and give our conclusions. II. THEORETICAL ANALYSIS
Here we develop a theory to predict p ck from P ∞ ( p ),and develop a scaling analysis that allows one to predictthe convergence exponents for the p ck . A. Relation to P ∞ The first assumption is that we must be in the super-critical state, since only then will the k points be ableto connect together via the infinite network. At p c , theinfinite cluster is tenuous and fractal, and does not con-nect to given points with a significant probability (for alarge system), and below p c the clusters are all small andit would be virtually impossible for points far apart toconnect together.Thus, for k widely separated points to be all con-nected together, we hypothesize that they must be partof the infinite cluster in the supercritical state. Theprobability a single point belongs to the infinite clusteris denoted as P ∞ ( p ); for a finite system we can define P ∞ ( p, L ) = s max /L where s max is the number of sitesin the largest cluster in the system. Thus, we conjec-ture that the probability that k widely separated pointsare connected must be equal to [ P ∞ ( p )] k . The probabil-ity density that they first connect when the occupationprobability is p is then P r = ( d/dp )[ P ∞ ( p )] k = k [ P ∞ ( p )] k − P (cid:48)∞ ( p ) (1)and the average value of p at which the k points firstconnect will be given by p ck = (cid:104) p (cid:105) = (cid:90) p ( d/dp )[ P ∞ ( p )] k dp (2)Integrating by parts, we find p ck = 1 − (cid:90) [ P ∞ ( p )] k dp = (cid:90) (1 − [ P ∞ ( p )] k ) dp (3)For the problem of a single site connected to the bound-ary (corresponding to p c ), the above formulas also ap-ply, taking k = 1. In this case, the largest cluster surelyconnects to the boundary, so we are asking for just theprobability that a point connects to the largest cluster,which is given by P ∞ ( p ). Note, for the case of k = 1, wedo not use periodic boundary conditions.For k > p ck should be independent ofthe exact configuration of the k points, as long as theirrelative distances grow with L , so that they become in-finitely far apart as L → ∞ and greater than the corre-lation length ξ , which is finite for any given p > p c . Forfinite systems, the specific configuration of the points willbe relevant for the precise threshold.We can make a very useful approximation for calculat-ing p ck from P ∞ ( p ) for finite systems by simply assuming P ∞ ( p ) = 0 for p < p c , which is true for an infinite sys-tem. Then the integrand in the second form of equation(3) is exactly 1 in the interval 0 < p < p c , and we canwrite as an alternative to (3) p ck = p c + (cid:90) p c (1 − [ P ∞ ( p )] k ) dp (4)where p c = 1 / L → ∞ , but itwill turn out that (4) gives a much better estimate of p ck for finite L . B. Scaling of the estimates
If we assume that the mapping of our problem to[ P ∞ ( p )] k is correct for finite systems characterized by P ∞ ( p, L ), we can then estimate the scaling behavior ofthe estimates from finite-size scaling theory. That theorystates that for L → ∞ and p − p c → p − p c ) L /ν constant, P ∞ ( p, L ) ∼ aL − β/ν F ( b ( p − p c ) L /ν ) (5)where a and b are system-dependent constants (“metricfactors”) while β , ν and F ( z ) are universal quantities,having the same values and behavior for all systems ofa given dimensionality, and also a given system shapefor the case of F ( z ). For d = 2, one has β = 5 /
36 and ν = 4 / p ck givenby Eq. (3). First we consider the interval p = (0 , p c ).In this interval, we assume that the finite-size effects areessentially those given by the scaling function F ( z ), be-cause when p < p c , P ∞ ( p, ∞ ) = 0. That is, we assumethe non-scaling corrections are unimportant for large L for p < p c .Putting (5) into the integral in Eq. (3) over the interval p = (0 , p c ), we find (cid:90) p c [ P ∞ ( p )] k dp = a k (cid:90) p c L − kβ/ν [ F ( b ( p − p c ) L /ν )] k dp (6)and a change of variables yields (cid:90) p c [ P ∞ ( p )] k dp = a k b − L − ( kβ +1) /ν (cid:90) − bp c L /ν [ F ( z )] k dz ≈ a k b − L − ( kβ +1) /ν (cid:90) −∞ [ F ( z )] k dz (7)where z = b ( p − p c ) L /ν . In the second integral in (7) weextended the lower limit to −∞ , valid for large L becausethe integrand decays exponentially for negative z .Therefore, this contribution to the integral in (3)should scale as L − /ν k with1 /ν k = ( kβ + 1) /ν = 36 + 5 k
