FIG. 3: (color online) Average over 8 samples of ln( S ( π/ 64)) for 64 × 64 × 64 lattices as a function of T/J . The error bars indicate approximately three standard deviation statis- tical errors. FIG. 2: (color online) Average over 8 samples of ln( S ( k )) for 64 × 64 × 64 lattices with h r = 2 and p = 0 . 5 at various temperatures. The error bars indicate approximately three standard deviation statistical errors, and the x -axis is scaled logarithmically. hysteresis which is associated with a normal first-order phase transition. As was discussed in somewhat more detail in the earlier work,[12] the author believes that this phase transition should be described by a theory of the Anderson-Yuval[13] type, where the order parameter jumps to zero discontinuously at T$_{c}$ . A phase transition of this type is also believed to occur in the k -core percolation model,[14] which has been suggested as model for the ordinary glass transition. no longer true for T/J < 1 . 3125, and thus results for these lower temperatures are not reported. For each sample at each value of T/J , the data shown in the figure were obtained by averaging S ( k ) over 16 spin states taken at intervals of 51,200 Monte Carlo steps per spin (MCS), after letting the sample equilibrate. Thus the length of each data-gathering run was 768,000 MCS. The values shown for S ( k ) are obtained by averaging over all of the values of ⃗ k which have length k . This multi- plicity, which is not the same for all values of k , was not taken into account in calculating the error bars shown in the figures. Thus each error bar shown in the figures represents approximately three standard deviations. The procedure used for the Type B samples was essentially identical. The qualitative behavior which we see in Fig. 1 and Fig. 2 is that the average over samples of ln( S ( k )) shows a peak at small k which grows in size as we lower T , until it reaches a maximum at T$_{c}$ . The value of T$_{c}$/J is about 1.40625 for the Type A samples and 1.5625 for the Type B samples. Upon lowering T below T$_{c}$ , the size of the peak becomes somewhat smaller. This is seen more clearly in Fig. 3, which shows the temperature depen- dence of av[ln( S ( π/ 64))] for both types of samples. (The value of π/ 64 is approximately 0.049.) 3 For T > T$_{c}$ both types display a substantial range of T over which S ( π/ 64) is increasing exponentially as T is lowered, and this increase slows very close to T$_{c}$ . This kind of temperature dependence is well known phenomenologically[15] to be a characteristic of glassy freezing behavior. However, the reader should keep in mind that one expects the usual glass transition to cor- respond to n = 3 spins, and not the n = 2 spins studied here. If one generalizes Eqn. (1) to the n = 3 case, sub- stantially more computing resources would be required to make the equivalent calculation feasible. Below T$_{c}$ , S ( π/ 64) first decreases, and then becomes In Fig. 2 we show the corresponding data for the Type B samples, for T/J between 1.4375 and 2.00. The data for T/J > 1 . 5625 are taken from the hot start runs, and the data for lower values of T/J are taken from the cold start run which gave the lower average value for the en- ergy, 〈 E 〉 . It was also true for these Type B samples that the hot start samples were unable to equilibrate on accessible time scales for T/J ≤ 1 . 5625. We can define T$_{c}$ in a number of ways. For instance, it can be defined as the highest temperature for which we have been unable to equilibrate the hot start runs. It is also the temperature for which the equilibration time for the cold start runs becomes very long. The fact that equilibration times get very long at the same T$_{c}$ for both the hot start and the cold start initial conditions that the behavior of this model can not be explained by the usual 1.3GLYPH<13> 1.4GLYPH<13> 1.5GLYPH<13> 1.6GLYPH<13> 1.7GLYPH<13> 1.8GLYPH<13> 1.9GLYPH<13> 3GLYPH<13> 4GLYPH<13> 5GLYPH<13> 6GLYPH<13> 7GLYPH<13> 8GLYPH<13> random-field ZGLYPH<13> 12GLYPH<13> SC L=64GLYPH<13> av[ln(S(0.049))]GLYPH<13> T/JGLYPH<13>  hr= 1.5GLYPH<13>  hr= 2.0 dil= 0.5GLYPH<13> 0.04GLYPH<13> 0.06GLYPH<13> 0.08GLYPH<13> 0.1GLYPH<13> 0.2GLYPH<13> 0GLYPH<13> 2GLYPH<13> 4GLYPH<13> 6GLYPH<13> 8GLYPH<13> random-field ZGLYPH<13> 12GLYPH<13> SC L=64GLYPH<13> hr= 2.0 dil= 0.5GLYPH<13> 8 samplesGLYPH<13> av[ln(S(k))]GLYPH<13> kGLYPH<13>  1.4375 GLYPH<13>  1.75GLYPH<13>  1.5 GLYPH<13>  1.8125GLYPH<13>  1.5625 GLYPH<13>  1.875GLYPH<13>  1.625 GLYPH<13>  1.9375GLYPH<13>  1.6875 GLYPH<13>  2.0GLYPH<13>