Lattice trails. I. Exact results

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Abstract
Considers the problem of lattice trails, introduced by Malakis (1975, 1976). The author proves the existence of a finite connective constant, and establishes a result for the growth of n-step trails tn analogous to the best known result for self-avoiding walks, tn= lambda n expt(O(/n)). For the honeycomb (d=2) and Lave's lattice (d=3) the author establishes a counting theorem, from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice, lambda 2=2+ square root 2. Further, it follows that the trail problem is in the same universality class as the self-avoiding walk problem for those lattices. An exact amplitude relation between trails and self-avoiding walks, and between dumb-bells, and trails and self-avoiding walks is established. The non-existence of a counting theorem for arbitrary lattices is established. An inequality for the triangular lattice connective constant is proved. A high-density expansion for lambda for the d-dimensional hybercubic lattice is also obtained. The author argues that, contrary to recent suggestions, the model is in the same universality class as the self-avoiding walk model.

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