Percolation in the canonical ensemble

Percolation in the canonical ensemble

Hao Hu, Henk W. J. Blöte and Youjin Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230027, China Instituut Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
July 15, 2019

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent both in energy-like and magnetic quantities, where is the thermal renormalization exponent and is the spatial dimension. Furthermore, we find that while most of universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has .

05.50.+q, 64.60.ah,, 64.60.F-

I Introduction

The bond-percolation model SA () can be considered as the limit of the -state Potts model RBP (); FW (). Consider a lattice with () the vertex (edge) set, the reduced (divided by ) Hamiltonian of the Potts model reads


where is the coupling strength. By introducing bond variables on the edges and integrating out the spin degrees of freedom, one can map the Potts model onto the random-cluster (RC) model. This is the celebrated Kasteleyn-Fortuin transformation KF (). The partition sum of the RC model thus assumes the form


where the sum is over all the spanning subgraphs of , is the number of occupied bonds in , and is the number of connected components (clusters). The Kasteleyn-Fortuin mapping provides a generalization of the Potts model to non-integer values . In the limit, the RC model reduces to the bond percolation problem KF (); FW (). In this limit, the partition sum assumes the non-singular form , where is the number of the lattice edges. Nevertheless, rich physics still exists, e.g. in the derivatives of the partition sum involving .

Percolation in two dimensions has been extensively studied. Percolation thresholds on many two-dimensional lattices are exactly known or have been determined to very high precision (see e.g. Ref. Ziff11 () and references therein). Most of the critical exponents are also exactly known. For instance, the leading and subleading thermal renormalization exponents are and , respectively; the leading two critical exponents in the magnetic sector are and BNDL (); JCDL (). There still exist some critical exponents whose exact values are unknown, including the backbone exponent and the shortest-path fractal dimension GB (); ZYDZ (). Much progress has recently been made in the context of the stochastic-Löwner evolution WKBN (); JC ().

The present work provides a study of another aspect of percolation. We introduce an “energy-like” constraint that the total number of occupied bonds is fixed and study the effects of such a constraint on the critical finite-size scaling (FSS). Since occupied bonds can formally be considered as particles, we shall refer to percolation under this constraint as percolation in the canonical ensemble, and to the case without the constraint as percolation in the grand-canonical ensemble. Two of us have, for several years, been studying the effects of such constraints DB (); DHB (); DB1 (). The leading FSS in systems under energy-like constraints is derived on the basis of the so-called Fisher renormalization MF () procedure, which was originally formulated for systems in the thermodynamic limit. Consider an energy-density-like observable in the critical RC model that scales as in the grand-canonical ensemble, where is the linear system size and is the spatial dimensionality. The constraint in the canonical ensemble implies:
(i), for a given size is fixed at the expectation value in the thermodynamic limit. This expectation value is normally different from the grand-canonical finite-size average .
(ii), fluctuations of are forbidden—i.e., .
The effect of (i) is accounted for by the Fisher renormalization. The bond density is such an observable. For bond percolation, however, since the bond variables on different edges are independent of each other, does not depend on the system size —i.e., if we write , then the amplitude . Thus, percolation provides an ideal system to study the effects of energy-like constraints due to the suppression of fluctuations.

The remainder of this work is organized as follows. Section II describes the sampled quantities and the FSS of physical observables for percolation in the grand-canonical ensemble. Section III derives the effects of the “energy-like” constraint in the critical FSS. The numerical results are presented in Sec. IV. A brief discussion is given in Sec. V.

Ii Sampled quantities and finite-size scaling

ii.1 Simulation and sampled quantities

We consider the bond-percolation problem on an square lattice with periodic boundary conditions. The grand-canonical simulations follow the standard procedure: each edge is occupied by a bond with probability , after which the percolation clusters are constructed. For simulations in the canonical ensemble, a Kawasaki-like scheme Kawasaki () is used. Given an initial configuration with a total number of occupied bonds , where , an update is defined as the random selection of two edges and the exchange of their occupation states. Each sweep consists of updates. Quantities are sampled after every sweep. Note that although the bond number happens to be an integer for the critical square-lattice bond percolation, is generally a real number. In this case, one can simulate at the ceiling and the floor integer of and apply an interpolation.

Given a configuration as it occurs in the simulation, we denote the sizes of clusters as (); is reserved for the largest cluster. The following observables were sampled.

  1. Energy-like quantities:

    • The bond-occupation density .

    • The cluster-number density .

  2. Specific-heat-like quantities:

    • .

    • .

    • .

    • .

    Here, the factor in the definition of arises from the fact that the critical line of the RC model (2) on the square lattice is described by , which leads to a weight for a subgraph along the critical line. In the canonical ensemble, the bond number is fixed, and and reduce to .

  3. Magnetic quantities:

    • The largest-cluster size .

