  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.
