Chelkak, Laslier, and Russkikh introduce a new type of graph embedding called a t-embedding, and use it to prove the convergence of dimer model height fluctuations to a Gaussian Free Field (GFF) in a naturally associated metric, under certain technical assumptions. Building on a work of Chelkak and Ramassamy, we study the properties of "perfect" t-embeddings of uniform Aztec diamond graphs, and in particular utilize the integrability of the “shuffling algorithm” on these graphs to give new exact formulas for the t-embeddings. We use these to provide a precise asymptotic analysis of the t-embeddings, in order to verify the validity of the technical assumptions required for convergence to the GFF. As a consequence, we complete a new proof of GFF fluctuations for the dimer model height function on the uniformly weighted Aztec diamond.
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