Complete graph asymptotics for the Ising and random cluster models on 5D grids with cyclic boundary
The finite size scaling behaviour for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject for a long running debate. The older papers have been based on ideas from e.g. field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FK-random-cluster model on a complete graph. Based on those results we identify several distinct scaling regions in an -dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first order phase transition for the same Ising model.
The Ising model is one of the most studied models in statistical physics. We now have a well developed mathematical theory for it’s behaviour in 2-dimensions Smirnov (2010); Chelkak and Smirnov (2012); Grimmett (2004), extensive numerical work in 3-dimensions, and it is known that the upper critical dimension for the model if . For it is known that the model takes on it’s mean field critical exponents in the thermodynamic limit. For finite systems far less is known rigorously and there has been a long running debate on differences in scaling for systems with free and cyclic boundaries Binder (1985); Binder et al. (1985); Binder (1992); Rudnick et al. (1985a, b); Rickwardt et al. (1994); Mon (1996); Parisi and Ruiz-Lorenzo (1996); Luijten and Blöte (1996); Blöte and Luijten (1997); Luijten et al. (1999); Jones and Young (2005). In a previous paper Lundow and Markström (2011) we found an hitherto overlooked way in which for the behaviour of the model depends strongly on the boundary condition, in fact so strongly that at the critical point the difference is not believed to vanish as the side of the system grows. This conclusion has been questioned Berche et al. (2008) by a group of authors which has earlier been promoting an alternative scaling picture for the case with free boundary conditions. However, in a recent paper we revisited this question with much larger system sizes and found a good agreement with our earlier results, and the standard scaling picture, where the susceptibility of the model with free boundary grows as and for cyclic boundary grows as .
In view of these results a natural question is why we should see a different behaviour for the case with cyclic boundary, and what the exact form of the scaling behaviour for this case should be for above the critical dimension. Our approach to the answer of this question is via the Random-cluster model. The Fortuin-Kasteleyn random cluster model, or RC-model for short, is a natural extension of the Ising, Potts and edge percolation models, all captured by varying , one of its two parameters, the other corresponding to the temperature in the first two models and the edge probability in the last. Many properties of the Ising model have direct interpretations in terms of the numbers and sizes of connected clusters in the RC-model. In particular the susceptibility corresponds to the average cluster size.
For percolation, the case of the RC-model, Aizenman Aizenman (1997) conjectured that the largest clusters should scale as for with cyclic boundary, instead of for free boundary. One of the reasons for this conjecture was that this gives the same scaling as for the largest connected in the Erdös-Renyi, or ER for short, random graph with vertices at it’s critical point, where the largest component has size proportional to . Note that the ER-random graph can be seen as the percolation model on the complete graph on vertices. This conjecture was proved in Heydenreich and van der Hofstad (2007, 2011), and in Borgs et al. (2005a, b, 2006) it was proved that asymptotics of the same type as on the complete graph can be expected on a wider class of finite graphs as well.
As we have pointed out percolation is a special case of the RC-model and the latter has also been studied on the complete graph by mathematicians. In Bollobás et al. (1996) the critical probability was identified, the exponential asymptotics of the partition function was studied, and it was proved that the phase transition is of second order for and of first order for . Later Luczak and Luczak (2006) a more detailed study of the cluster structure was carried out, and it was found that there are three ranges of with distinct behaviour , and .
Our aim in this paper is to compare the behaviour of the largest and second largest cluster in the RC-model for , corresponding to the Ising model, for 5-dimensional lattices with side and cyclic boundary. For the case Luczak and Luczak (2006) identified no less than five distinct scaling regions near the critical probability for the complete graph, each with its own asymptotic behaviour for and , the sizes of the largest and second largest clusters respectively. Some of the regions are difficult to study, since in order to obtain correct scaling they would require far larger graphs than those used here. Thus we have focused on three cases: i) the high-temperature case (fixed for ), ii) near at fixed , and iii) near at fixed , where and are different -dependent couplings. As we shall see, for these regions we have an excellent agreement between the scaling for the complete graph and for the 5-dimensional Ising model with cyclic boundary.
