  Trails (bond-avoiding walks) provide an alternative lattice model of polymers
to self-avoiding walks, and adding self-interaction at multiply visited sites
gives a model of polymer collapse. Recently, a two-dimensional model
(triangular lattice) where doubly and triply visited sites are given different
weights was shown to display a rich phase diagram with first and second order
collapse separated by a multi-critical point. A kinetic growth process of
trails (KGT) was conjectured to map precisely to this multi-critical point. Two
types of low temperature phases, globule and crystal-like, were encountered.
Here, we investigate the collapse properties of a similar extended model of
interacting lattice trails on the simple cubic lattice with separate weights
for doubly and triply visited sites. Again we find first and second order
collapse transitions dependent on the relative sizes of the doubly and triply
visited energies. However we find no evidence of a low temperature crystal-like
phase with only the globular phase in existence.
  Intriguingly, when the ratio of the energies is precisely that which
separates the first order from the second-order regions anomalous finite-sized
scaling appears. At the finite size location of the rounded transition clear
evidence exists for a first order transition that persists in the thermodynamic
limit. This location moves as the length increases, with its limit apparently
at the point that maps to a KGT. However, if one fixes the temperature to sit
at exactly this KGT point then only a critical point can be deduced from the
data. The resolution of this apparent contradiction lies in the breaking of
crossover scaling and the difference in the shift and transition width
(crossover) exponents.
