Abstract
We consider a model of particles jumping on a row of cells, called in physics the one dimensional totally asymmetric exclusion process with open boundaries (TASEP). From the point of view of combinatorics a remarkable feature of this Markov chain is that Catalan numbers are involved in several entries of its stationary distribution.
In a companion paper, we gave a combinatorial interpretation and a simple proof of these observations in the simplest case where the particles enter,jump and exit at the same rate. To do this we revealed a second row in which particles travel backward and defined on these two row configurations a Markov chain with uniform stationary distribution which is a covering of the TASEP.
In this paper we show how to deal with general rates. The covering chain is still defined on two row configurations, but its stationary distribution is not uniform anymore. Instead it is described in terms of two natural combinatorial statistics.
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References
B. Derrida, E. Domany, and D. Mukamel. An exact solution of a one dimensional asymmetric exclusion model with open boundaries. J. Stat. Phys., 69:667–687, 1992.
B. Derrida, C. Enaud, and J. L. Lebowitz. The asymmetric exclusion process and Brownian excursions. Available electronically as arXiv:cond-mat/0306078.
B. Derrida, M.R. Evans, V. Hakim, and V. Pasquier. Exact solution of a one-dimensional asymmetric exclusion model using a matrix formulation. J. Phys. A: Math., 26:1493, 1993.
B. Derrida, J. L. Lebowitz, and E. R. Speer. Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process. Available electronically as arXiv: cond-mat/0205353.
E. Duchi and G. Schaeffer. A combinatorial approach to jumping particles I: maximal flow regime. In proceedings of FPSAC’04, 2004.
O. Häggström.Finite Markov Chains and Algorithmic Applications. Cambridge University Press, 2002.
T. M. Liggett. Interacting Particle Systems. Springer, New York, 1985.
R. Stanley. Enumerative Combinatorics,volume II. Cambridge University Press, 1999.
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Duchi, E., Schaeffer, G. (2004). A Combinatorial Approach to Jumping Particles II: General Boundary Conditions. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds) Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7915-6_40
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DOI: https://doi.org/10.1007/978-3-0348-7915-6_40
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9620-7
Online ISBN: 978-3-0348-7915-6
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