48 (8)so that 1 /ν = 41 /
48 = 0 . /ν = 23 /
24 =0 . p > p c , it is not clear how to attack the finite-size corrections of the integral in (3) because there arelarge non-scaling contributions to P ∞ whose behavior wedo not know, but it seems reasonable to assume thatthe finite-size corrections for p > p c scale the same as (a)(b)FIG. 1: Two illustrations of the system are presented for alattice of size 64 ×
64 with periodic boundary conditions, andwith k = 2 anchor points (marked by filled blue circles) sep-arated by a distance of 32 lattice units. Bonds of the largestoccupied cluster are shown in red (grey), and all other occu-pied bonds are shown in black. (a) A system where the num-ber of bonds is exactly 4096 or p = 4096 / / p c ,without connection between the two anchor points. (b) Thesame system where the number of occupied bonds is increasedto 4415 bonds or p c = 4415 / ≈ . p c = 0 . (a)(b)FIG. 2: (a) Here the density of occupied bonds at the firstconnection occurs at p = p c = 3660 / . p is increased to p c = 1 / p c , wherethe point connecting cluster (magenta or light grey bonds) isdifferent from the largest cluster (red or grey bonds). In fact,the spanning cluster is relatively small and does not extendover the whole system. Such behavior where spanning occursbelow p c can only happen in smaller systems. In most cases,the individual values of p c are larger than p c and the clusterconnecting them is the “infinite” cluster that spreads overvirtually the entire system as in Fig. 1(b). p c2 P r ( p c , L ) -2 0 2 4 (p c2 - p c )L ν P r ( p c , L ) L ( β− ) / ν (a)(b) FIG. 3: Simulation result: (a) The probability distribution P r ( p c , L ) of the percolation threshold p c of connecting twoanchor points has been plotted against p c for L = 128 (blackor lower curve), 256 (green or middle curve), 512 (red or up-per, most peaked curve). (b) The scaling plot of the proba-bility distribution P r ( p c , L ) L (2 β − /ν against ( p c − p c ) L /ν with β = 5 / , ν = 4 / p c = 1 /
2. Bottom to top L = 128, 256, and 512. those we found for p < p c , so we conjecture that theexponents ν k above should characterize the full finite-sizecorrections to p ck . That is, we conjecture p ck ( L ) = p ck + cL − /ν k (9)where c is a constant and ν k is given by Eq. (8). Theconstant term on the right-hand side, p ck , derives fromthe non-scaling parts of P ∞ for p > p c .Note that it also follows from the scaling argumentsabove that P r = k [ P ∞ ( p )] k − P (cid:48)∞ ( p ) behaves with L inthe scaling regime as P r ∼ ka k − L − ( k − β/ν [ F ( b ( p − p c ) L /ν )] k − × aL − β/ν F (cid:48) ( b ( p − p c ) L /ν ) bL /ν ∼ L − ( kβ − /ν G ( b ( p − p c ) L /ν ) (10) III. SIMULATION METHODSA. Simulation method to find p ck We carried out computer simulations of these processeson systems of size L × L for bond percolation, with pe-riodic boundary conditions. For the case k = 1, we con- p P ∞ P ∞ ′ FIG. 4: Plots of P r ( p, L ) = 2 P ∞ ( p ) P (cid:48)∞ ( p ) for L = 128 , P ∞ ( p ) ratherthan by direct simulation of P r . Note here p is equivalent to p c used in Fig. 3. - - ( p - p c ) L ν L ( β - ) / ν P ∞ P ∞ ' FIG. 5: Scaling plot of L (2 β − /ν P ∞ ( p ) P (cid:48)∞ ( p ) vs. ( p − p c ) L /ν for L = 128 ,
256 and 512 (bottom to top). Thecurves collapse well to a universal curve, except for the tailfor large ( p − p c ) L /ν which represents the non-scaling partof this quantity. This plot is comparable with Fig. 3(b), hereevaluated through P ∞ rather than the direct measurement of P r . Here p is equivalent to p c in Fig. 3. sider L odd and add bonds until the center point con-nects to the boundary for p = p c . Repeating this pro-cess many times, we average the values of p c to find p c . For k = 2, 3 and 4, we consider periodic L × L systems with L = 2 n , n = 5 , , . . .
