Statistical Dependence and Related Topics

TL;DR

This paper introduces the statistical dependence time to better understand dynamical correlations in Monte Carlo simulations, providing a new method to estimate relaxation times and critical exponents without needing correlation functions.

Contribution

The paper proposes a novel quantity, the statistical dependence time, and a new method for calculating equilibrium relaxation times that avoids using time-displaced correlation functions.

Findings

Estimated the dynamical critical exponent z for Ising models.

Analyzed systematic errors in response functions due to short simulations.

Validated the new method on 2D and 3D Ising models.

Abstract

On the basis of the dynamical interpretation of Monte Carlo simulations, we discuss the relation of the equilibrium relaxation time, the susceptibility and the statistical error. We introduce a new quantity called {\it the statistical dependence time} $\tau_{dep}$, which gives the reduction factor for the statistical degree of freedom due to the dynamical correlations between the data. A new method is proposed for calculating equilibrium relaxation time using $\tau_{dep}$, the method which does not require knowledge of any time-displaced correlation function. We apply this method to the critical dynamics of Ising models in two and three dimensions, and estimate the dynamical critical exponent $z$ precisely. Systematic errors in response functions due to short simulations are also discussed from the viewpoint of the statistical dependence.