# The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension

### 2013/08/13

(with

Ivan Corwin)

*Journal of Statistical Physics, 160 (2015), no. 4, 1005–1026* •

arXiv:1308.3124 [math.PR]

We introduce a new interacting (stochastic) particle system $q$-PushASEP which
interpolates between the q-TASEP introduced by Borodin and Corwin (see
arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716)
and the $q$-PushTASEP introduced recently by Borodin and Petrov
(arXiv:1305.5501). In the $q$-PushASEP, particles can jump to the left or to the
right, and there is a certain partially asymmetric pushing mechanism present.
This particle system has a nice interpretation as a model of traffic on a
one-lane highway in which cars are able to accelerate or slow down.

Using the quantum many body system approach, we explicitly compute the
expectations of a large family of observables for this system in terms of
nested contour integrals. We also discuss relevant Fredholm determinantal
formulas for the distribution of the location of each particle, and connections
of the model with a certain two-sided version of Macdonald processes and with
the semi-discrete stochastic heat equation.