From generalized directed animals to the asymmetric simple exclusion process

TL;DR

This paper establishes a connection between generalized directed animals and the ASEP, showing how geometric features of directed animals correspond to phase transitions and shocks in ASEP, supported by analytical and numerical methods.

Contribution

It introduces a novel link between directed lattice animals and ASEP, providing new insights into phase transitions and a method to compute ASEP generating functions.

Findings

Partition function of directed animals matches ASEP stationary configurations

Shocks at first order transition line correspond to geometric features of animals

Derived ASEP generating function using generalized weighted paths

Abstract

Using the generalized normally ordered form of words in a locally-free group of $n$ generators, we show that in the limit $n\to\infty$, 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.