[Paper Review] Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains

Fabian H. L. Eßler, Werner Krauth|arXiv (Cornell University)|Jun 22, 2023

Markov Chains and Monte Carlo MethodsMathematics3 citations

TL;DR

This paper introduces the 'lifted TASEP' as a novel, exactly solvable non-reversible Markov chain model that extends the totally asymmetric simple exclusion process (TASEP) via a lifted state space. Using an unconventional coordinate Bethe ansatz, the authors establish integrability and provide strong evidence that the model exhibits faster relaxation than the KPZ universality class, offering a paradigm for efficient non-reversible MCMC algorithms.

ABSTRACT

Markov-chain Monte Carlo (MCMC), the field of stochastic algorithms built on the concept of sampling, has countless applications in science and technology. The overwhelming majority of MCMC algorithms are time-reversible and satisfy the detailed-balance condition, just like physical systems in thermal equilibrium. The underlying Markov chains typically display diffusive dynamics, which leads to a slow exploration of sample space. Significant speed-ups can be achieved by non-reversible MCMC algorithms exhibiting non-equilibrium dynamics, whose steady states exactly reproduce the target equilibrium states of reversible Markov chains. Such algorithms have had successes in applications but are generally difficult to analyze, resulting in a scarcity of exact results. Here, we introduce the "lifted" TASEP (totally asymmetric simple exclusion process) as a paradigm for lifted non-reversible Markov chains. Our model can be viewed as a second-generation lifting of the reversible Metropolis algorithm on a one-dimensional lattice and is exactly solvable by an unusual kind of coordinate Bethe ansatz. We establish the integrability of the model and present strong evidence that the lifting leads to faster relaxation than in the KPZ universality class.

Motivation & Objective

• To develop a paradigm for non-reversible Markov chains that relax faster than diffusive processes.
• To construct a lifted version of the TASEP that maintains exact solvability while breaking detailed balance.
• To demonstrate integrability of the lifted TASEP using a non-standard coordinate Bethe ansatz.
• To provide evidence that the model relaxes faster than systems in the KPZ universality class.
• To establish a framework for analyzing non-reversible MCMC algorithms beyond the reversible, equilibrium paradigm.

Proposed method

• The model is constructed by lifting the state space of the standard TASEP to include directional information, forming a larger state space Ω^l-TASEP.
• The dynamics involve transitions that preserve particle order and include a direction-flip mechanism with probability α and 1−α.
• Integrability is established via a generalized coordinate Bethe ansatz, solving eigenvalue equations for multi-particle sectors with specific boundary conditions.
• The spectral parameters satisfy a set of nonlinear equations (Eq. 77) involving the transfer matrix T(z_a, z_b), derived from periodic boundary conditions.
• The master equation is mapped to an imaginary-time Schrödinger equation with a non-Hermitian Hamiltonian H, defined via non-local permutation operators.
• The solution involves expressing wavefunction amplitudes in terms of a single amplitude A via recursive relations (Eq. 72), ensuring consistency across sectors.

Experimental results

Research questions

• RQ1 Can a non-reversible Markov chain be constructed that is exactly solvable and relaxes faster than diffusive systems?
• RQ2 Does lifting the TASEP via state-space extension preserve integrability and allow for a Bethe ansatz solution?
• RQ3 What is the relaxation behavior of the lifted TASEP compared to the KPZ universality class?
• RQ4 How can a non-Hermitian quantum spin chain representation be used to describe the stochastic dynamics of the lifted TASEP?
• RQ5 Can the Bethe ansatz be adapted to non-reversible processes with non-local dynamics and non-trivial boundary conditions?

Key findings

• The lifted TASEP is exactly solvable via a novel coordinate Bethe ansatz, extending integrability to non-reversible stochastic processes.
• The model's eigenvalue spectrum is determined by a set of nonlinear equations (Eq. 77) involving spectral parameters and a transfer matrix T(z_a, z_b).
• All wavefunction amplitudes in multi-particle sectors are expressed in terms of a single amplitude A through recursive relations (Eq. 72), ensuring consistency.
• The periodic boundary conditions lead to a closed system of equations that define the allowed eigenvalues and spectral parameters.
• The non-Hermitian Hamiltonian representation confirms the model's structure as a quantum spin chain with non-local interactions.
• Strong evidence is presented that the relaxation time of the lifted TASEP is faster than that of the KPZ class, suggesting improved mixing for MCMC applications.