Statistics of Peaks in Chi-Squared Fields

TL;DR

This paper provides a detailed statistical analysis of the extrema of chi-squared random fields, focusing on their neighborhood properties, spherical symmetry, and efficient sampling methods.

Contribution

It introduces a harmonic decomposition approach to analyze stationary points of chi-squared fields and develops metrics for their spherical symmetry.

Findings

Expected profile of biased chi-squared fields computed

Harmonic modes used to assess neighborhood symmetry

Efficient sampling methods for fields around stationary points

Abstract

Chi-squared random fields arise naturally from the study of fluctuations in field theories with SO(n) symmetry. The extrema of chi-squared fields are of particular physical interest. In this paper, we undertake a statistical analysis of the stationary points of chi-squared fields, with particular emphasis on extrema. We begin by describing the neighborhood of a stationary point in terms of a biased chi-squared random field, and then compute the expected profile of this field, as well as a variety of associated statistics. We are interested in understanding how spherically symmetric the neighborhood of a stationary point is, on average. To this end, we decompose the biased field into its spherical harmonic modes about this point, and explore their statistics. Using these mode statistics, we construct a metric to gauge the degree of spherical symmetry of the field in this neighborhood. Finally, we show how to leverage the harmonic decomposition to efficiently sample both Gaussian and chi-squared fields about a stationary point.