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Weakly directed self-avoiding walks

Abstract : We define a new family of self-avoiding walks (SAW) on the square lattice, called $\textit{weakly directed walks}$. 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.
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Axel Bacher, Mireille Bousquet-Mélou. Weakly directed self-avoiding walks. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.473-484, ⟨10.46298/dmtcs.2883⟩. ⟨hal-00653979v2⟩



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