Optimality of zero index-space drift for autocorrelation

Establish whether setting the factor-field parameter so that the index-space velocity satisfies v_idx = 0 optimises the autocorrelation time of the q = 1 structure factor in factor-field event-chain Monte Carlo for the generalised harmonic-chain model.

Background

The paper studies factor-field event-chain Monte Carlo (ECMC) for particles on a one-dimensional torus, including the generalised harmonic-chain model with pairwise polynomial spring potentials. The factor-field parameter b controls the lifting dynamics on the auxiliary index space, and the associated index-space velocity v_idx measures the long-time drift of the active-particle index.

For the p = 2 generalised harmonic chain, the authors derive that choosing b = Lk/N yields zero pressure and therefore v_idx = 0. They state that this condition appears to minimise the autocorrelation time of the q = 1 structure factor, but formulate the relationship as a hypothesis rather than proving it. The question is consequently whether zero index-space drift is generally sufficient to guarantee optimal autocorrelation, rather than merely providing an empirically supported predictor in the analysed case.

References

The hypothesis that $v_{\rm idx} = 0$ optimises autocorrelation then predicts that the autocorrelation time $\tau$ of the $q=1$ structure factor $S(q) = \left| \sum_j \exp(i q x_j) \right|2 / N$ is minimised at $b=Lk/N$, as supported by the results in Fig. \ref{fig:HarmonicChain}(b) (which present $\tau$ as a function of $b$ and $N$ at fixed $\rho$).

— Optimal sampling strategies in event-chain Monte Carlo  (2610.01659 - Gulliford et al., 1 Oct 2026) in Section 3.2, “Generalised harmonic chain” (subsection “Generalised harmonic chain”)