Ii Terminology, definitions and sampling details
For a graph on vertices and edges the random cluster model’s partition function is
where is the number of (connected) components, or clusters as we will call them, of the graph , i.e. the graph with vertex set and edge set . Note that and are parameters to the distribution. The Ising model without an external field, on the other hand, has the partition function
where the sum is taken over all functions from to . Thus vertex has spin . Here the energy is , summed over all edges in . The parameter is the dimensionless coupling, or inverse temperature.
The two models are actually equivalent for when setting . Assuming we have a state from the random cluster distribution in Equation (1), we can obtain a state from the Ising distribution (2) in the following way: for each component, pick a spin uniformly at random and assign this spin to all vertices of the component. Going in the other direction is also easy. Starting with a state from the Ising distribution at coupling , let . Now add each satisfied edge, i.e. edges with , to the set with probability . For more information on this see Grimmett (2004).
This second scheme is a surprisingly efficient way of obtaining random cluster data. Just start up your trusted Ising state generator, whether it be Metropolis, heat-bath or Wolff cluster Newman and Barkema (1999), and convert the Ising states to correctly distributed random cluster states.
All graphs studied here are 5-dimensional grid graphs with periodic boundary conditions, i.e. cartesian products of five cycles on vertices, so that and , and we have used , 6, 8, 10, 12, 16, 20 and 24. We collected the data by generating Ising states with the Wolff cluster method Wolff (1989) and then converting them to random cluster states using the scheme described above.
Throughout we use the critical coupling Lundow and Markström (2011), thus corresponding to the random cluster critical probablity , and denote one scaled temperature by and another by . Recall that the critical temperature for a complete graph on vertices approaches zero as and hence . We will denote a scaled probability by .
The th central moment of a distribution is denoted by , the mean value by and the variance by . The standard deviation is then . The median is written . Given a random cluster state the largest and second largest cluster size, i.e. the number of vertices in these clusters, is denoted and respectively.
ii.1 Scaling for the complete graph
We will here give a very brief description of some of the relevant scaling results from Luczak and Luczak (2006) for the complete graph on vertices. Where relevant further details will be given in later sections. As proven in Bollobás et al. (1996) the critical probability is given by . Now if then the model is not critical and depending on whether is less than 1 or greater than 1 we see behaviour corresponding to the high- and low-temperature regions respectively, with only small clusters in the first case and a large cluster, linear in sized, plus some small ones in the latter case.
If we are inside the critical window and need a finer parameterization of . We thus assume that , where can depend on . We now see five distinct regions inside the critical window:
if then, asymptotically, all clusters are trees and small.
if , where is a constant, then is roughly of order .
if but there exists a unique largest component and it’s size is , for a value .
if , where is a constant, then is of order and is bounded by .
if and then is of order and of order .
Iii The high-temperature region
Strictly speaking, the complete graph version of the high-temperature case only requires that . Of course, a fixed will satisfy this condition. Hence, for the 5D case we will simply test the case of a fixed for .
given some parameters . Extreme value distributions have skewness and kurtosis . In Figure 1 we show how the skewness and excess kurtosis scales with and how they approach these values for two different temperatures. The right plot of Figure 1 shows the distribution of at for . In short, we have good reason to think that the complete graph behaviour also holds for 5d in this case.
Iv Negative scaled temperature
Here we consider the case of a fixed negative scaled temperature, . This falls under the complete graph case . In Luczak and Luczak (2006) it was shown that for this particular case the two largest clusters behave almost surely as
for all functions tending to infinity. Since is allowed to grow as slowly as we like, thus boxing in and , it is quite possible that in fact , though with different prefactors. Let us test this for the 5D case. In Figure 2 we show the normalised mean and for different negative . The right plot shows a zoom-in for .
Note here that when whereas the counterpart actually has a local maximum. The right panel of Figure 2 shows a zoom-in for , both the sampled data for and also an estimated limit function based on fitting second degree polynomials to the values at versus . The has a distinct local maximum for all and its location may possibly have zero as limit, approaching it very slowly. It would take considerably bigger graphs to settle that question. However, a rough estimate gives that the limit has the local maximum at . To conclude this section, we find that the 5D behaviour matches that of the complete graph case.
