Determinantal structures in space inhomogeneous dynamics on interlacing arrays

TL;DR

This paper extends space inhomogeneous dynamics on interlacing arrays, showing determinantal point processes and explicit correlation kernels, and connects these to inhomogeneous particle systems like TAZRP and PushTASEP.

Contribution

It introduces a space inhomogeneous generalization of interlacing array dynamics and derives explicit correlation kernels for specific initial conditions.

Findings

Correlation functions are determinantal for certain initial conditions.

Explicit double contour integral formulas for the correlation kernel.

Provides a coupling between inhomogeneous TAZRP and PushTASEP systems.

Abstract

We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari. We show that for a certain class of initial conditions the point process associated to 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.