12. For k = 2 we con-sider the connectivity between a point at the origin (0,0)and a point at (0 , L/ k = 3, the connectivity be-tween the three points (0,0), ( L/ , , L/ k = 4, the connectivity between the four points (0,0),( L/ , , L/ L/ , L/
2) is considered. Notethat for k = 3, the three points are the vertices of aright triangle rather than an equilateral triangle, so thedistances between pairs of points are not identical, butthis is not important — all that matters is that the threepoints are relatively far apart from each other. Theaverage value of p at the first connection gives p ck .It is clear from Eq. (1) that the values of the thresholds p ck should depend only on the value of k , and not on theactual distribution of the k points. We have numericallyverified this issue for k = 2 by randomly distributingthese two points on the lattice for every configuration.Our simulation results show that the values of p c remainunchanged.We also studied the average p at which the origin con-nects to point x = 1, x = 2, . . . , x = L/ y = 0 forsystems of different L . We discovered that p c ( x ) doesnot noticeably depend upon L as along as x (cid:28) L , in-dicating that the size of the system is unimportant forshorter-range connections. B. Simulation method to find P ∞ To test the conjecture relating p ck to P ∞ , we carriedout measurements of P ∞ ( p ) using the method of New-man and Ziff (NZ) [22, 23], which involves adding bondsone at a time to the system and using the union-findprocedure to merge clusters and keep track of the clus-ter distribution. This method allows one to effectivelymeasure a quantity Q ( p ) (such as P ∞ ( p )) for all val-ues of p in a single simulation. In this method, onefirst determines the “microcanonical” Q n (here P ∞ ,n )when exactly n bonds have been placed down, and thendetermines the “canonical” Q ( p ) (here P ∞ ( p )) by car-rying out a convolution with the binomial distribution B ( N, n, p ) = (cid:0) Nn (cid:1) p n (1 − p ) N − n : Q ( p ) = N (cid:88) n =0 (cid:18) Nn (cid:19) p n (1 − p ) N − n Q n (11)where N is the total number of bonds in the system, inthis case 2 L . For large systems, the differences betweenthe microcanonical Q n with n = pN and Q ( p ) are small,except for regions of high curvature or second derivative,but the convolution serves a further purpose of smoothingout the data, and connecting it with a continuous curve,rather than the discrete values p = 1 /N, /N, . . . . Tointegrate P ∞ ( p ) (as required for p c according Eqs. (3) or(4)), one can just as well sum the microcanonical values,because of the identity [24] (cid:90) Q ( p ) dp = N (cid:88) n =0 (cid:18) Nn (cid:19) Q n (cid:90) p n (1 − p ) N − n dp = 1 N + 1 N (cid:88) n =0 Q n (12)Likewise it follow that (cid:90) pQ ( p ) dp = 1( N + 1)( N + 2) N (cid:88) n =0 ( n + 1) Q n (13)To integrate [ Q ( p )] k = [ P ∞ ( p )] k for k > p , it is most straightforward to first carry out the con-volution to find P ∞ ( p ), and then numerically integratethe [ P ∞ ( p )] k at equally spaced values of p .Derivatives of Q ( p ) can also be found directly from the Q n [24]: Q (cid:48) ( p ) = N (cid:88) n =0 (cid:18) Nn (cid:19) Q n ddp (cid:0) p n (1 − p ) N − n (cid:1) = 1 p (1 − p ) N (cid:88) n =0 ( n − N p ) (cid:18) Nn (cid:19) ( p n (1 − p ) N − n ) Q n = (cid:104) ( n − N p ) Q n (cid:105) p (1 − p ) (14)and likewise Q (cid:48)(cid:48) ( p ) = (cid:104) n Q n (cid:105)− (2( N − p +1) (cid:104) nQ n (cid:105) + N ( N − p (cid:104) Q n (cid:105) p (1 − p ) = (cid:104) ( n − Np ) Q n (cid:105) +(2 p − (cid:104) ( n − Np ) Q n (cid:105)− Np (1 − p ) (cid:104) Q n (cid:105) p (1 − p ) (15)where the averages are over the binomial distribution B ( N, n, p ). Note that in Ref. [24], there is a typo inEq. (32) for Q (cid:48)(cid:48) ( p ), in which the last term should havethe factor ( N − n −
