  We present the outline of a proof for the 3-d phase transition which we hope
to carry forth. At the same time this paper provides some physical
understanding of the phase transition, in the flavor of relatively simple
arguments from an undergraduate course. A number of directions for mathematical
research, interesting in their own right, will be suggested by aspects of the
development. We hope and believe that readers will be enticed by the
naturalness and beauty of the path; some perhaps even, big game veterans,
sniffing the quarry, will be ready to join the hunt.
  The central construct views the trace, Tr(exp(-beta*H)), as a lattice gas of
polymers, each representing a cycle in the permutation group, with hard core
interactions. The activities of the polymers have expressions as arising from
the main conjecture of the paper. The estimates lead to a phase transition in
3-d, but not 2-d. This occurs via the same argument that a random walk in 2-d
has certain return to the origin, but not so for a random walk in 3-d.