V Scaled temperature
The next case is that of , for constant . This fits under the complete graph case in Luczak and Luczak (2006) of for real constants . In this region the complete graph has different scalings for and and we treat them separately in the following two subsections.
v.1 The largest cluster
In the complete graph case it was shown Luczak and Luczak (2006) that
and in fact can be replaced by with some positive limit value . Note that the -distribution essentially has the same form as the Ising magnetisation distribution of a complete graph, which is , see Lundow and Rosengren (2010), except that has support only for . This follows since the spin-state coming from an RC-state is obtained by assigning a random spin value to each cluster in the RC-state. Above all this means that for constant . In Figure 3 we show the normalised mean and the normalised variance over the interval . The plot also shows the complete graph values computed from Equation (5) when setting . Note how close the complete graph values are to their 5D counterparts.
The limit of the mean is easily found by plotting them for some versus and then fitting a polynomial to the points. There is some very mild correction to scaling at work for but it is easily captured by a second degree polynomial. A slightly more careful analysis on the case , i.e. at , based on fitting second degree polynomials to all 4-subsets of the data points for , results in a median value , and the error corresponds to the interquartile range. The complete graph value at is .
What about the distribution of ? Figure 4 shows a normalised form of the -distribution, where , for a few values of with . Between roughly the distribution actually goes through a bimodal phase and this property is not found in the complete graph case of Equation (5). However, it agrees with the finding in Lundow and Markström (2011) that for the Ising model with cyclic boundary in 5d the energy distribution at the critical temperature becomes bimodal. The second plot in Figure 4 shows the distribution at for and it clearly shows the distribution retaining its bimodal form with increasing .
The skewness and kurtosis of the -distribution are shown in Figure 5 together with the complete graph case. Clearly there are distinct limits for each , occasionally with some mild corrections to scaling, easily captured by a second degree polynomial. Perhaps unexpectedly, the data suggest a limit value of the skewness of about as . There is no conflict in the existence of such a limit and our earlier claim of the skewness taking the extreme value distribution value of in the high-temperature case. It is simply a matter of taking limits in the right order, i.e. and the kurtosis limit is approximately . This is in fact also the behaviour of the complete graph case though it takes different values. As its skewness approaches and the kurtosis has the limit . The two models agree well on the case though. Here the 5D case has skewness and kurtosis while the complete graph values are and respectively.
v.2 The second largest cluster
In the complete graph case it was shown Luczak and Luczak (2006) that
for . Note that may be replaced by function having some limit . This gives a density function
which implies an infinite expectation value. We will thus instead consider the median value. So, when we move close enough to , from to , the second largest cluster drops in size from to . This also seems to be the case for the 5D case though the corrections to scaling are quite significant for .
Figure 6 shows the normalised mediam versus for 5D and the complete graph case which demonstrates this effect. Note that for it would require enormous graphs to get anything close to the complete graph case. For , however, the two curves quickly agree on the complete graph behaviour, if not on the actual value. The right plot of the figure shows a normalised form of the -distribution, with , for a range of together with the complete graph distribution in Equation (7) at where the two cases largely agree.
If the 5D -distribution has a fat tailed distribution like the complete graph case of Equation (7) this must eventually show up in some higher moment, and indeed it does. Consider the skewness and kurtosis of the -distribution in Figure 7. Both show a divergent behaviour around . In fact, a very rough estimate suggests that the peak skewness grows as and the peak kurtosis as .
Our aim has been to compare the scaling of the sizes of the largest and second largest clusters for the 5D random cluster model with that seen for the corresponding value of for the complete graph. For the complete graph there are two non-critical regions and inside the critical window five distinct scaling regions. Due to the limitations coming from the range of system sizes for which we can simulate the model we have chosen to work with the high temperature region and two of the regions from the critical window.
In each of the tested regions we have found that the scaling from the complete graph agrees well with the observed values from the 5D model, and in many cases not only the scaling but also the probability distributions agree well. The most notable exception is the bimodal distribution of for close to 0 for the 5D model. This feature corresponds to the bimodal energy distribution seen for the Ising model on the same graphs in Lundow and Markström (2011), which also clearly separates the 5D model with cyclic boundary from the case with free boundary and the infinite system thermodynamic limit.
We expect this agreement to hold for as well. If we in the scaling for and on the complete graph case replace by we conjecture that we get the correct scaling for the -dimensional case with cyclic boundary.
The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N) and at Chalmers Centre for Computational Science and Engineering (C3SE).
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