Understanding stochastic perturbation theory: toy models and statistical analysis

TL;DR

This paper investigates the validity of stochastic perturbation theory using toy models, highlighting issues with estimator distributions and proposing bootstrap methods for reliable error estimation, with implications for lattice gauge theory calculations.

Contribution

It systematically analyzes the statistical properties of estimators in stochastic perturbation theory and demonstrates effective error estimation techniques for complex models.

Findings

Large deviations from normal distribution in estimators ('Pepe effect')

Bootstrap method provides reliable error estimates in certain models

Potentially fewer issues at moderate loop orders in lattice gauge theory

Abstract

The numerical stochastic perturbation method based on Parisi-Wu quantisation is applied to a suite of simple models to test its validity at high orders. Large deviations from normal distribution for the basic estimators are systematically found in all cases (``Pepe effect''). As a consequence one should be very careful in estimating statistical errors. We present some results obtained on Weingarten's ``pathological'' model where reliable results can be obtained by an application of the bootstrap method. We also present some evidence that in the far less trivial application to Lattice Gauge Theory a similar problem should not arise at moderately high loops (up to O(\alpha^{10})).