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We study the dynamics of a certain discrete model of interacting particlesthat comes from the so called shuffling algorithm for sampling a random tilingof an Aztec diamond. It turns out that the transition probabilities have aparticularly convenient determinantal form. An analogous formula in acontinuous setting has recently been obtained by Jon Warren studying certainmodel of interlacing Brownian motions which can be used to construct Dyson'snon-intersecting Brownian motion.  We conjecture that Warren's model can be recovered as a scaling limit of ourdiscrete model and prove some partial results in this direction. As anapplication to one of these results we use it to rederive the known result thatrandom tilings of an Aztec diamond, suitably rescaled near a turning point,converge to the GUE minor process.

On the Shuffling Algorithm for Domino Tilings