Counting walks by their last erased self-avoiding polygons using sieves

P.-L Giscard (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

Let G be an infinite, vertex-transitive lattice with degree λ and fix a vertex on it. Consider all cycles of length exactly l from this vertex to itself on G. Erasing loops chronologically from these cycles, what is the fraction F_p /λ^L(p) of cycles of length L whose last erased loop is some chosen self-avoiding polygon p of length L(p), when L → ∞ ? We use combinatorial sieves to prove an exact formula for F_p /λ^L(p) that we evaluate explicitly. We further prove that for all self-avoiding polygons p, F_p ∈ Q[χ] with χ an irrational number depending on the lattice, e.g. χ = 1/π on the infinite square lattice. In stark contrast we current methods, we proceed via purely deterministic arguments relying on Viennot's theory of heaps of pieces seen as a semi-commutative extension of number theory. Our approach also sheds light on the origin of the difference between exponents stemming from loop-erased walk and self-avoiding polygon models, and suggests a natural route to bridge the gap between both.

Mots clés: Self-avoiding polygons; Heaps of pieces; Sieves 2000 MSC: 05C38; 05A15; 11N36

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

P.-L Giscard. Counting walks by their last erased self-avoiding polygons using sieves. Discrete Mathematics, 2021, 344 (4), pp.112305. ⟨10.1016/j.disc.2021.112305⟩. ⟨hal-03597248⟩