Optimized replica gas estimation of absolute integrals and partition functions

TL;DR

This paper introduces an optimized replica gas estimation method that improves the accuracy and convergence of calculating absolute integrals and partition functions, especially demonstrated on a 2D Ising model.

Contribution

It generalizes replica gas identities with an arbitrary weighting function and derives an optimal form with minimal asymptotic variance, enhancing Monte Carlo integration.

Findings

Improved convergence of partition function estimates in a 2D Ising model

Derived an optimal weighting function for replica gas identities

Demonstrated reduced variance in integral estimates

Abstract

In contrast with most Monte Carlo integration algorithms, which are used to estimate ratios, the replica gas identities recently introduced by Adib enable the estimation of absolute integrals and partition functions using multiple copies of a system and normalized transition functions. Here, an optimized form is presented. After generalizing a replica gas identity with an arbitrary weighting function, we obtain a functional form that has the minimal asymptotic variance for samples from two replicas and is provably good for a larger number. This equation is demonstrated to improve the convergence of partition function estimates in a 2D Ising model.