Forget them not
Writing in Physical Review Letters, Martin Weigel of the Johannes-Gutenberg-Universität and Wolfhard Janke of the Universität Leipzig, both in Germany, examine the effect of cross-correlations on the computational output of Monte Carlo simulations. The particular setting is the finite-size scaling analysis in the vicinity of the critical point for the Ising model in two and three dimensions, respectively. The authors focus on determining the exponent , a key parameter that is central to the scaling theory of phase transitions. This exponent describes the way the correlation length of a system diverges with temperature on approaching the transition; it also affects all other relevant scaling exponents, e.g., the scaling dimension of the magnetic field.
Cross-correlations are always present, essentially by design, whenever same-run data are used to determine different quantities from which the targeted exponents are then extracted. Most simulations ignore cross-correlations and use uncorrelated estimates. In contrast, Weigel and Janke demonstrate how to harness cross-correlations and produce a better overall estimate of the quintessential exponent. At the same time, their proposed extension of conventional analysis leads to a significant reduction of statistical errors, in one case reducing the error by a factor of three at essentially no extra computational cost. The proposed procedure is easy to grasp: Given a number of different common-origin estimates that lead to the critical exponent in question, the optimal choice is given by means of an appropriate averaging of the estimated quantities with explicitly prescribed weighted coefficients.
It might be tempting to conclude that the rest is merely an algorithmic task, especially since the outlined technique is generic and is likely to apply to all kinds of computer simulations. On the other hand, for most of the prospective problems a reference analytic solution would not be available. The results of Weigel and Janke show that much knowledge and intuition will be needed in any nontrivial implementation of their technique. – Yonko Millev