Mean-field tricritical random walks
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.