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

- 150 Downloads
- 7 Citations

## 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.

## References

- 1.Alimohammadi, M., Karimipour, V., Khorrami, M.: A two-parametric family of asymmetric exclusion processes and its exact solution. J. Stat. Phys.
**97**(1–2), 373–394 (1999). arXiv:cond-mat/9805155 - 2.Balász, M., Komjáthy, J., Seppäläinen, T.: Microscopic concavity and fluctuation bounds in a class of deposition processes. Ann. Inst. H. Poincaré B
**48**, 151–187 (2012)ADSCrossRefGoogle Scholar - 3.Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain). Zeitschrift fur Physik
**71**, 205–226 (1931)ADSCrossRefGoogle Scholar - 4.Borodin, A.: Schur dynamics of the Schur processes. Adv. Math.
**228**(4), 2268–2291 (2011). arXiv:1001.3442 [math.CO] - 5.Borodin, A., Corwin, I.: Macdonald processes. Probab. Theor. Relat. Fields
**158**(1–2), 225–400 (2014). arXiv:1111.4408 [math.PR] - 6.Borodin, A., Corwin, I.: Discrete time q-TASEPs. Int. Math. Res. Not.
**2**, 499–537 (2015). doi: 10.1093/imrn/rnt206, arXiv:1305.2972 [math.PR] - 7.Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary KPZ equation (2013, preprint). arXiv:1308.3475 [math-ph]
- 8.Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes. Trans. Am. Math. Soc. (2013, to appear). arXiv:1306.0659 [math.PR]
- 9.Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for the q-Boson particle system. Compos. Math.
**151**(1), 1–67 (2015). arXiv:1308.3475 [math-ph] - 10.Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab.
**42**(6), 2314–2382 (2014). arXiv:1207.5035 [math.PR] - 11.Borodin, A., Ferrari, P.: Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab.
**13**, 1380–1418 (2008). arXiv:0707.2813 [math-ph] - 12.Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. (2013, to appear). arXiv:1305.5501 [math.PR]
- 13.Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Eur. Phys. Lett.
**90**(2), 20002 (2010)ADSCrossRefGoogle Scholar - 14.Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equation. McGraw Hill, New York (1955)Google Scholar
- 15.Dotsenko, V.: Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers. J. Stat. Mech. (07), P07010 (2010). arXiv:1004.4455 [cond-mat.dis-nn]
- 16.Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley-Interscience, New York (1986)CrossRefzbMATHGoogle Scholar
- 17.Liggett, T.: Interacting Particle Systems. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
- 18.Liggett, T.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Grundlehren de mathematischen Wissenschaften, vol. 324. Springer, New York (1999)CrossRefGoogle Scholar
- 19.Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)zbMATHGoogle Scholar
- 20.Matveev, K., Petrov, L.: In preparationGoogle Scholar
- 21.O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab.
**40**(2), 437–458 (2012). arXiv:0910.0069 [math.PR] - 22.O’Connell, N., Pei, Y.: A q-weighted version of the Robinson–Schensted algorithm. Electron. J. Probab.
**18**(95), 1–25 (2013). arXiv:1212.6716 [math.CO] - 23.O’Connell, N., Yor, M.: Brownian analogues of Burke’s theorem. Stoch. Process. Appl.
**96**(2), 285–304 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Povolotsky, A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A
**46**(465205) (2013). arXiv:1308.3250 [math-ph] - 25.Povolotsky, A., Mendes, J.F.F.: Bethe ansatz solution of discrete time stochastic processes with fully parallel update. J. Stat. Phys.
**123**(1), 125–166 (2006). arXiv:cond-mat/0411558 [cond-mat.stat-mech] - 26.Sasamoto, T., Wadati, M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A
**31**, 6057–6071 (1998)MathSciNetADSCrossRefzbMATHGoogle Scholar - 27.Spitzer, F.: Interaction of Markov processes. Adv. Math.
**5**(2), 246–290 (1970)MathSciNetCrossRefzbMATHGoogle Scholar