Lifted TASEP: long-time dynamics,generalizations, and continuum limit

TL;DR

This work analyzes the lifted TASEP and its generalization GL-TASEP through an exact Bethe-ansatz treatment of the transition-matrix spectrum, revealing a crossover in the relaxation-time scaling from Δ ∼ N^{-5/2} (α ≠ α_{ m crit}) to Δ ∼ N^{-2} (α = α_{ m crit}) as N grows. It reconciles observed MC autocorrelation scalings by showing that the states responsible for the N^{-2} scaling have vanishing overlaps with the large-scale density modes tracked in simulations, making the asymptotic relaxation difficult to detect numerically. The authors construct a continuum, integrable limit linked to the hard-sphere event-chain Monte Carlo algorithm, where α maps to the system pressure and α_{ m crit} corresponds to vanishing pressure. They extend the model to a broad class of nearest-neighbor interactions (GL-TASEP) with Boltzmann stationary states, and demonstrate that appropriate tuning of the pullback parameter again yields polynomial speedups to steady-state convergence, while noting the potential integrability of GL-TASEP. Overall, the results provide a deep connection between spectral properties, continuum limits, and non-reversible lifting strategies for efficient sampling in interacting particle systems.

Abstract

We investigate the lifted TASEP and its generalization, the GL-TASEP. We analyze the spectral properties of the transition matrix of the lifted TASEP using its Bethe ansatz solution, and use them to determine the scaling of the relaxation time (the inverse spectral gap) with particle number. The observed scaling with particle number was previously found to disagree with Monte Carlo simulations of the equilibrium autocorrelation times of the structure factor and of other large-scale density correlators for a particular value of the pullback $α_{\rm crit}$. We explain this discrepancy. We then construct the continuum limit of the lifted TASEP, which remains integrable, and connect it to the event-chain Monte Carlo algorithm. The critical pullback $α_{\rm crit}$ then equals the system pressure. We generalize the lifted TASEP to a large class of nearest-neighbor interactions, which lead to stationary states characterized by non-trivial Boltzmann distributions. By tuning the pullback parameter in the GL-TASEP to a particular value we can again achieve a polynomial speedup in the time required to converge to the steady state. We comment on the possible integrability of the GL-TASEP.

Figures (13)

• Figure 1: Autocorrelation functions of the structure factor at $L = 2N$ (with $\alpha_\text{crit} = 0.5$), for different values of the pullback $\alpha$. For $\alpha < \alpha_\text{crit}$, the asymptotic scaling is as $\sim N^{5/2}$, but for $\alpha \lesssim \alpha_\text{crit}$, this is reached only for very large $N$. The scalings $\sim N^{3/2}$ and $\sim N^{5/2}$ are indicated through straight lines.
• Figure 2: Sets of roots $\{u_j|1\leq j\leq L/2\}$ (left panel) and scaling of the eigenvalue of the transition matrix with system size (right panel) for the excited state of Eq. (\ref{['ints1']}) for $L=2N$ and $\alpha=0.1$, with $L=172$ (blue) and $L=92$ (yellow). For large $L$, the roots approach a non-trivial contour in the complex plane. The orange line in the right panel is the fit of Eq. (\ref{['fit1_01']}) to the functional form of Eq. (\ref{['eofl']}).
• Figure 3: Sets of roots $\{u_j|1\leq j\leq L/2\}$ (left panel) and scaling of the eigenvalue of the transition matrix with system size (right panel) for the excited state of Eq. (\ref{['ints1']}) for $L=2N$ and $\alpha=0.2$, with $L=248$ (blue), $L=172$ (yellow) and $L=92$ (green). The orange line in the right panel is the fit of Eq. (\ref{['fit1_02']}) to the functional form of Eq. (\ref{['eofl']}).
• Figure 4: Sets of roots $\{u_j|1\leq j\leq L/2\}$ (left panel) and scaling of the eigenvalue of the transition matrix with system size (right panel) for the excited state of Eq. (\ref{['ints']}) for $L=2N$ and $\alpha=0.4$, with $L=484$ (blue), $L=244$ (yellow), $L=124$ (green) and $L=24$ (red). For large $L$, the roots approach a non-trivial contour in the complex plane.
• Figure 5: Sets of roots $\{u_j|1\leq j\leq L/2\}$ (left panel) and scaling of the eigenvalue of the transition matrix with system size (right panel) for the excited state of Eq. (\ref{['ints1']}) for $L=2N$ and $\alpha=0.5$. The orange line in the right panel is the fit of Eq. (\ref{['fit1_05']}) to the functional form of Eq. (\ref{['eofl_05a']}).