4 peratures. In agreement with prior results[12] on weaker and stronger random fields, we argue that our results in- dicate a phase transition into a long-range ordered low temperature phase, with a jump in the order parameter at T$_{c}$ . The behavior above T$_{c}$ strongly resembles well known phenomenology for glassy phase transitions.[15] approximately constant, at least over the range of T shown in the figure, where the samples are believed to be equilibrated. This behavior below T$_{c}$ is consistent with the existence of a δ -function peak in S at k = 0 in the limit L → ∞ , indicative of true long-range order. One would not expect to see this behavior if there were only power-law correlations in S ( k ), with no δ -function peak. The reader should understand that we are not claim- ing the existence of some mathematical error in the analysis[16] of the elastic glass, which leads to the power- law correlated Bragg glass phase. That analysis begins with the Larkin Hamiltonian,[1] which approximates the spin-exchange term of Eqn. (1) with a harmonic interac- tion. What we are saying is that the long-range behav- ior of the Larkin Hamiltonian is different from that of Eqn. (1), at least for the types of P ( h$_{i}$ ) distributions we have studied. If we fit the data for av[ln( S ( k ))] at T$_{c}$ over the range π/ 64 ≤ k ≤ π/ 8, as shown in Fig. 1 and Fig. 2, we find good straight line fits, with slopes of -3.20(3) for the Type A samples and -3.10(2) for the Type B samples. Because both of these slopes are more negative than -3, it is clearly not correct to extrapolate these fits to k = 0. S ( k ) satisfies a sum rule, and thus nonintegrable singu- larities cannot occur. Thus if we were able to obtain data using these P ( h$_{i}$ ) distributions for even larger val- ues of L , we would expect to find that the behavior of av[ln( S ( k ))] at T$_{c}$ for small enough nonzero k would look like the data for the case h$_{r}$ = 2 with p = 0 found in the earlier work.[12] In this work we have performed Monte Carlo studies of the 3D RFXYM on L = 64 simple cubic lattices, with isotropic random field distributions of two types: h$_{r}$ = 1.5 with p = 0, and h$_{r}$ = 2 with p = 0.5. We present re- sults for the structure factor, S ( k ), at a sequence of tem- The author thanks the Physics Department of Prince- ton University for providing use of computers on which the data were obtained. ∗ ron@princeton.edu [1] A. I. Larkin, Zh. Eksp. Teor. Fiz. 58 , 1466 (1970) [Sov. Phys. JETP 31 , 784 (1970)]. [2] M. J. P. Gingras and D. A. Huse, Phys. Rev. B 53 , 15193 (1996). [3] R. Fisch, Phys. Rev. B 55 , 8211 (1997). [4] P. J. Steinhart et al. , Nature 396 , 55 (1998). [5] Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35 , 1399 (1975). [6] G. Grinstein, Phys. Rev. Lett. 37 , 944 (1976). [7] A. Aharony, Y. Imry and S.-K. Ma, Phys. Rev. Lett. 36 , 1364 (1976). [8] G. Parisi and N. Sourlas, Nucl. Phys. B206 , 321 (1982). [9] J. Z. Imbrie, Phys. Rev. Lett. 53 , 1747 (1984). [10] J. Bricmont and A. Kupiainen, Phys. Rev. Lett. 59 , 1829 (1987). [11] R. B. Griffiths, Phys. Rev. 152 , 240 (1966). [12] R. Fisch, Phys. Rev. B 76 , 214435 (2007). [13] P. W. Anderson and G. Yuval, J. Phys. C 4 , 607 (1971). [14] J. M. Schwarz, A. J. Liu and L. Q. Chayes, Europhys. Lett. 73 , 560 (2006). [15] M. L. Williams, R. F. Landel and J. D. Ferry, J. Am. Chem. Soc. 77 , 3701 (1955). [16] D. S. Fisher, Phys. Rev. Lett. 78 , 1964 (1997).