A numerical and theoretical study of multilevel performance for two-point correlator calculations

TL;DR

This paper evaluates the effectiveness of multilevel algorithms in the Ising model near criticality, revealing their limitations as correlation lengths grow, supported by numerical and theoretical analysis.

Contribution

It provides a combined numerical and theoretical analysis of multilevel algorithm performance in the Ising model, especially near critical points.

Findings

Multilevel performance deteriorates as correlation length increases.

Multilevel reduces error when correlation length is less than one-tenth of lattice size.

Theoretical model accurately predicts performance scaling.

Abstract

An investigation of the performance of the multilevel algorithm in the approach to criticality has been undertaken using the Ising model, performing simulations across a range of temperatures. Numerical results show that the performance of multilevel in this system deteriorates as the correlation length is increased with respect to the lattice size. The statistical error of the longest correlator in the system is reduced in a multilevel setup when the correlation length is less than one-tenth of the lattice size, while for longer correlation lengths multilevel performs more poorly than a computer-time equivalent single level algorithm. A theoretical model of this performance scaling is outlined, and shows remarkable accuracy when compared to numerical results. This theoretical model may be applied to other systems with more complex spectra to predict if multilevel techniques are likely to result in improved statistics.