Analytic spectral asymptotics, integrable structure, and integrable generalizations of the lifted TASEP

The lifted TASEP, a non-reversible lifted Markov chain introduced by Essler and Krauth as a one-dimensional lattice paradigm for event-chain Monte Carlo, is exactly solvable by an unusual nested coordinate Bethe ansatz, but three of the four scientific issues its discoverers left open remain unresolved: (i) the Bethe equations have only been analyzed numerically, and still need to be recast as nonlinear integral equations (thermodynamic/Baxter-type or Destri–de Vega-type) to derive analytically the asymptotic large-system scaling of the low-lying eigenvalues and their finite-size corrections; (ii) the algebraic integrable structure — the $R$ -matrix, L-operator, and transfer-matrix framework underlying the model — remains unidentified, and it is unclear whether the spatially non-local transition matrix fits into existing classifications of quantum-integrable systems; (iii) it is unknown whether integrable liftings exist for the partially asymmetric exclusion process (PASEP), which boundary conditions preserve integrability, whether the open TASEP admits a lifting that reproduces its non-trivial steady state with significantly shorter relaxation time, and whether integrable extensions to several active particles exist. The fourth original issue, the integrability of the density-to-zero continuum limit, was resolved affirmatively by Essler, Gipouloux, and Krauth in 2025, who constructed the integrable continuum limit and showed that the critical pullback equals the system pressure.

Background

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic stochastic model of $N$ hard-core particles hopping unidirectionally on an $L$ -site lattice; its relaxation belongs to the Kardar–Parisi–Zhang (KPZ) universality class, and its spectrum has been studied since Gwa and Spohn (1992) by the coordinate Bethe ansatz, a method that diagonalizes transition matrices through $N$ -particle scattering wavefunctions parametrized by complex rapidities $z_j$ satisfying algebraic 'Bethe equations'. For the ordinary TASEP the relaxation time scales with system size as $N^{3/2}$ in continuous time (Gwa–Spohn; Golinelli–Mallick), or as $N^{5/2}$ in the discrete-time one-move-attempt-per-step convention in which the lifted TASEP is defined below. For such chains one standard route to exact finite-size and asymptotic information is to recast the Bethe equations as nonlinear integral equations (NLIEs) of thermodynamic (Baxter-type) or Destri–de Vega type — integro-differential equations for counting functions from which the large- $L$ spectrum can be extracted analytically, a technique used throughout the later exact spectral analysis of exclusion processes. 'Lifting' (Diaconis–Holmes–Neal 2000) augments a reversible Markov chain with directed velocity states, producing a non-reversible chain with the same stationary distribution but faster relaxation. Essler and Krauth (Phys. Rev. X 14, 041035 (2024)) introduced the 'lifted TASEP': $N$ particles on an $L$ -site ring at density $N/L=1/2$ , with a single 'active' particle carrying the activity (the pointer). At each time step the active particle attempts a forward hop by one site, which succeeds if the target site is empty; in addition, with probability $\alpha$ the activity is transferred to the nearest particle to the left of the active one (the 'pullback'), a transfer that can act across arbitrarily large distances and makes the transition matrix spatially non-local. The model is the one-dimensional lattice reduction of the event-chain Monte Carlo (ECMC) algorithm, a non-reversible Markov-chain Monte Carlo method built on lifting. It is exactly solvable by a nested coordinate Bethe ansatz despite this non-locality; its steady state is exactly the equal-probability mixture of configurations, the same as for the reversible symmetric simple exclusion process (SSEP), so the lifting provably preserves the equilibrium state while changing the spectrum drastically. Averaged over the dynamics, the activity drifts with velocity $1-\alpha L/N$ , which vanishes at the critical pullback $\alpha_{\rm crit}=N/L$ (equal to $1/2$ at density $1/2$ ). Numerically, the relaxation time (inverse spectral gap) scales as $N^{5/2}$ for $\alpha\neq\alpha_{\rm crit}$ and $N^{2}$ for $\alpha=\alpha_{\rm crit}$ , to be compared with $N^{3}$ for the SSEP and $N^{5/2}$ for the TASEP in the same discrete-time convention — so that at the critical pullback the lifting relaxes faster than the KPZ scaling of the TASEP. The 2025 follow-up by Essler, Gipouloux and Krauth confirmed this gap crossover numerically and resolved the apparent disagreement with Monte Carlo autocorrelation times ( $\tau\sim N^{3/2}$ ) by showing that the slow modes' matrix elements in typical observables are exponentially small, making them invisible in simulations; it also constructed the continuum limit of the lifted TASEP, proved it integrable, showed that the critical pullback equals the system pressure (the zero-pressure point of the activity drift), and connected it to hard-sphere event-chain Monte Carlo. By contrast, the analytic and structural questions about the discrete model remain open. The open issues below were listed by Essler and Krauth in their conclusions, with the continuum-limit-integrability sub-issue subsequently settled; the rest frame a single objective: completing the Bethe-ansatz and integrability characterization of the lifted-TASEP paradigm, the first many-particle Markov chain with spatially non-local moves to be solved exactly by the Bethe ansatz.

