# Inhomogeneous exponential jump model

### 2017/03/10

(with

Alexei Borodin)

*Probability Theory and Related Fields, to appear* •

arXiv:1703.03857 [math.PR]

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:1703.03857 [math.PR].

Limit shape and fluctuations in the inhomogeneous exponential jump model