    • The cluster-size moments (). Defining we sampled and .

  4. Dimensionless quantities:

    • Wrapping probabilities , and . The probability counts the events that configuration connects to itself along the direction, but not along the direction; is for simultaneous wrapping in both directions; is for the direction, irrespective of the direction; is for wrapping in at least one direction. They are related as and , thus only two of them are independent NZ (); NZ1 ().

    • Universal ratio .

    • Universal ratio .

ii.2 Finite-size scaling in the grand-canonical ensemble

The critical FSS of the sampled quantities can be obtained from derivatives of the free-energy density ( is given by Eq. (2)) with respect to the thermal scaling field , the magnetic scaling field , or the parameter . In the grand-canonical ensemble, the FSS of is expected to behave as


where higher-order scaling fields have been neglected, and and denote the regular and the singular part of the free-energy density, respectively. The thermal scaling field is approximately proportional to in Eq. (2), where is the critical line of the -state RC model on the square lattice.

FSS for energy-like quantities. From the partition sum in Eq. (2) and Eq. (3), it can be derived that at criticality


where is the free-energy density along the critical line . The last equality in Eq. (4) reflects the analyticity of (including the amplitude of its finite-size dependence) along the critical line KZ (). Moreover, the correction amplitude is universal ZFA (); KZ (), which is also referred to as the excess cluster number for percolation. The term follows from the exact results for the critical Potts free energy on the square lattice Temperley-71 (). An exact result is also available for the triangular lattice BTA (). The last equality in Eq. (5) arises from Eq. (3), and accounts for the background contribution; the self-duality of the square-lattice model yields . In the limit, the amplitude vanishes as . Therefore, one has for critical percolation


FSS for specific-heat-like quantities. Similarly, one has the second derivatives (at ) as


where the last equalities in Eq. (10) and (11) arise from the fact that the parameter in Eqs. (5) and (6) behaves as . The analyticity of along the critical line KZ () is also reflected in Eq. (8). It is also derived that the amplitude is a universal quantity KZ (), called the excess fluctuation of the number for critical bond percolation.

FSS for magnetic quantities. The critical FSS of magnetic quantities can be obtained by differentiating the free-energy density with respect to the magnetic scaling field . It can be shown that at criticality,


Corrections to FSS. At criticality, the asymptotic behavior of a quantity is supposed to follow the form


where is the leading critical exponent and is the amplitude. For our observables in bond percolation, values of are given by the previous analysis ( for dimensionless quantities). The FSS theory predicts several types of “corrections” in Eq. (13) (see e.g. Refs.GOPS () and FDB () for a review and references therein). These terms include corrections from the irrelevant scaling fields; the leading one has exponent for percolation. A regular background term, which appears e.g. as for and for , contributes to the “corrections”. These regular background terms are also present in the associated dimensionless ratios. For the energy-like quantities ( and ) and the specific-heat-like quantities, is negative, thus the regular background (which is here) included in the “corrections” in Eq. (13) is actually not a correction, instead it describes the leading behavior, which is the thermodynamic limit of the sampled quantity.

Iii Finite-size scaling in the canonical ensemble

As mentioned in the Introduction, the critical FSS for bond percolation in the canonical ensemble is due to the suppression of the fluctuation of occupied-bond density . In this section, we demonstrate how the critical FSS is affected by such a suppression.

Without the constraint, any configuration with bond number occurs with probability (). Thus, the value of an observable in the grand-canonical ensemble can be written as


where is the occupied-bond density and is obtained by averaging over all the configurations with , of which the total number is . Namely, is the canonical-ensemble value of observable . For a sufficiently large system, the binomial distribution in Eq. (14) is well approximated by a Gaussian distribution


where is the average bond density for occupation probability , and the variance decreases as . Thus, Eq. (14) can be re-written as


In principle, based on the known scaling behavior of in the grand-canonical ensemble, the critical FSS of can be obtained by the inverse orthogonal transformation of Eq. (16). Here, we take an approximation approach by Taylor expanding in Eq. (16) in powers of about , and term-by-term evaluation of the integrals:


where the derivatives and are taken at , and the average is over the Gaussian distribution in Eq. (15). It is easily derived that and . Combination of Eqs. (16) and (17) yields


with a non-universal constant.

Near criticality , we denote . Since , we can equivalently denote the distance to the percolation threshold as with and . Thus


where new finite-size corrections with exponent are allowed in Eq. (20). It is known from Eq. (20) that . Taylor-expanding the r.h.s of Eqs. (19-20) and substituting them in Eq. (18) give


Assuming that , the term with acts as a finite-size correction. By comparing the scaling behavior of the l.h.s and the r.h.s. of Eq. (21), we obtain

  1. and ,

  2. , (for ), and etc. In other words, the scaling functions in Eqs. (19) and (20) are identical up to some correction terms.

