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

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## Abstract

We introduce a new interacting (stochastic) particle system \(q\)-PushASEP which interpolates between the \(q\)-TASEP of Borodin and Corwin (Probab Theory Relat Fields 158(1–2):225–400, 2014; see also Borodin et al., Ann Probab 42(6):2314–2382, 2014; Borodin and Corwin, Int Math Res Not 2:499–537, 2015; O’Connell and Pei, Electron J Probab 18(95):1–25, 2013; Borodin et al., Comput Math, 2013) and the \(q\)-PushTASEP introduced recently (Borodin and Petrov, Adv Math, 2013). 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. 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.

### Keywords

Integrable probability Kardar–Parisi–Zhang universality class## Notes

### Acknowledgments

The authors would like to thank Alexei Borodin for very helpful discussions and remarks. IC was partially supported by the NSF through DMS-1208998 as well as by Microsoft Research through the Schramm Memorial Fellowship, and by the Clay Mathematics Institute through a Clay Research Fellowship. LP was partially supported by the RFBR-CNRS Grant 11-01-93105.

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