For Aztec diamond tilings with i.i.d. one-periodic edge weights, height function fluctuations are, in the critical regime, GFF plus independent Brownian motion, and in the fixed-variance regime, Brownian motion alone at scale sqrt M.
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Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions
For Aztec diamond tilings with i.i.d. one-periodic edge weights, height function fluctuations are, in the critical regime, GFF plus independent Brownian motion, and in the fixed-variance regime, Brownian motion alone at scale sqrt M.