  3. new correction terms appear in the canonical ensemble, and they have exponents and (if ). This is because the leading finite-size correction in the l.h.s of Eq. (21) is described by an exponent , and thus the terms with and have to be cancelled by the newly included correction terms with . It can be shown that corrections with exponent with can also occur if higher-order terms are kept in the above Taylor expansions.

  4. for the case of , does not hold. Instead, one has . This predicts that the excess cluster number in Eq. (7) is changed by the constraint; further, since and are non-universal, the canonical-ensemble value of is no longer universal.

Note that the assumption holds since for percolation in two and higher dimensions.

Finally, we mention that although the constraint is “energy-like”, the derivation applies to both the energy-like and the magnetic observables.

Iv Numerical results

To examine the above theoretical derivations, we carry out Monte Carlo simulations for the critical bond percolation on the square lattice, with a fixed bond number , and for sizes in range . The number of samples was about for , for , for , and for . For the purpose of comparison, additional simulations were also carried out in the grand-canonical ensemble.

iv.1 Evidence for the correction exponent

We first examine the critical FSS of the wrapping probabilities. In the grand-canonical ensemble, the finite-size corrections arise only from the irrelevant scaling fields for which the leading exponent . This is illustrated in the inset of Fig. 1. In the canonical ensemble, the existence of the newly induced correction exponent is clearly demonstrated by the approximately linear behavior for large in Fig. 1, where is plotted versus .

Figure 1: (Color online). Plot of the wrapping probability versus in the canonical ensemble (main plot). The inset shows versus in the grand-canonical ensemble. The straight lines represent the leading finite-size dependence as obtained from the fit. The error bars of the data are smaller than the size of the data points.

According to the least-squares criterion, we fitted the data for these wrapping probabilities by


where represents the universal value for , and the correction exponent is fixed at . As a precaution against higher-order correction terms not included in the fit formula, the data points for small were gradually excluded to see how the residual changes with respect to . In general, we use results of the fit corresponding to a for which the quality of fit is reasonable, and for which subsequent increases of do not cause the value to drop by vastly more than one unit per degree of freedom. In practice, the word “reasonable” means here that is less than or close to the number of degrees of freedom, and that the fitted parameters become stable.

Fit using





Table 1: Fit results for wrapping probabilities. The exact values of  Pinson (); ZLK (), and are used here. The abbreviations “GE” and “CE” stand for grand- and canonical ensemble, respectively. Entries “-” indicate the absence of a result, and the numbers without error bars are fixed in the fits. Error margins are quoted as two times of the statistical errors in the fits.

The results are shown in Table 1. The error margins are quoted as two times of the statistical errors in the fits, which also applies to other tables, in order to account for possible systematic errors. The universal values can be exactly obtained Pinson (); ZLK (). In the grand-canonical ensemble, these exact values of were used and the amplitudes and were set at , so that the fit formula in Eq. (22) has only a single free parameter. Indeed, such a simple formula can well describe the data with for all the wrapping probabilities (, , , and ); further, for the amplitude of is very small. As expected, including terms with and/or only yields messy information and does not improve the quality of the fits. In the canonical ensemble, we also fitted the data by Eq. (22) with a single correction term (). As shown in Tab. 1, the data up to sufficiently large (except ) have to be discarded for a reasonable residual . Nevertheless, we find that (i), the correction term has an exponent close to , and (ii), the estimates of agree well with the exact values. A better description of the data can be obtained by including correction terms with and/or ; for simplicity the exact values of are used. As a result, the estimates of become more accurate, and are in good agreement with the prediction . We note that the correction coefficient takes different values in the grand-canonical and the canonical ensemble. This is also in agreement with the theoretical expectation. As predicted in Sec. III, in addition to those with exponent , correction terms with exponent can also exist in the canonical ensemble. For the wrapping probabilities, one has and thus such a correction term has the same exponent as . Finally, in order to illustrate the term with , we plot versus as in Fig. 2.

Figure 2: (Color online). Plot of versus for the canonical wrapping probability . The straight line represents the leading finite-size dependence of , i.e. . Parameter and are fixed at the exact value and respectively, and the amplitudes , are obtained from the fit.

In order to demonstrate that the correction exponent for the grand-canonical wrapping probabilities is indeed , we also fitted the data by Eq. (22) with being free, being fixed at the exact values and . The fit results are also included in Table 1. The result is , , and , for , , and , respectively. These results are in compatible with the existing numerical result  NZ1 () for , and agree well with the exact value . It is interesting to mention that although correction of form ( is a constant) has been observed for other observables in the square-lattice bond percolation FDB (), there is no evidence for such a logarithmic factor in the finite-size corrections for the wrapping probabilities.

Fit using
Table 2: Fit results for , , and the universal ratios and .

