# Irreversible local Markov chains with rapid convergence towards equilibrium

## TL;DR

This paper introduces irreversible local Markov chains for the 1D hard-sphere model that converge to equilibrium faster than traditional reversible algorithms, revealing two universality classes with distinct rapid mixing properties.

## Contribution

The paper presents new irreversible Markov chains, including the lifted TASEP, that achieve faster mixing times and extend to higher dimensions and general interactions.

## Key findings

- Irreversible chains outperform reversible ones in mixing speed.
- Event-chain algorithm belongs to the fastest universality class.
- Lifted TASEP extends the class of efficient Markov chains.

## Abstract

We study the continuous one-dimensional hard-sphere model and present irreversible local Markov chains that mix on faster time scales than the reversible heatbath or Metropolis algorithms. The mixing time scales appear to fall into two distinct universality classes, both faster than for reversible local Markov chains. The event-chain algorithm, the infinitesimal limit of one of these Markov chains, belongs to the class presenting the fastest decay. For the lattice-gas limit of the hard-sphere model, reversible local Markov chains correspond to the symmetric simple exclusion process (SEP) with periodic boundary conditions. The two universality classes for irreversible Markov chains are realized by the totally asymmetric simple exclusion process (TASEP), and by a faster variant (lifted TASEP) that we propose here. Lifted Markov chains and the recently introduced factorized Metropolis acceptance rule extend the irreversible Markov chains discussed here to general pair interactions and to higher dimensions.