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

Anomalous critical behaviour in the polymer collapse transition of three-dimensional lattice trails