Bethe ansatz solvable interacting particle systems on Z

We study the transition probabilities and the distributions of time-integrated currents of the Bethe ansatz solvable interacting particle systems: the asymmetric simple exclusion process (ASEP), the PushASEP, the asymmetric avalanche process and the asymmetric zero range process (AZRP) with constant rates. For the ASEP with the alternating initial condition, we give the distribution of the position of the particle at time t which was at m = 2k -- 1, k ∈ Z at time 0. In the ASEP with the alternating initial condition, there arises a new combinatorial identity which simplify the sum of N! terms to a single term. Also, we present the exact expressions of the transition probabilities of the other models with N particles. These transition probabilities are expressed as contour integrals similar to Tracy-Widom's integral formula for the ASEP [32]. Using these transition probabilities, we revisit the mapping between the states of the ASEP and the AZRP with constant rates. This dissertation is a combined version of the author's two published papers [16, 17].