高维奇割集的增长常数
The growth constant of odd cutsets in high dimensions
摘要：
割集是$\mathbb{Z}^d$的非空有限子集，它既连通又共连通。一个割集是奇的，如果它的顶点边界位于$\mathbb{Z}^d$的奇二划分类中。Peled提出，包含原点并具有$n$边界边的奇割集的数目可能是$e^{Θ(n/d)}$阶为$d \to \infty$，比Lebowitz和Mazel证明的一般割集的数目要小得多，一般割集的数目是$d^{Θ(n/d)}$阶。在本文中，我们验证了这一点，表明这样的奇割集的数量是$(2+o(1))^{n/2d}$。
英文摘要：
A cutset is a non-empty finite subset of $\mathbb{Z}^d$ which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of $\mathbb{Z}^d$. Peled suggested that the number of odd cutsets which contain the origin and have $n$ boundary edges may be of order $e^{Θ(n/d)}$ as $d \to \infty$, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel to be of order $d^{Θ(n/d)}$. In this paper, we verify this by showing that the number of such odd cutsets is $(2+o(1))^{n/2d}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
