  There have been separate studies of the polymer collapse transition, where
the collapse was induced by two different types of attraction. In each case,
the configurations of the polymer were given by the same subset of random walks
being self-avoiding trails on the square lattice.
  Numerical evidence shows that when interacting via nearest-neighbour
contacts, this transition is different from the collapse transition in
square-lattice trails interacting via multiply visited sites. While both
transitions are second-order, when interacting via nearest-neighbour contacts,
the transition is relatively weak with a convergent specific heat, while when
interacting via multiply visited sites, the specific heat diverges strongly.
Moreover, an estimation of the crossover exponent for the nearest-neighbour
contact interaction provides a value close to that of the canonical polymer
collapse model of interacting self-avoiding walks, which also interact via
nearest-neighbour contacts.
  From computer simulations using the flatPERM algorithm, we extend these
studies by considering a model of self-avoiding trails on the square lattice
containing both types of interaction, and which therefore contains all three of
the models discussed above as special cases. We find that the strong
multiply-visited site collapse is a singular point in the phase diagram and
corresponds to a higher order multi-critical point separating a line of weak
second-order transitions from a line of first-order transitions.