1) rather than ( N − n + 1). p ( d / dp ) [ P ∞ ( p ) ] k FIG. 6: A plot of ( d/dp )[ P ∞ ( p )] k vs. p for k = 1 (red,highest peak)), k = 2 (orange, second-highest peak), k = 3(green, second-lowest peak), and k = 4 (blue, lowest peak)for a system with L = 512, calculated from the results ofthe numerical simulations of P ∞ ( p ), including using Eq. (14)to find P (cid:48)∞ ( p ). The estimates of p ck are the means of thesedistributions according to Eq. (2), and it can be seen thatthe distribution spreads to the right as k increases, yieldinglarger values of p ck . To find the accurate values of p ck , onehas to consider systems of different L and take the limit that L → ∞ , although the change is small for systems larger than L = 512. Note that the distribution is broad and the largefluctuations in the individual values of p ck persist as L → ∞ . To find P ∞ ( p ) we simulated 10 samples each for L = 64 , ,
256 and 512 on L × L periodic systems, sav-ing the 2 L microcanonical values of s max in a file. Forthe largest system L = 512, the simulations took severaldays on a laptop computer. Then we used a separateprogram to read the files and calculate P ∞ ( p ) = s max /L for 10 points p = 0 , . , . . . , . P (cid:48)∞ ( p ) and P (cid:48)(cid:48)∞ ( p ) usingthe formulas of Eqs. (14) and (15). We used the recursivemethod described in Ref. [23] to calculate the binomial distribution for each p . To find the integrals of [ P ∞ ( p )] k for Eqs. (3) and (4), we carried out numerical integra-tion of the 10 points using the trapezoidal rule (namelycounting the two endpoints with relative weight 1/2 andall other points with weight 1). We compared some ofthe integrals using 10 and 10 points and did not findsignificant difference in the results, and used 10 valuesof p in our calculations. IV. RESULTS
Figure 3(a) shows the probability distribution P r ( p c , L ) of the percolation threshold p c of connect-ing two anchor points, from direct measurements. Note p c is the value of p at which the connection first takesplace in a given sample, as opposed to p c which is theaverage value over many samples. Figure 3(b) shows ascaling plot of the data, using the scaling implied in Eq.10.Figure 4 shows the predicted behavior of P r from theansatz of Eq. 1, using the simulation results of P ∞ ratherthan measuring P r directly. These curves can be com-pared with those of Fig. 3(a), and the two can be seen toagree.Figure 5 shows the predicted scaling behavior of P r from the ansatz of Eq. 1, and the results can be seen tobe similar to the scaling plot of the directly measured P r given in Fig. 3(b).Figure 6 shows plots of the predicted distributions ofthe probabilities of first connection, ( d/dp )[ P ∞ ( p )] k , for k = 1, 2, 3, and 4, based upon measurements P ∞ ( p ), fora system of L = 512. As can be seen, the distributionsare broad, meaning that the thresholds we find p ck havelarge fluctuations from system to system and persist as L → ∞ .In Fig. 7 we plot estimates for p c found from di-rect simulations with the point in the center of an( L + 1) × ( L + 1) system, for L = 64 , , . . . , k = 1based upon P ∞ ( p ). The data are plotted based on thepredicted scaling L − / from Eq. (8). We do not expectthat the values of p c ( L ) would be the same for finite L from the two methods (direct simulation and via P ∞ );however, we expect that the extrapolation as L → ∞ should be the same, because in that limit the probabil-ity the point connects to the boundary should exactlybe the probability the point belongs to the largest clus-ter, namely P ∞ . Furthermore, we expect the two esti-mates of p c should scale with L with the same exponent1 /ν = 41 /
48, and indeed that plot confirms that expec-tation. The two different approaches suggest a thresholdof p c = 0 . P ∞ ( p, L ) = 0 for p < p c , converges muchmore quickly than the estimate based upon (3). On amore expanded scale, the convergence to this estimate isalso shown to obey the L − / scaling, but is not shown L -41/48 p c ( L ) FIG. 7: Values of p c ( L ) found from simulations of con-nections of a point at the center to the boundary of an( L + 1) × ( L + 1) square system (triangles), by integrat-ing P ∞ ( p ) on L × L periodic systems using Eq. (3) for k = 1 (squares), and by integrating P ∞ ( p ) using Eq. (4) (cir-cles). The estimates are all plotted vs. L − / according tothe prediction of Eq. (8). The equations of the linear fitsthrough the points are p c = a + bL − / with a = 0 . b = − . a = 0 . b = − . a = 0 . b = − . here. The results for k = 2, 3 and 4 are shown in Figs.8, 9, and 10. Our values of p ck are given in Table I. L -46/48 p c ( L ) FIG. 8: Values of p c ( L ) found from direct simulations on an L × L periodic system with the two points at (0,0) and (0, L/ P ∞ ( p ) on an L × L periodic system, all plotted as a function of L − / = L − / as predicted by Eq. (8). Here L = 32, 64, 128, 256, and 512for the upper two sets of data, and also L = 1024 for the lowerset. V. CORRELATIONS
