Random surface growth with a wall and Plancherel measures for O(infinity)

Markov processes on the path space of the Gelfand–Tsetlin graph and on its boundary

“…Certain Markov chain on interlacing arrays with a similar block-push mechanism have been studied in [BF08+], see also [BK10]. In those examples the jump rates are constant though.…”

“…This work is a result of interaction of two circles of ideas. The first one deals with a certain class of random growth models in two space dimensions [War07], [Nor10], [BF08+], [BG09], [BGR09+], [BK10] [Bor10+], while the second one addresses constructing and analyzing stochastic dynamics on spaces of point configurations with distinguished invariant measures that are often given by, or closely related to, determinantal point processes [BO06a], [BO06b], [BO09], [Ols10], [Ols10+].…”

“…typically depends only on a few surrounding coordinates, one can argue that we have a model of local random growth. It should be compared to the models treated in [BF08+], [BK10], where a similar block-push mechanism was considered with constant jumps rates, and in [Bor10+], where the jump rates were also dependent on the location and numbering of the coordinates.…”

Two Speed TASEP

“…The large time fluctuations are then given by the antisymmetric GUE(M) process (for fixed time it was characterized in [22], see also [14], [15]). This is also closely related to the asymptotics of a certain Markovian dynamics for two-dimensional interlacing particle systems with a wall, see Section 2.3 of [38] and [9]. Using the relation between last passage directed percolation with exp(1) random variables and TASEP, one can predict that there should be a relation between the maximum process for the largest eigenvalue of the Dyson's Brownian Motion and systems of nonintersecting paths with a wall.…”

Universal shocks in the Wishart random-matrix ensemble

“…This was done in the context of non-intersecting squared Bessel paths in [39] where the integral representation of the limiting kernel was derived with Riemann-Hilbert techniques. Finally, this kernel reduces to the so-called symmetric Pearcey kernel identified through the studies of random growth with a wall [40,41]. Our work differs from those above by the use of completely different methods.…”