Abstract
We introduce a space-inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari (Commun Math Phys 325:603–684, 2014). We show that for a certain class of initial conditions the point process associated with the dynamics has determinantal correlation functions, and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this, we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson’s Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array, respectively.
Highlights
These dynamics can equivalently be viewed as growth of random surfaces; see [2,7,12] or as random fields of Young diagrams; see [20,21]
By construction since the projections on any sub-pattern are autonomous, the processes (XN (t; MN ) ; t ≥ 0)N≥1 are consistent as well: πNN+1 ∗ Law [XN+1(t; MN+1)] = Law [XN (t; MN )], ∀t ≥ 0, ∀N ≥ 1, 5We believe that this is the most general model of continuous-time Borodin–Ferrari dynamics with determinantal correlations and which can be treated with the methods developed here
By applying an extension of the famous Eynard–Mehta theorem [28] to interlacing particle systems, see [9,19], it is fairly standard; see Sect. 3.1, that under some Gibbs initial conditions the point process associated with XN (t; MN ) has determinantal correlation functions
Summary
The study of stochastic dynamics, in both discrete and continuous time, on interlacing arrays has seen an enormous amount of activity in the past two decades, see for example [1,2,3,4,7,8,10,11,12,15,16,17,52,53]. The one that we will be concerned with in this contribution is due to Borodin and Ferrari2 [7] (see the independent related work of Warren and Windridge [53] and Warren’s Brownian analogue of the dynamics [52]) based on some ideas from [23] This is the simplest out of the three approaches to describe (see Definition 1.2 for a precise description) and in some sense, see [20], the one with “maximal noise”. We give the necessary background in order to introduce the model and state our main results precisely
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