论文标题
使用筛子来计算他们最后一次擦除的自我避免的多边形
Counting walks by their last erased self-avoiding polygons using sieves
论文作者
论文摘要
令$ g $为一个无限的，顶点传播的晶格，$λ$，并在其上修复一个顶点。考虑所有从该顶点$ l $的长度周期，以$ g $。从这些周期中按时间顺序擦除循环，$ f_p/λ^{\ ell（p）}的分数是什么，该循环的长度$ l $的循环是$ l $的循环，其最后一个删除的循环是某些选择的自避免的polygon $ p $ length $ p $ length $ \ ell（我们使用组合筛子来证明我们明确评估的$ f_p/λ^{\ ell（p）} $的精确公式。我们进一步证明，对于所有避开自我的多边形$ p $，$ f_p \ in \ mathbb {q} [χ] $带有$χ$的$一个不合理的数字，取决于晶格，例如。 $χ= 1/π$在无限的方格上。相比之下，我们当前的方法是通过纯粹的确定性论点进行的，这些论点依靠Viennot的一堆碎片理论，被视为数字理论的半共同扩展。我们的方法还阐明了来自基于环路的步行和自我避免多边形模型的指数之间差异的起源，并提出了一种自然的途径来弥合两者之间的间隙。
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 $F_p/λ^{\ell(p)}$ 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 $F_p/λ^{\ell(p)}$ that we evaluate explicitly. We further prove that for all self-avoiding polygons $p$, $F_p\in\mathbb{Q}[χ]$ 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.
