We consider a random walk on the complete graph. The walk experiences competing self-repulsion and self-attraction, as well as a variable length. Variation of the parameters governing the self-attraction and the variable length leads to a rich phase diagram containing a tricritical point (known as the "theta" point in chemical physics). We discuss the phase diagram, as well as the method of proof used to determine the phase diagram. The method involves a supersymmetric representation for the random walk, together with the Laplace method for an integral with large parameter. This is joint work with Roland Bauerschmidt.