Non-Gaussian numerical errors versus mass hierarchy

TL;DR

This paper investigates the non-Gaussian nature of numerical errors in renormalization group calculations using Dyson's hierarchical model, revealing significant deviations from Gaussian behavior and explaining their origin.

Contribution

It introduces a model showing how short-distance errors dominate, leading to non-Gaussian error distributions and challenging the applicability of the central-limit theorem in this context.

Findings

Numerical errors deviate significantly from Gaussian distribution.

Average errors are comparable to standard deviation, indicating non-Gaussian behavior.

Short-distance errors have a larger impact than long-distance errors.

Abstract

We probe the numerical errors made in renormalization group calculations by varying slightly the rescaling factor of the fields and rescaling back in order to get the same (if there were no round-off errors) zero momentum 2-point function (magnetic susceptibility). The actual calculations were performed with Dyson's hierarchical model and a simplified version of it. We compare the distributions of numerical values obtained from a large sample of rescaling factors with the (Gaussian by design) distribution of a random number generator and find significant departures from the Gaussian behavior. In addition, the average value differ (robustly) from the exact answer by a quantity which is of the same order as the standard deviation. We provide a simple model in which the errors made at shorter distance have a larger weight than those made at larger distance. This model explains in part the non-Gaussian features and why the central-limit theorem does not apply.