From Vicious Walkers to TASEP

TL;DR

This paper introduces a semi-vicious walkers model that bridges the totally asymmetric simple exclusion process and vicious walkers, providing new asymptotic results for particle survival probabilities and current distributions.

Contribution

It develops a unified model interpolating between two known processes and derives new universal distribution results for particle currents in large time limits.

Findings

Derived asymptotics of survival probability for multiple particles.

Obtained a new universal non-Gaussian distribution for particle current fluctuations.

Established a scaling function describing the transition between limiting cases.

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 $.