Distribution of a Particle's Position in the ASEP with the Alternating Initial Condition

In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find P(X_m(t)≤x) where X_m(t) is the position of the particle at time t which was at m=2k-1, k∈ Z at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand of the integral formula for P(X_m(t)≤x) to a determinantal form together with an extra product.