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 $F_p∕{\lambda }^{\ell \left(p\right)}$ of cycles of length $l$ whose last erased loop is some chosen self-avoiding polygon $p$ of length $\ell \left(p\right)$, when $l\to \infty$ ? We use combinatorial sieves to prove an exact formula for $F_p∕{\lambda }^{\ell \left(p\right)}$ that we evaluate explicitly. We further prove that for all self-avoiding polygons $p$, $F_p\in \mathbb{Q}\left[\chi \right]$ 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.

让 $G$ 是具有度数的无穷顶点可传递晶格 $\lambda$并在其上固定一个顶点。准确考虑所有长度的循环$升$ 从这个顶点到自身 $G$。从这些周期中按时间顺序清除循环，分数是多少$F_p∕{\lambda }^{\ell （p）}$ 长度的周期 $升$ 其最后擦除的循环是一些选择的自动回避多边形 $p$ 长度 $\ell （p）$， 什么时候 $升\to \infty$？我们使用组合筛来证明$F_p∕{\lambda }^{\ell （p）}$我们明确评估。我们进一步证明，对于所有自我避免的多边形$p$， $F_p\in \mathbb{问}\left[\chi \right]$ 与 $\chi$ 取决于晶格的无理数，例如 $\chi =\mathrm{1个}∕\pi$在无限的方格上。与目前的方法形成鲜明对比的是，我们依靠纯粹的确定性论证来进行研究，这些论证依赖于维恩诺特的碎片堆理论，被视为数论的半交换扩展。我们的方法还阐明了环路擦除的行走模型和自避免多边形模型产生的指数之间差异的起源，并提出了一条弥合两者之间差距的自然路线。