From generalized directed animals to the asymmetric simple exclusion process

Using the generalized normally ordered form of words in a locally-free group of $n$ generators, we show that in the limit $n\to \mathrm{\infty }$, the partition function of weighted directed lattice animals on a semi-infinite strip coincides with the partition function of stationary configurations of the asymmetric simple exclusion process (ASEP) with arbitrary entry/escape rates through open boundaries. We relate the features of the ASEP in the different regimes of the phase diagram to the geometric features of the associated generalized directed animals by showing the results of numerical simulations. In particular, we show how the presence of shocks at the first order transition line translates into the directed animal picture. Using the evolution equation for generalized, weighted Lukasiewicz paths, we also provide a straightforward calculation of the known ASEP generating function.