A Fredholm Determinant Representation in ASEP

Abstract

In previous work the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers, and consider the distribution function for the m'th particle from the left. In the previous work an infinite series of multiple integrals was derived for this distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.

12 Pages. Version 3 includes a scaling conjecture

References in corpus (5)

• Shape Fluctuations and Random Matrices
• Exact Solution of the Master Equation for the Asymmetric Exclusion Process
• Integral Formulas for the Asymmetric Simple Exclusion Process
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• $t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$

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