Cross Correlations in Scaling Analyses of Phase Transitions

Thermal or finite-size scaling analyses of importance sampling Monte Carlo computer simulations 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 often results in 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.

Figure 1

Estimates of the critical exponent $\nu$ of the 2D Ising model from the FSS of the indicated observables (circles). The (almost identical) dotted and dashed lines indicate the plain average ${\overline{\nu }}_{\mathrm{plain}}$ and the error-weighted mean ${\overline{\nu }}_{\mathrm{err}}$, respectively. The covariance-weighted mean ${\overline{\nu }}_{\mathrm{cov}}$ corresponds to the solid line. The shaded areas indicate the corresponding $1\sigma$ environments $\overline{\nu }\pm {\sigma }_{\mathrm{corr}}$.