Title: Inhomogeneous exponential jump model,
Abstract: We introduce and study the inhomogeneous exponential jump model - an
integrable stochastic interacting particle system on the continuous half line
evolving in continuous time. An important feature of the system is the presence
of arbitrary spatial inhomogeneity on the half line which does not break the
integrability. We completely characterize the macroscopic limit shape and
asymptotic fluctuations of the height function (= integrated current) in the
model. In particular, we explain how the presence of inhomogeneity may lead to
macroscopic phase transitions in the limit shape such as shocks or traffic
jams. Away from these singularities the asymptotic fluctuations of the height
function around its macroscopic limit shape are governed by the GUE Tracy-Widom
distribution. A surprising result is that while the limit shape is
discontinuous at a traffic jam caused by a macroscopic slowdown in the
inhomogeneity, fluctuations on both sides of such a traffic jam still have the
GUE Tracy-Widom distribution (but with different non-universal normalizations).
The integrability of the model comes from the fact that it is a degeneration
of the inhomogeneous stochastic higher spin six vertex models studied earlier
in arXiv:1601.05770 [math.PR]. Our results on fluctuations are obtained via an
asymptotic analysis of Fredholm determinantal formulas arising from contour
integral expressions for the q-moments in the stochastic higher spin six vertex
model. We also discuss "product-form" translation invariant stationary
distributions of the exponential jump model which lead to an alternative
hydrodynamic-type heuristic derivation of the macroscopic limit shape.