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

Abstract

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 Fp∕λℓ(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 Fp∕λℓ(p) that we evaluate explicitly. We further prove that for all self-avoiding polygons p, Fp∈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.