Improvement of Monte Carlo estimates with covariance-optimized finite-size scaling at fixed phenomenological coupling

In the finite-size scaling analysis of Monte Carlo data, instead of computing the observables at fixed Hamiltonian parameters, one may choose to keep a renormalization-group invariant quantity, also called phenomenological coupling, fixed at a given value. Within this scheme of finite-size scaling, we exploit the statistical covariance between the observables in a Monte Carlo simulation in order to reduce the statistical errors of the quantities involved in the computation of the critical exponents. This method is general and does not require additional computational time. This approach is demonstrated in the Ising model in two and three dimensions, where large gain factors in CPU time are obtained.

Figure 1

Decomposition of the vector $\widehat{\delta O}$ into two components orthogonal $\widehat{\delta O_{\perp }}$ and parallel $\widehat{\delta O_{\parallel }}$ to the space $\mathcal{V}=\mathrm{Span}<\delta R_1,...,\delta R_N>$. The affine subspace $\mathcal{W}$ is given by Eq. (7), which represents the normalization on the parameters $\left\{{\lambda }_i\right\}$. The minimum length for $\widehat{\delta O_{\parallel }}-\widehat{\delta v}$ is obtained when it is orthogonal to $\mathcal{W}$.