We also considered a related question for two- andthree-point correlations. Studying this problem shedslight on the correlations that occur in the system in thecritical vs. the post-critical regime where the connectivitybetween the anchor points mainly occurs. L -51/48 p c ( L ) FIG. 9: Values of p c ( L ) found from direct simulations on an L × L periodic system with the three points at (0,0), (0 , L/ L/ ,
0) (triangles), and the predictions from Eqs. (3)(squares) and (4) (circles) based upon measurements of P ∞ ( p )on an L × L periodic system, all plotted as a function of L − / = L − / as predicted by Eq. (8). L -56/48 p c ( L ) FIG. 10: Values of p c ( L ) found from direct simulations onan L × L periodic system with the four points at the corners ofa square of length L/ P ∞ ( p ) using Eq. (3) (squares) and Eq. (4) (circles),both based upon measurements of P ∞ ( p ) on an L × L periodicsystem. All data are plotted as a function of L − / = L − / as predicted by Eq. (8). Lines show linear fits through thedata. It can be seen that estimates based upon Eq. (4) exhibitthe fastest convergence with system size. In [20, 21] the following ratio was considered: R ( p ) = P ( r , r , r ) (cid:112) P ( r , r ) P ( r , r ) P ( r , r ) (16)where r , r and r are three points in the system, P ( r i , r j ) is the probability that points r i and r j connect,and P ( r , r , r ) is the probability that all three pointsconnect.This ratio has previously been studied, to our knowl-edge, only at p = p c , where the value of R ( p c ) approachesthe value C = 1 . C wasfirst observed numerically in [20] and then derived ana-lytically from conformal field theory in [21]. The fact thatthis ratio is unequal to 1 implies a correlation betweenthe three points in the system. If we make the assump- - ( p - p c ) L ν R ( p ) FIG. 11: (color online) Scaling plot of R ( p ) vs. ( p − p c ) L /ν where R ( p ) is given in Eq. (16), for three points at positions(0 , , L/n ), and ( L/n, n = 2 (red, the curve withthe highest peak), n = 4, (green, the curve with the second-highest peak), n = 8 (blue, the curve with the third-highestpeak), and n = 16 (black, the curve that does not reach a peakin this interval), for L = 64, 128, 256, and 512 respectively.Other runs show that there is a small L -dependence on theresults, but the main variation is due to n . At p = p c = 1 / R ( p c ) approaches the theoretical value C = 1 . n gets large, in which case the three points are closetogether compared to the size of the system. The meaning ofthe crossing point for ( p − p c ) L /ν ≈ − . tion that P ( r i , r j ) = P ∞ ( p ) and P ( r , r , r ) = P ∞ ( p ) ,which we expect to be the case for p > p c , then we wouldhave R = 1. At p c , where the infinite cluster does notspan throughout the system, one would not expect thisto be valid and indeed R ( p ) (cid:54) = 1, although it turns outquite close to 1.Here we consider the three points in a right triangle,(0 , , L/n ), and ( L/n, L × L periodic system,for n = 2 , n increases for large L (that is, asthe separation of the three points is small compared tothe size of the system), R ( p c ) approaches the value C .Using the NZ method, we were able to calculate R ( p ) as afunction of p after executing a microcanonical simulationwhere we found the P ( r i , r j ) and P ( r , r , r ) as a func-tion of the number of bonds added. We then carried outthe convolution to the canonical ( p -dependent) functionsfor all P ’s separately, and calculated R ( p ) according toEq. (16). The results are shown in Fig. 11.As can be seen, at p = p c , R ( p c ) approaches C as n increases (in which case the points get closer togethercompared to the size of the system). In the limit that L → ∞ , R ( p ) evidently becomes a discontinuous functionof p , with R ( p c ) = C for p = p c , and R ( p ) = 1 for p > p c .The behavior for p < p c is not clear. Notice in Fig. 11that there is a maximum for R ( p ) in finite systems at z = ( p − p c ) L /ν ≈ − .
