  We provide an introductory account of a tricritical phase diagram, in the
setting of a mean-field random walk model of a polymer density transition, and
clarify the nature of the density transition in this context. We consider a
continuous-time random walk model on the complete graph, in the limit as the
number of vertices $N$ in the graph grows to infinity. The walk has a repulsive
self-interaction, as well as a competing attractive self-interaction whose
strength is controlled by a parameter $g$. A chemical potential $\nu$ controls
the walk length. We determine the phase diagram in the $(g,\nu)$ plane, as a
model of a density transition for a single linear polymer chain. A dilute phase
(walk of bounded length) is separated from a dense phase (walk of length of
order $N$) by a phase boundary curve. The phase boundary is divided into two
parts, corresponding to first-order and second-order phase transitions, with
the division occurring at a tricritical point. The proof uses a supersymmetric
representation for the random walk model, followed by a single block-spin
renormalisation group step to reduce the problem to a 1-dimensional integral,
followed by application of the Laplace method for an integral with a large
parameter.