For magnetic quantities, we consider the largest-cluster size and the second cluster-size moment , as well as the dimensionless ratios. We fitted the data by


with and being fixed at the exactly known value for the two-dimensional percolation. The term with is the correction from the regular background, with for and , and for and . The results are shown in Table 2. Again, we find that (i), the values of remain unchanged in the canonical ensemble, irrespective of whether they are universal (for and ) or non-universal (for and ), and (ii), new correction terms with an exponent are introduced. The estimate in the grand-canonical ensemble agrees with the existing result  DB1 ().

Fit using


Table 3: Fit results for the cluster-number density as determined for the bond (BP) and site (SP) percolation problem. Some fits make use of the exact results  Temperley-71 () and  KZ ().

iv.2 Change of the universal excess cluster number

For energy-like quantities, we consider the cluster-number density , and analyze the data by


with being fixed. The background term can be exactly obtained as for the critical square-lattice bond percolation Temperley-71 (). In the grand-canonical ensemble, the excess cluster number is known to be universal and the value has been obtained as  KZ (), with subscript for the grand-canonical ensemble.

The fit results are given in Table 3. As in the above subsection, the correction term with is observed in the canonical ensemble, and the background contribution remains unchanged. However, the excess cluster number is now , clearly different from the universal value  KZ () in the grand-canonical ensemble. This indeed confirms the theoretical prediction for the case of in Sec. III. In other words, one expects that , where is a constant and is the second derivative of the regular part of with respect to the bond density. Since both and are non-universal, the canonical-ensemble value is no longer universal. As an illustration, we plot versus in Fig. 3, where the difference of and is reflected by the different slopes of the data lines.

Figure 3: (Color online). Plot of the cluster-number density versus for the bond percolation model. The straight lines represent the leading finite-size dependence of as obtained from the fit.

In order to examine the “non-universal” nature of the excess cluster number in the canonical ensemble, we also performed simulations of square-lattice site-percolation problem at the percolation threshold  Lee (); FDB (); Ziff11 (). In the grand-canonical ensemble, the simulations used 15 system sizes in range , and the results of the fits by Eq. (24) are given in Tab. 3. The estimate of agrees well with the universal value  KZ ().

In the canonical ensemble, the total number of occupied sites is fixed. However, is not an integer, and thus the actual simulations were carried out for the total occupied-site number and , with for the floor integer. The Monte Carlo results at were then obtained by linear interpolation. The simulations used system sizes in range . The results of fits using Eq. (24) are also given in Tab. 3. As expected, one has a correction term with exponent and . The “non-universal” property of is demonstrated by the fact .

In addition, from our fits, one gets , which is in good agreement with the existing result  ZFA (), and reduces the error margin significantly.

We also study the FSS of the specific-heat-like quantity for the bond percolation model and find that due to the limited precision, the data for are well described by . As already mentioned in Section II, the excess fluctuation is universal. In the grand-canonical ensemble, the fit yields and ; the former is consistent with the existing result  DY (), the latter agrees well with the exactly known value  KZ (). In the canonical ensemble, the result is and . It is interesting to observe that not only the magnitude of the excess fluctuation is modified, but that also its sign has changed.

iv.3 Universality of scaling functions

In Sec. III, we state that the scaling functions in Eqs. (19) and (20) are identical up to some correction terms, which reflects the universality of the functions. In order to demonstrate this, we carried out simulations near the critical point. We plot the grand-canonical versus and the canonical versus as in Fig. 4. As expected, data points in both the ensembles nicely collapse to the same curve.

Figure 4: (Color online). Plot of versus for bond percolation. Parameter represents in the grand-canonical ensemble (GE), and in the canonical ensemble (CE). The excellent collapse demonstrates the universality of scaling functions near the critical point.

V Discussion

We derive the critical finite-size scaling behavior of percolation under the constraint that the total number of occupied bonds/sites is fixed, and confirm theoretical predictions for the two-dimensional percolation by means of Monte Carlo simulation in two dimensions. In particular, it is found that with the constraint, new finite-size corrections with exponent are induced and the excess cluster number becomes non-universal. We note that our theory in Sec. III can be used to explain the observed correction exponent for the canonical wrapping probabilities in simulating the two-dimensional percolation by the Newman-Ziff algorithm NZ (); NZ1 (). The predictions should be valid in any dimension with . We believe that this work provides an additional useful reference for percolation—a pedagogical system in the field of statistical mechanics. Furthermore, our work can help to understand the critical finite-size-scaling properties of other statistical systems in the canonical ensemble with a fixed total number of particles, which is the usual situation during experiments.

This work was supported by NSFC (under grant numbers 10975127, 91024026 and 11275185), CAS, and Fundamental Research Funds for the Central Universities No. 2340000034. We would like to thank R. M. Ziff for helpful comments. H. Hu is grateful for the hospitality of the Instituut Lorentz of Leiden University.


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