Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model

TL;DR

This paper investigates large deviations in skew-detailed-balance lifted Markov processes applied to the Curie-Weiss model, demonstrating their efficiency in sampling equilibrium distributions compared to traditional detailed-balance methods.

Contribution

It extends the analysis of lifted Markov chains with skew-detailed-balance to the Curie-Weiss model, providing insights into their large deviation properties and sampling efficiency.

Findings

Lifted Markov chains outperform detailed-balance chains in sampling efficiency.

Large deviations analysis reveals differences in empirical observables.

Skew-detailed-balance processes improve convergence to equilibrium.

Abstract

Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of 'Lifting' consists in duplicating the configuration space into two copies $\sigma=\pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie-Weiss mean-field ferromagnetic model, where the dynamics for the magnetization is closed. The large deviations at various levels for empirical time-averaged observables are analyzed and compared with their detailed-balance counterparts, both for the discrete extensive magnetization $M$ and for the continuous intensive magnetization $m=\frac{M}{N}$ for large system-size $N$.