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
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
. A chemical
potential
controls the walk length. We determine the phase diagram in the
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
) 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.
Keywords
polymer model, complete graph, mean field, phase
transition, tricritical point, theta point