The spectral boundary of the Asymmetric Simple Exclusion Process (ASEP) -- free fermions, Bethe ansatz and random matrix theory

Abstract

In non-equilibrium statistical mechanics, the Asymmetric Simple Exclusion Process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length $L$, under periodic (pbc) and open boundary conditions (obc). Notably, the spectral boundary exhibits $L$ spikes for pbc and $L+1$ spikes for obc. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the non-interacting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For pbc, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit, and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.

Figures (9)

• Figure 1: Spectrum of the generator matrix $H$ of TASEP (a,b) and the non-interacting TASEP (c,d) on $L=11$ sites. The spectrum shows $L$ spikes in (a,c) for pbc with $N=5$ particles and $L+1$ spikes in (b,d) for obc. Red solid lines in (c,d) denote the spectral boundary according to Eq. \ref{['eq:H0_boundary']}.
• Figure 2: Spectrum of the non-interacting TASEP $H_0$ on $L=11$ sites with pbc. Single-body eigenvalues with $p=1$ and $q=0$ in (a) and $p=0.7$ and $q=0.3$ in (b). In (c) we show part of the many-body spectrum with $N=5$ particles highlighting the tips of the spikes (red) and other boundary eigenvalues (blue). All boundary eigenvalues are located on circles of radius $1$, with crosses marking the midpoints.
• Figure 3: Spectrum of the generator matrix $H$ of TASEP (a) and the "non-interacting" TASEP (b) on $L=40$ sites with $N=2$ particles (dilute limit). The red solid line in (b) denotes the spectral boundary according to Eq. \ref{['eq:H0_boundary']}. The spectral boundary appears smooth and non-spiky in both panels.
• Figure 4: Many-body spectrum of the non-interacting TASEP with obc on (a)$L=6$ and (b)$L=7$ sites. Similar to pbc in Fig. \ref{['fig:H0_pbc']}, all boundary eigenvalues lie on circles, with midpoints denoted by crosses.
• Figure 5: All $N\cdot\binom{L}{N}$ Bethe roots $Z_j$ of the TASEP. (a)-(d)$L=8$, $N=4$, for different values of $U$. (e)$L=14$, $N=7$, $U=1$.