From Vicious Walkers to TASEP

Abstract

We propose a model of semi-vicious walkers, which interpolates between the totally asymmetric simple exclusion process and the vicious walkers model, having the two as limiting cases. For this model we calculate the asymptotics of the survival probability for $m$ particles and obtain a scaling function, which describes the transition from one limiting case to another. Then, we use a fluctuation-dissipation relation allowing us to reinterpret the result as the particle current generating function in the totally asymmetric simple exclusion process. Thus we obtain the particle current distribution asymptotically in the large time limit as the number of particles is fixed. The results apply to the large deviation scale as well as to the diffusive scale. In the latter we obtain a new universal distribution, which has a skew non-Gaussian form. For $m$ particles its asymptotic behavior is shown to be $e^{-\frac{y^2}{2m^2}}$ as $y\to -\infty$ and $e^{-\frac{y^2}{2m}}{y}^{-\frac{m(m-1)}{2}}$ as $y\to \infty$.

37 pages, 4 figures, Corrected references

References in corpus (21)

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Cited by in corpus (6)

• Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport
• Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature
• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Some Exact Results for the Exclusion Process
• Correlation Functions of XX0 Heisenberg Chain, q-Binomial Determinants, and Random Walks
• Determinant representation for some transition probabilities in the TASEP with second class particles