From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.