Walks, trials and polymers with loops

Lattice models interpolating between free and self-avoiding random walks are investigated. Generating functions are constructed for k-tolerant walks, various trail problems, etc. For trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension d_c=4. Their asymptotic large-scale behaviours are the same as those of the self-avoiding walk. Arguments are presented to support the same conclusion for much more general classes of walks including k-tolerant walks. One of the models exhibits a new tricritical point of order epsilon ^1/2 if the fugacity for crossing is increased.