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Integral Formulas for the Asymmetric Simple Exclusion Process

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An Erratum to this article was published on 14 April 2011

Abstract

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice \({\mathbb{Z}}\) with nearest neighbor hopping rates p to the right and q = 1−p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the \(N\to\infty\) limit. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schütz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m th left-most particle is at x at time t. In one of these formulas we can take the \(N\to\infty\) limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, CA, 95064, USA

    Harold Widom

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  1. Craig A. Tracy
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  2. Harold Widom
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Correspondence to Harold Widom.

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Communicated by H. Spohn

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-011-1249-2.

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Tracy, C.A., Widom, H. Integral Formulas for the Asymmetric Simple Exclusion Process. Commun. Math. Phys. 279, 815–844 (2008). https://doi.org/10.1007/s00220-008-0443-3

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