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

Let $G$ be an infinite, vertex-transitive lattice with degree $\lambda$ 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 $F_p/\lambda^{\ell(p)}$ of cycles of length $l$ whose last erased loop is some chosen self-avoiding polygon $p$ of length $\ell(p)$, when $l\to\infty$ ? We use combinatorial sieves to prove an exact formula for $F_p/\lambda^{\ell(p)}$ that we evaluate explicitly. We further prove that for all self-avoiding polygons $p$, $F_p\in\mathbb{Q}[\chi]$ with $\chi$ an irrational number depending on the lattice, e.g. $\chi=1/\pi$ 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.