  A new method involving particle diagrams is introduced and developed into a
rigorous framework for carrying out embedded random matrix calculations. Using
particle diagrams and the attendant methodology including loop counting it
becomes possible to calculate the fourth, sixth and eighth moments of embedded
ensembles in a straightforward way. The method, which will be called the method
of particle diagrams, proves useful firstly by providing a means of classifying
the components of moments into particle paths, or loops, and secondly by giving
a simple algorithm for calculating the magnitude of combinatorial expressions
prior to calculating them explicitly. By confining calculations to the limit
case $m \ll l\to\infty$ this in many cases provides a sufficient excuse not to
calculate certain terms at all, since it can be foretold using the method of
particle diagrams that they will not survive in this asymptotic regime.
Applying the method of particle diagrams washes out a great deal of the
complexity intrinsic to the problem, with sufficient mathematical structure
remaining to yield limiting statistics for the unified phase space of random
matrix theories.
  Finally, since the unified form of random matrix theory is essentially the
set of all randomised k-body potentials, it should be no surprise that the
early statistics calculated for the unified random matrix theories in some
instances resemble the statistics currently being discovered for quantum spin
hypergraphs and other randomised potentials on graphs [HMH05,ES14,KLW14]. This
is just the beginning for studies into the field of unified random matrix
theories, or embedded ensembles, and the applicability of the method of
particle diagrams to a wide range of questions as well as to the more exotic
symmetry classes, such as the symplectic ensembles, is still an area of
open-ended research.
