The current distribution of the multiparticle hopping asymmetric diffusion model

In this paper we treat the \textit{multiparticle hopping asymmetric diffusion model} (MADM) on $\mathbb{Z}$ introduced by Sasamoto and Wadati in 1998. The transition probability of the MADM with $N$ particles is provided by using the Bethe ansatz. The transition probability is expressed as the sum of $N$-dimensional contour integrals of which contours are circles centered at the origin with restrictions on their radii. By using the transition probability we find $\mathbb{P}(x_m(t) =x)$, the probability that the $m$th particle from the left is at $x$ at time $t$. The probability $\mathbb{P}(x_m(t) =x)$ is expressed as the sum of $|S|$-dimensional contour integrals over all $S \subset \{1,...,N\}$ with $|S| \geq m$, and is used to give the current distribution of the system. The mapping between the MADM and the pushing asymmetric simple exclusion process (PushASEP) is discussed.