5, meaning for some values of p < p c , R ( p ) is greater than the value at p c . However, itis not clear what the behavior is as n → ∞ (for large L );it is possible that the peak for negative z disappears andthe peak occurs only at z = 0 or p = p c . The behavior TABLE I: Our best estimates for the extrapolated values ofthe average percolation thresholds p ck from direct measure-ments (second column) and from P ∞ via Eqs. (3) and (4) fordifferent values of k . The averages of these values are quotedin the abstract. k p ck measured Eq. (3) Eq. (4)1 0.51749(5) 0.51761(3) 0.51755(3)2 0.53212(5) 0.53220(3) 0.53226(3)3 0.54450(5) 0.54458(3) 0.54461(3)4 0.55520(5) 0.55530(3) 0.55531(3) for p < p c needs further investigation.At the point p c ≈ . R ( p ) approaches 1, since thatwould correspond to ( p − p c ) L /ν going to infinity as L goes to infinity. This result reiterates that at the placeswhere multiple points connect, there are no correlationsamong connections between different pairs of widely sep-arated points. VI. DISCUSSION
We have shown that exploring the average value ofthe probability p of bond occupation at which a certainnumber of separated points first connect leads to a newset of average thresholds. The distribution of the valuesof p is broad, so that this threshold is not sharp as in theusual case of thresholds in percolation. For example, themedian rather than the mean of the distribution wouldgive a different value. We have shown that the valuescan be related to P ∞ ( p ), and confirm this relation bysimulation. From this theory it is apparent that while thepercolation thresholds p ck indeed depend on the number k of points, their values are robust with respect to theactual spatial distribution of the k points. For example,the k points may either be symmetrically placed on thelattice or, they can be randomly distributed (for L → ∞ ).This work suggests further research in a variety of ar-eas. It might be interesting to study these thresholdsin higher dimensions, where the relations to P ∞ ( p ) inEqs. (3) and (4), and the scaling in (8) (but with ν and β being the three-dimensional result) should still hold,for connections to points as we considered here. Fur-thermore, connections between higher-dimensional ob-jects (lines, surfaces, ...) can also be considered. Onequestion to consider is whether the thresholds continueto have broad distributions as found here, and how thosethresholds scale with L .With respect to the correlations R ( p ), one can con-sider a point in the center of a surface of a cylinder (thatis, the center of a square with periodic b.c. in one direc-tion), and find the probability of connecting the centerto one boundary or to both boundaries of the cylinder.At p c , the corresponding R ( p ) should go to the value C = 2 / − / π / Γ(1 / − / = 1 . . . . [20] whilethe behavior away from p c has not been studied before.Likewise, similar correlations in higher dimensions havenot been studied. Many aspects of correlations in perco-lation are yet to be explored. VII. ACKNOWLEDGMENT
The authors would like to thank Deepak Dhar for acareful reading and constructive comments on the paper. [1] Dietrich Stauffer and Ammon Aharony.
Introduction toPercolation Theory, 2nd ed.
CRC press, 1994.[2] 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.[3] L. S. Ramirez, P. M. Centres, and A. J. Ramirez-Pastor.Standard and inverse bond percolation of straight rigidrods on square lattices.
