Oriented self-avoiding walks with orientation-dependent interactions

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Abstract
We consider oriented self-avoiding walks on the square lattice with different energies between steps that are oriented parallel or antiparallel across a face of the lattice. Rigorous bounds on the free energy and exact enumeration data are used to study the statistical mechanics of this model. We conjecture a phase diagram in the parallel-antiparallel interaction plane, and discuss the order of the associated phase transitions. The question, raised by previous field theoretical considerations, of the existence of an exponent that varies continuously with the energy of interaction is discussed at length. In connection with this we have also studied two oriented walks fixed at a common origin; this being the simplest model of branched oriented polymers in two dimensions. The evidence, although not conclusive, tends to support the field theoretic prediction.