Problem Statement

For the lifted TASEP of Essler and Krauth ( $N$ particles on an $L$ -site ring at density $N/L=1/2$ , i.e. $L=2N$ so that $\alpha_{\rm crit}=1/2$ , single active particle carrying the activity, pullback parameter $\alpha$ , solved by a nested coordinate Bethe ansatz), establish the analytic characterization of its spectrum and integrable structure left open by the original analysis and not settled by the 2025 follow-up — with one sub-issue (the continuum-limit integrability at vanishing density, resolved affirmatively) removed from the original four-part list. Concretely, the problem comprises four linked deliverables toward the single objective of completing the Bethe-ansatz/integrability theory of the lifted-TASEP paradigm: (1) Analytic spectral asymptotics — recast the Bethe equations as nonlinear integral equations of thermodynamic (Baxter-type) or Destri–de Vega type, and use them to derive analytically the scaling behavior of the low-lying eigenvalues (the spectral gap $\Delta$ with $\Delta\sim N^{-5/2}$ for $\alpha\neq\alpha_{\rm crit}$ and $\Delta\sim N^{-2}$ for $\alpha=\alpha_{\rm crit}$ , so far established only numerically in this half-filling $L=2N$ setting), including the leading finite-size corrections, and identify which eigenvalue(s) constitute the true slowest relaxation mode(s). (2) Integrable structure — identify the $R$ -matrix and L-operator that generate the lifted TASEP's spatially non-local transition matrix as a transfer matrix, thereby clarifying whether and how the model fits existing classifications of quantum-integrable systems. (3) Integrable generalizations — determine whether the partially asymmetric exclusion process (PASEP) admits an integrable lifting analogous to the lifted TASEP; determine which boundary conditions (open chain, reservoirs) preserve integrability; and determine whether there exists a lifting of the open TASEP that reproduces its non-trivial steady state while having a significantly shorter relaxation time. (4) Multi-active-particle extensions — determine whether integrable extensions of the model to several simultaneously active particles exist. Deliverables (2)–(4) are linked to (1): the algebraic structure identified in (2) is the expected tool for constructing the PASEP, boundary, and multi-active-particle extensions requested in (3) and (4), which is why the source presents them as one coherent set of open issues.

References

F. H. L. Essler and W. Krauth, 'Lifted TASEP: A Solvable Paradigm for Speeding up Many-Particle Markov Chains', Physical Review X 14, 041035 (2024), DOI: 10.1103/PhysRevX.14.041035; preprint arXiv:2306.13059 (titled 'Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains'), https://arxiv.org/abs/2306.13059
F. Essler, J. Gipouloux and W. Krauth, 'Lifted TASEP: Long-time dynamics, generalizations, and continuum limit', SciPost Physics Core 8, 063 (2025), DOI: 10.21468/SciPostPhysCore.8.4.063, arXiv:2502.16549, https://arxiv.org/abs/2502.16549
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