A fredholm determinant representation in ASEP

In previous work (Tracy and Widom in Commun. Math. Phys. 279: 815-844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP)on the integer lattice Z. 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 Z(+) and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279: 815-844, 2008) an infinite series of multiple integrals was derived for the 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.