  Recent laser-plasma experiments report the existence of dynamically
significant magnetic fields, whose statistical characterisation is essential
for understanding the physical processes these experiments are attempting to
investigate. In this paper, we show how a proton imaging diagnostic can be used
to determine a range of relevant magnetic field statistics, including the
magnetic-energy spectrum. To achieve this goal, we explore the properties of an
analytic relation between a stochastic magnetic field and the image-flux
distribution created upon imaging that field. We conclude that features of the
beam's final image-flux distribution often display a universal character
determined by a single, field-scale dependent parameter - the contrast
parameter - which quantifies the relative size of the correlation length of the
stochastic field, proton displacements due to magnetic deflections, and the
image magnification. For stochastic magnetic fields, we establish the existence
of four contrast regimes - linear, nonlinear injective, caustic and diffusive -
under which proton-flux images relate to their parent fields in a qualitatively
distinct manner. As a consequence, it is demonstrated that in the linear or
nonlinear injective regimes, the path-integrated magnetic field experienced by
the beam can be extracted uniquely, as can the magnetic-energy spectrum under a
further statistical assumption of isotropy. This is no longer the case in the
caustic or diffusive regimes. We also discuss complications to the
contrast-regime characterisation arising for inhomogeneous, multi-scale
stochastic fields, as well as limitations currently placed by experimental
capabilities on extracting magnetic field statistics. The results presented in
this paper provide a comprehensive description of proton images of stochastic
magnetic fields, with applications for improved analysis of given proton-flux
images.
