We study random tilings of the square lattice by horizontal and vertical trimers. Such tilings can be thought as four-dimensional interfaces where each site is given a height in a two-dimensional space. We measure the fluctuations both in this height function and a non-Abelian one, and find that they decay more slowly as a function of wave number than they would if the interface were governed by a Gaussian effective field theory. We also measure the spatial correlations in the system, and find that they decay as r^-eta where eta ~ 1.5 +- 0.1, whereas we argue that the Gaussian theory gives eta = 2. These data suggest that this system is close to, or at, its roughening transition.
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