2 small. This will occur when the energy required for re- orienting 〈 ⃗ M 〉 is nonzero, but not large compared to T . If this were to occur, then there would be no local order parameter, and thus no terms in the Landau-Ginsburg free-energy functional which depended on gradients of 〈 ⃗ M 〉 . Monte Carlo calculations for Eqn. (1) on simple cubic lattices were performed some time ago by Gingras and Huse[2] and Fisch[3]. More recently, additional calcula- tions were carried out.[12] The results of these studies, which used several different choices for the probability distribution P ( h$_{i}$ ), indicate that for typical samples the order parameter 〈 M $^{2}$〉 becomes positive at some posi- tive temperature T$_{c}$ , as long as the random fields are not too strong. However, the precise nature of what hap- pens near T$_{c}$ remained unclear. In this work we report additional studies using new choices of P ( h$_{i}$ ), which dis- play phenomena which were not seen clearly in the earlier studies. The computer program which was used was an en- hanced version of the one used before.[12] The data were obtained from L × L × L simple cubic lattices with L = 64 using periodic boundary conditions. Some preliminary studies for smaller values of L were also done. The program approximates O (2) with Z 12 , a 12-state clock model. It was modified to enable the study of random- field probability distributions of the form initial conditions followed by slow warming. For the Type A samples three different cold start initial conditions were used, equally spaced around the circle. For the Type B samples, where the average random field is somewhat weaker, only two cold start initial conditions were used. In Fig. 1 we show data for the average over our eight Type A samples of ln( S ( k )) for T/J between 1.3125 and 1.75. The data for T/J ≥ 1 . 4375 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 lowest average value for the energy, 〈 E 〉 . When T = 1 . 4375 or more, all runs for a given sample converge in a rather short time to give similar values of 〈 E 〉 and 〈 ⃗ M 〉 . However, for lower values of T/J the hot start runs remain stuck in metastable states. This modification makes the program run a few percent slower than before. In this work we study two cases. Type A samples have h$_{r}$ = 1 . 5 with p = 0, and Type B samples have h$_{r}$ = 2 with p = 0 . 5. For both of these cases we find T$_{c}$/J near 1.5, which was suggested by Gingras and Huse[2] to be optimal for this sort of calculation. On a simple cubic lattice of size L × L × L , the magnetic structure factor, S ( ⃗ k ), for n = 2 spins is An attempt was made to drive the hot start runs into equilibrium by deep undercooling, followed by slow warming. This thermal cycling technique had proven successful earlier,[12] for the weak random field case of h$_{r}$ = 1 . 0 with p = 0. For the cases studied here, however, a single undercooling was only partially successful in driv- ing the samples to equilibrium. The author believes that repeating cycling would have worked successfully. How- ever, since a detailed study of the nonequilibrium behav- ior was not the purpose of the work described here, this was not attempted. Somewhat surprisingly, the cold start runs were much more successful in finding a single free energy minimum at T/J between 1.3125 and 1.40625. In the majority of samples, all of the cold start initial conditions were converging to the same free energy minimum. This was where ⃗ r$_{ij}$ is the vector on the lattice which starts at site i and ends at site j . Thus, setting ⃗ k to zero yields As we will see, the extrapolation of 〈 M $^{2}$( L ) 〉 to L = ∞ is not trivial. In this work, we present results for the average over angles of S ( ⃗ k ), which we write as S ( k ). Eight different L = 64 samples of the random fields θ$_{i}$ were used for each P ( h$_{i}$ ). The same samples of random fields were used for all values of T . The Monte Carlo calculations were performed using both hot start ( i.e. random) initial conditions followed by slow cooling, and also cold start ( i.e. ferromagnetic) P ( h$_{i}$ ) = (1 − p ) δ ( h$_{i}$ − h$_{r}$ ) + pδ ( h$_{i}$ ) . (5) S ( ⃗ k ) = L − 3 ∑ i,j cos( ⃗ k · ⃗ r$_{ij}$ ) 〈 cos( φ$_{i}$ − φ$_{j}$ ) 〉 , (6) S (0) /L 3 = 〈 M $^{2}$( L ) 〉 . (7) FIG. 1: (color online) Average over 8 samples of ln( S ( k )) for 64 × 64 × 64 lattices with h r = 1 . 5 and p = 0 at various temperatures. The error bars indicate approximately three standard deviation statistical errors, and the x -axis is scaled logarithmically. 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= 1.5GLYPH<13> 8 samplesGLYPH<13> av[ln(S(k))]GLYPH<13> kGLYPH<13>  1.3125 GLYPH<13>  1.5GLYPH<13>  1.34375 GLYPH<13>  1.5625GLYPH<13>  1.375 GLYPH<13>  1.625GLYPH<13>  1.40625 GLYPH<13>  1.6875GLYPH<13>  1.4375 GLYPH<13>  1.75GLYPH<13>  1.46875GLYPH<13>