Infinite Variance in Monte Carlo Sampling of Lattice Field Theories

TL;DR

This paper investigates the issue of infinite variance in Monte Carlo sampling of lattice quantum field theories, demonstrating divergence in certain estimators and proposing alternative methods for accurate estimation.

Contribution

It identifies the problem of infinite variance in Monte Carlo estimators for lattice field theories and introduces two new sampling techniques to address this challenge.

Findings

Variance of estimators diverges in the Gross-Neveu model.

Proposed alternative sampling methods reduce variance issues.

Numerical tests validate the effectiveness of new methods.

Abstract

In Monte Carlo calculations of expectation values in lattice quantum field theories, the stochastic variance of the sampling procedure that is used defines the precision of the calculation for a fixed number of samples. If the variance of an estimator of a particular quantity is formally infinite, or in practice very large compared to the square of the mean, then that quantity can not be reliably estimated using the given sampling procedure. There are multiple scenarios in which this occurs, including in Lattice Quantum Chromodynamics, and a particularly simple example is given by the Gross-Neveu model where Monte Carlo calculations involve the introduction of auxiliary bosonic variables through a Hubbard-Stratonovich (HS) transformation. Here, it is shown that the variances of HS estimators for classes of operators involving fermion fields are divergent in this model and an even simpler zero-dimensional analogue. To correctly estimate these observables, two alternative sampling methods are proposed and numerically investigated.