Phys. Rev. E , 97:042113, 2018.[4] Wenxiang Xu, Zhigang Zhu, Yaqing Jiang, and YangJiao. Continuum percolation of congruent overlap-ping polyhedral particles: Finite-size-scaling analysis andrenormalization-group method.
Phys. Rev. E , 99:032107,2019.[5] Sumanta Kundu and S. S. Manna. Percolation modelwith an additional source of disorder.
Phys. Rev. E ,93:062133, 2016.[6] Sayantan Mitra, Dipa Saha, and Ankur Sensharma. Per-colation in a distorted square lattice.
Phys. Rev. E ,99:012117, 2019.[7] M. G. Slutskii, L. Yu. Barash, and Yu. Yu. Tarasevich.Percolation and jamming of random sequential adsorp-tion samples of large linear k -mers on a square lattice. Phys. Rev. E , 98:062130, 2018.[8] Shogo Mizutaka and Takehisa Hasegawa. Disassortativ-ity of percolating clusters in random networks.
Phys.Rev. E , 98:062314, 2018.[9] Sumanta Kundu, Nuno A. M. Ara´ujo, and S. S. Manna.Jamming and percolation properties of random sequen-tial adsorption with relaxation.
Phys. Rev. E , 98:062118,2018.[10] Yunqing Ouyang, Youjin Deng, and Henk W. J. Bl¨ote.Equivalent-neighbor percolation models in two dimen-sions: Crossover between mean-field and short-range be-havior.
Phys. Rev. E , 98:062101, 2018.[11] Wei Huang, Pengcheng Hou, Junfeng Wang, Robert M.Ziff, and Youjin Deng. Critical percolation clusters inseven dimensions and on a complete graph.
Phys. Rev.E , 97:022107, 2018.[12] Stephan Mertens and Cristopher Moore. Percolationthresholds and fisher exponents in hypercubic lattices.
Phys. Rev. E , 98:022120, 2018. [13] Stephan Mertens and Cristopher Moore. Series expansionof the percolation threshold on hypercubic lattices.
J.Phys. A: Math. Th. , 51(47):475001, 2018.[14] Cesar I. N. Sampaio Filho, Jos´e S. Andrade, Hans J. Her-rmann, and Andr´e A. Moreira. Elastic backbone definesa new transition in the percolation model.
Phys. Rev.Lett. , 120:175701, 2018.[15] M. M. H. Sabbir and M. K. Hassan. Product-sum univer-sality and rushbrooke inequality in explosive percolation.
Phys. Rev. E , 97:050102(R), 2018.[16] S. M. Flores, J. J. H. Simmons, P. Kleban, and R. M. Ziff.A formula for crossing probabilities of critical systemsinside polygons.
J. Phys. A: Math. Th. , 50(6):064005,2017.[17] John C. Wierman. On bond percolation thresholdbounds for archimedean lattices with degree three.
J.Phys. A: Math. Th. , 50(29):295001, 2017.[18] Robert M. Ziff and Salvatore Torquato. Percolation ofdisordered jammed sphere packings.
J. Phys. A: Math.Th. , 50(8):085001, 2017.[19] Ivan Kryven, Robert M. Ziff, and Ginestra Bianconi.Renormalization group for link percolation on planar hy-perbolic manifolds.
Phys. Rev. E , 100:022306, 2019.[20] Jacob J. H. Simmons, Robert M. Ziff, and Peter Kleban.Factorization of percolation density correlation functionsfor clusters touching the sides of a rectangle.
J. Stat.Mech. Th. Exp. , 2009(02):P02067, 2009.[21] Gesualdo Delfino and Jacopo Viti. On three-point con-nectivity in two-dimensional percolation.
J. Phys. A.:Math. Th. , 44(3):032001, 2010.[22] M. E. J. Newman and R. M. Ziff. Efficient MonteCarlo algorithm and high-precision results for percola-tion.
Phys. Rev. Lett. , 85(19):4104–4107, 2000.[23] M. E. J. Newman and R. M. Ziff. Fast Monte Carloalgorithm for site or bond percolation.
Phys. Rev. E ,64(1):016706, 2001.[24] R. M. Ziff and M. E. J. Newman. Convergence of thresh-old estimates for two-dimensional percolation.