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

Abstract

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.

Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects

“…Substituting this expression into (18), rewrite the partition function in the following approximate form:…”

“…Indeed, a random walk on a tree-like graph can be mapped on a biased random walk on a half-line [ 15 ], and if the external field is strong enough to compensate for that bias [ 16 ], condensation occurs in a similar way to a random walk on a half-line that is biased in the direction of the origin. Contrary to that, localization in the path-counting problem occurs as a result of a point-like local disorder [ 8 ] or, equivalently, as noted in [ 17 , 18 ], as a result of an external field applied only to the endpoint of the trajectory. For Brownian motion, such localization is impossible; it is easy to see that in a potential composed of a localized well at the origin and a constant bias steering away from it, the Brownian walker always escapes.…”