Height representation and long-range order in random trimer tilings of the square lattice

Cristopher Moore (DRAFT!)

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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Cris Moore <moore@santafe.edu>