Partition Function Zeros of Paths and Normalization Zeros of ASEPS

TL;DR

The paper establishes a unified framework to obtain the thermodynamic-limit locus of partition-function (Lee–Yang) zeros for random-allocation and lattice-path models, and shows its exact equivalence to the normalization zeros of the ASEP. By leveraging generating functions and conformal mappings, the authors derive the zero loci as images of convergence-circle boundaries under z ↦ 1/f(z), and they validate this via Dyck-path mappings and the ASEP grand-canonical normalization F(z,α,β). They characterize the phase boundaries (including second-order and first-order lines) with explicit zero-density behavior near critical points and connect these results to electrostatic analogies and existing numerical findings. The approach unifies equilibrium and non-equilibrium zero analyses and suggests extensions to more complex constrained urn-like models. Overall, the work provides exact, analytically tractable loci for zeros and links them to conformal mappings and generating-function singularities, with implications for understanding phase structure in non-equilibrium steady states.

Abstract

We exploit the equivalence between the partition function of an adsorbing Dyck walk model and the Asymmetric Simple Exclusion Process (ASEP) normalization to obtain the thermodynamic limit of the locus of the ASEP normalization zeros from a conformal map. We discuss the equivalence between this approach and using an electrostatic analogy to determine the locus, both in the case of the ASEP and the random allocation model.

Figures (6)

• Figure 1: A Dyck walk with contact fugacity $c$, and additionally with excursion fugacity $u$.
• Figure 2: The analytically calculated locus of zeros for adsorbing Dyck walks in the complex $v$ plane. The transition at $v_{cr}=1/2$ is clearly visible, as is its second-order nature, since the impact angle of the locus of zeros is $\pm 3\pi/4$.
• Figure 3: (Totally) Asymmetric Exclusion Process on a line with insertion rate $\alpha$ and removal rate $\beta$. The internal jump rate to the right is $1$ and only single site occupancy is allowed.
• Figure 4: The ASEP phase diagram. Region (1) is a low-density phase, region (2) is a high-density phase and region (3) is the maximal current phase. The dotted transition line along $\alpha=\beta<1/2$ is first order and the $\alpha=1/2, \beta=1/2$ transition lines are second order.
• Figure 5: Analytically calculated locus of zeros in the thermodynamic limit from ${\cal Z} (z, \alpha,\beta)$ and numerically determined zeros from $Z_N$ with $N=1000$ for the ASEP normalization with $\beta=3/4$ (the second-order transition regime). The analytic locus is identical to that for the adsorbing Dyck walks in Figure \ref{['fig:dyck']}.