Weakly directed self-avoiding walks

Authors:
Axel Bacher;Mireille Bousquet-Mélou
Affiliations:
LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France;CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

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Abstract

We define a new family of self-avoiding walks (SAW) on the square lattice, called weakly directed walks. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.