Cross-correlations in scaling analyses of phase transitions

TL;DR

This paper highlights the importance of accounting for cross-correlations in scaling analyses of phase transitions, proposing an improved method that reduces statistical fluctuations without extra computational cost.

Contribution

It introduces a simple extension to conventional scaling analysis methods that accounts for cross-correlations, improving estimator accuracy.

Findings

Cross-correlations often cause underestimation of fluctuations in scaling analyses.

The proposed method effectively reduces statistical fluctuations in estimates.

Implementation requires minimal additional computational effort.

Abstract

Thermal or finite-size scaling analyses of importance sampling Monte Carlo time series in the vicinity of phase transition points often combine different estimates for the same quantity, such as a critical exponent, with the intent to reduce statistical fluctuations. We point out that the origin of such estimates in the same time series results in often pronounced cross-correlations which are usually ignored even in high-precision studies, generically leading to significant underestimation of statistical fluctuations. We suggest to use a simple extension of the conventional analysis taking correlation effects into account, which leads to improved estimators with often substantially reduced statistical fluctuations at almost no extra cost in terms of computation time.