  In this thesis, we have investigated the possibility of large cross sections
at large multiplicity in weakly coupled three dimensional $\phi^4$ theory using
Monte Carlo Simulation methods. We have built a Beowulf Supercomputer for this
purpose. We use spectral function sum rules to derive a bound on the total
cross section where the quantity determining the bound can be measured by Monte
Carlo simulation in Euclidean space. We determine the critical threshold energy
for large high multiplicity cross section according to the analysis of M.B.
Volosion and E.N. Argyres, R.M.P. Kleiss, and C.G. Papadopoulos. We compare the
simulation results with the perturbation results and see no evidence for large
cross section in the range where tree diagram estimates suggest they should
exist.
