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

SIGMA 16 (2020), 133, 6 pages      arXiv:2007.02913      https://doi.org/10.3842/SIGMA.2020.133

Determinantal Expressions in Multi-Species TASEP

Jeffrey Kuan
Texas A&M University, Department of Mathematics, Mailstop 3368, College Station, TX 77843-3368, USA

Received July 14, 2020, in final form December 03, 2020; Published online December 11, 2020

Abstract
Consider an inhomogeneous multi-species TASEP with drift to the left, and define a height function which equals the maximum species number to the left of a lattice site. For each fixed time, the multi-point distributions of these height functions have a determinantal structure. In the homogeneous case and for certain initial conditions, the fluctuations of the height function converge to Gaussian random variables in the large-time limit. The proof utilizes a coupling between the multi-species TASEP and a coalescing random walk, and previously known results for coalescing random walks.

Key words: determinantal; multi-species; TASEP; coalescing.

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References

1. Assiotis T., Random surface growth and Karlin-McGregor polynomials, Electron. J. Probab. 23 (2018), 106, 81 pages, arXiv:1709.10444.
2. Baik J., Ferrari P.L., Péché S., Limit process of stationary TASEP near the characteristic line, Comm. Pure Appl. Math. 63 (2010), 1017-1070, arXiv:0907.0226.
3. Borodin A., Bufetov A., Leading particles in multi-type TASEP, in preparation.
4. Borodin A., Ferrari P.L., Prähofer M., Fluctuations in the discrete TASEP with periodic initial configurations and the ${\rm Airy}_1$ process, Int. Math. Res. Pap. 2007 (2007), rpm002, 47 pages, arXiv:math-ph/0611071.
5. Borodin A., Ferrari P.L., Prähofer M., Sasamoto T., Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129 (2007), 1055-1080, arXiv:math-ph/0608056.
6. Borodin A., Ferrari P.L., Sasamoto T., Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP, Comm. Math. Phys. 283 (2008), 417-449, arXiv:0707.4207.
7. Borodin A., Ferrari P.L., Sasamoto T., Transition between ${\rm Airy}_1$ and ${\rm Airy}_2$ processes and TASEP fluctuations, Comm. Pure Appl. Math. 61 (2008), 1603-1629, arXiv:math-ph/0703023.
8. 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.
9. Chen Z., de Gier J., Hiki I., Sasamoto T., Exact confirmation of 1d nonlinear fluctuating hydrodynamics for a two-species exclusion process, Phys. Rev. Lett. 120 (2018), 240601, 6 pages, arXiv:1803.06829.
10. Johansson K., Discrete polynuclear growth and determinantal processes, Comm. Math. Phys. 242 (2003), 277-329, arXiv:math.PR/0206208.
11. Lee E., On the TASEP with second class particles, SIGMA 14 (2018), 006, 17 pages, arXiv:1705.10544.
12. Quastel J., Remenik D., KP governs random growth of a one dimensional substrate, arXiv:1908.10353.
13. Schütz G.M., Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427-445, arXiv:cond-mat/9701019.
14. Warren J., Dyson's Brownian motions, intertwining and interlacing, Electron. J. Probab. 12 (2007), no. 19, 573-590, arXiv:math.PR/0509720.