Abstract
We study the dynamics of a certain discrete model of interacting interlaced 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.
Highlights
There has been a lot of work in recent years connecting tilings of various planar regions with random matrices
One way of analysing that model, [Joh05; Joh01; JN06], is to define a particle process corresponding to the tilings so that the uniform measure on all tilings induces some measure on this particle process
In this article we study the so called shuffling algorithm, described in [EKLP92; Pro03], which in various variants can be used either to count or to enumerate all tilings of the Aztec diamond or to sample a random such tiling
Summary
There has been a lot of work in recent years connecting tilings of various planar regions with random matrices. The full process (X (t))t∈ 0 has remarkable similarities to, and is we believe a discretization of, a process studied recently by Warren, [War07] Warren in [War07] shows that Λ has the same distribution as X(1) To put this in perspective, let us note that a similar result for lozenge tilings is known from Okounkov and Reshetikhin [OR06]. They discuss the fact that, for quite general regions, that close to a so called turning point the GUE minor process can be obtained in a limit. Acknowledgements: The author would like to thank his supervisor Kurt Johansson for many useful discussions
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