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