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 to itself $G$. Erasing loops chronologically these cycles, what is the fraction $F_p/\lambda^{\ell(p)}$ whose last erased loop some chosen self-avoiding polygon $p$ $\ell(p)$, when $l\to\infty$ ? We use combinatorial sieves prove exact formula for that we evaluate explicitly. further polygons $p$, $F_p\in\mathbb{Q}[\chi]$ $\chi$ irrational number depending lattice, e.g. $\chi=1/\pi$ infinite square lattice. In stark contrast current methods, proceed via purely deterministic arguments relying Viennot's theory heaps pieces seen as semi-commutative extension theory. Our approach also sheds light origin difference between exponents stemming loop-erased walk models, suggests natural route bridge gap both.