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Monte Carlo Renormalization in Percolation
Xiao Zhang (Thomas)11Department of Physics, University of Michigan, Ann Arbor, MI 48109
In this paper we will dicuss how to use Monte Carlo Renormaliztion Group to determine the criticalconcentration pc for ordinary site percolation problems. We will also discuss how to percolationproblem on other geometries of such as the triangular lattice and kagome lattice.
I. INTRODUCTION
As discussed in class we can find the critical concentration pc for a percolation problem using Monte Carlo(MC)simulation. A example implementation is done in homework 4 and figure 1 is an example output running the algorithm:This gives us reasonally well approximation but it also has many problems. First it takes a fairly long time to run.
FIG. 1: An sample run using Monte Carlo simulation with system size L = 10, the critical concentration is determinedabout pc ≈ 0.6.
Even for a system size as small as L = 10 it takes a ridiculously long time to get a good answer and this making italmost impossible to go to larger system sizes. Given running time and limited to small sizes as barriers, this methodtherefore cannot give us other useful critical exponents other than the critical concetration.
One good way to encompass the difficulties we have in ordinary MC is to us Monte Carlo RenormalizationGroup(MCRG) method. The method is described at the end of Chapter 4 of the lecture notes and we go through itbriefly here in this paper:
1. Draw a set of random numbers uj for each set j.
2. Using binary search to determine the probability p̃ where a percolating cluster appears. We start with p̃ = 0.5(or essentially any number, but to get a more accurate answer and avoid large numbers of iterations, startingwith a number reasonably close to the anticipated value of pc is recommended), then depending on the latticepercolates or not we try p̃ = 0.75 or p̃ = 0.25. It is straight forward to show that after 10 iterations the accuracyis within 10−3 for a particular set of random numbers.
3. Now we get new set of uj and repeat the same iterations. This will gives us a very good estimation of ∆ and pc.
II. EXAMPLES ON SQUARE LATTICE
The result of running the algorithm discussed above gives us results as in Figure 2:
2
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
PHpL
L=1000
L=100
L=10
FIG. 2: Critical concentration pc determined with MCRGalgorithm of different system sizes
FIG. 3: standard deviaion ∆ fitted as a function of L. Plot inlog scale.
This algorithm runs much faster than the ordinary MC algorithm, for a system size with L = 1000 it can takeroughly the same running time. We can then very easily use to determine the critical exponents. In Figure 3 we findthat ν = 1.3285 compared with the true value ν = 4/3.
III. TRIANGULAR LATTICE
Now we look at a different lattice structure, the triangular lattice.The closed-packing with hexogonal tilings tri-angular lattice is depicted in Figure 4. Triangular lattice is interesting because the simplicity of its nature and thefact that the percolation problem in this case is exactly solvable. The difficulties in simulating percolation problemon a triangular lattice is that the position of the site is no longer easily described by integers. If we want to storethe location of the triangular lattice we will waste a humongus amount of memory to store the position of the pointswhere there is no lattice site at all. The solution to resolve this issue is to transform the triangular lattice into a squarelattice form. We shift horizontal the position of lattice sites if necessary while keeping the bonds to the neighborsunchanged,as shown in Figure 5.
FIG. 4: A closed-packed triangular latticeFIG. 5: Transformed triangular lattice in a form of square array
Now we only need the same amount of memory as for the case of square lattice for triangular lattice as well. Alsowhen we write the triangular lattice in this form, it becomes clear how we can change the original percolation codesfor square lattice to be adapted for triangular lattice. The only catch now is that not only we need to consider theleft and top neighbor for each site, but also the top-right neighbor as well. Making this change to Hoshen-Kopelmanalgorithm we can then simulate percolation problem on triangular lattice. After running a simulation using MCRGon a system size with L = 200 we get pc = 0.4962, which is in good agreement with the exact value pc = 0.5.
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IV. KAGOME LATTICE
Lastly we look at more exotic example Kagome lattice. The kagome lattice is semiregular tiling of a 2D plane.The different lattice sites have apparently different neighbors which make the percolation problem more complicated.However one should not be surprised that we can still use the idea of lattice transformation to reduce Kagome latticeto a square lattice. All one needs to be careful with is when writing down the Hoshen-Kopelman cluster updatealgorithm we should make sure for each fundamentally different site the neighbor we check corresponds to the correctones in the original lattice. The fundamentally different sites are shown in Figure 7. A simulation on a system sizeof L = 240 gives us pc = 0.64899 compared to the exact value of pc = 0.65270, which again is in good agreement.Notice we chose L to be a multiple of 4 so that in python the periodic condition is satisfied.
FIG. 6: Kagome lattice - a trihexagonal tilingFIG. 7: Six unique points we need to consider in Hoshen-Kopelman algorithm. Notice in the vertical direction the pe-riodicity is 4.
V. CONCLUSION
In this paper we have discussed how to use MCRG as a more efficient algorithm to simulate percolation problems.We also showed how could we apply the same algorithm to simulate percolation problem on different lattices as well.However percolation is a very broad subject and there is defnitely more to learn. The MCRG, though much moreefficient than ordinary MC but is still slow when we get more ambitious to try even larger systems. At that pointmore mordern implementation of percolation algorithm is needed. Further one can look at more both analyticallyand numerically at various different lattice structures and higher dimension problems. But this project has coveredthe most fundamental physics ideas and numerical methods.
[1] Suding, Paul N., and Robert M. Ziff. ”Site percolation thresholds for Archimedean lattices.” Physical Review E 60.1 (1999):275.
[2] Sykes, Mq F., and J. W. Essam. ”Exact critical percolation probabilities for site and bond problems in two dimensions.”Journal of Mathematical Physics 5 (1964): 1117.
[3] Stauffer, Dietrich, and Ammon Aharony. Introduction to percolation theory. CRC press, 1994.