### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 006, 17 pages      arXiv:1705.10544      https://doi.org/10.3842/SIGMA.2018.006

### On the TASEP with Second Class Particles

Eunghyun Lee
Department of Mathematics, Nazarbayev University, Kazakhstan

Received August 08, 2017, in final form January 08, 2018; Published online January 12, 2018

Abstract
In this paper we study some conditional probabilities for the totally asymmetric simple exclusion processes (TASEP) with second class particles. To be more specific, we consider a finite system with one first class particle and $N-1$ second class particles, and we assume that the first class particle is initially at the leftmost position. In this case, we find the probability that the first class particle is at $x$ and it is still the leftmost particle at time $t$. In particular, we show that this probability is expressed by the determinant of an $N\times N$ matrix of contour integrals if the initial positions of particles satisfy the step initial condition. The resulting formula is very similar to a known formula in the (usual) TASEP with the step initial condition which was used for asymptotics by Nagao and Sasamoto [Nuclear Phys. B 699 (2004), 487-502].

Key words: TASEP; Bethe ansatz; second class particles.

pdf (418 kb)   tex (21 kb)

References

1. Barraquand G., Corwin I., The $q$-Hahn asymmetric exclusion process, Ann. Appl. Probab. 26 (2016), 2304-2356, arXiv:1501.03445.
2. Borodin A., Corwin I., Sasamoto T., From duality to determinants for $q$-TASEP and ASEP, Ann. Probab. 42 (2014), 2314-2382, arXiv:1207.5035.
3. Chatterjee S., Schütz G.M., Determinant representation for some transition probabilities in the TASEP with second class particles, J. Stat. Phys. 140 (2010), 900-916, arXiv:1003.5815.
4. Johansson K., Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476, math.CO/9903134.
5. Kassel C., Turaev V., Braid groups, Graduate Texts in Mathematics, Vol. 247, Springer, New York, 2008.
6. Korhonen M., Lee E., The transition probability and the probability for the left-most particle's position of the $q$-totally asymmetric zero range process, J. Math. Phys. 55 (2014), 013301, 15 pages, arXiv:1308.4769.
7. Lee E., The current distribution of the multiparticle hopping asymmetric diffusion model, J. Stat. Phys. 149 (2012), 50-72, arXiv:1203.0501.
8. Lee E., Some conditional probabilities in the TASEP with second class particles, J. Math. Phys. 58 (2017), 123301, 11 pages, arXiv:1707.02539.
9. Lee E., Wang D., Distributions of a particle's position and their asymptotics in the $q$-deformed totally asymmetric zero range process with site dependent jumping rates, arXiv:1703.08839.
10. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015.
11. Mountford T., Guiol H., The motion of a second class particle for the TASEP starting from a decreasing shock profile, Ann. Appl. Probab. 15 (2005), 1227-1259, math.PR/0505216.
12. Nagao T., Sasamoto T., Asymmetric simple exclusion process and modified random matrix ensembles, Nuclear Phys. B 699 (2004), 487-502, cond-mat/0405321.
13. Povolotsky A.M., On the integrability of zero-range chipping models with factorized steady states, J. Phys. A: Math. Theor. 46 (2013), 465205, 25 pages, arXiv:1308.3250.
14. Schütz G.M., Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427-445, cond-mat/9701019.
15. Tracy C.A., Widom H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815-844, arXiv:0704.2633.
16. Tracy C.A., Widom H., On the distribution of a second-class particle in the asymmetric simple exclusion process, J. Phys. A: Math. Theor. 42 (2009), 425002, 6 pages, arXiv:0907.4395.
17. Tracy C.A., Widom H., On the asymmetric simple exclusion process with multiple species, J. Stat. Phys. 150 (2013), 457-470, arXiv:1105.4906.
18. Wang D., Waugh D., The transition probability of the $q$-TAZRP ($q$-bosons) with inhomogeneous jump rates, SIGMA 12 (2016), 037, 16 pages, arXiv:1512.01612.