Consider an infinite random matrix H = (h ij ) 0

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.

In 1962, Dyson [Dys62] introduced dynamics in random matrix models, in particular into GUE (also for β = 1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: β = 1, 2, 4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for β = 2, 4.2000 Mathematics Subject Classification. Primary: 60B20, 60G55; Secondary: 60J65, 60J10.

Consider an n × n Hermitean matrix valued stochastic process {H t } t≥0 where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect.In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the k × k in the upper left corner of H t . If you project this process to a space-like path it is a determinantal process and we compute the kernel. This kernel contains the well known GUE minor kernel, [JN06, OR06] and the Dyson Brownian motion kernel [NF98] as special cases.In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier's bead kernel [Bou09].We also compute the kernel for a process of intertwined Brownian motions introduced by Warren in [War07]. That too is a determinantal process along spacelike paths.

Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species j consists of ⌊j/2⌋ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge.