论文标题
自由统一森林的连接性与边缘重量的关系
Connectedness of the Free Uniform Spanning Forest as a function of edge weights
论文作者
论文摘要
令$ g $为普通树$ t $的笛卡尔产品和有限的连接及时图形$ h $。在Arxiv：2006.06387中显示了该图的自由统一跨度森林（$ \ Mathsf {fsf} $）可能无法连接，但是这种连接对$ H $的依赖性仍然有些神秘。我们研究了当$ g $的$ h $ copies的正重$ w $的情况下，并猜测$ \ mathsf {fsf} $的连接性表现出相位过渡。对于足够大的$ w $，我们表明$ \ mathsf {fsf} $已连接，而对于一个$ h $和$ t $的大家庭，$ \ mathsf {fsf} $在$ w $很小时断开了连接（依靠arxiv：2006.06387）。最后，我们证明，当$ h $是一个边缘的图表时，对于任何$ w $，$ \ mathsf {fsf} $是一棵树，我们给出了一个明确的公式，以在树内的两个点之间分配距离。
Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on $H$ remains somewhat mysterious. We study the case when a positive weight $w$ is put on the edges of the $H$-copies in $G$, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase transition. For large enough $w$ we show that the $\mathsf{FSF}$ is connected, while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove that when $H$ is the graph of one edge, then for any $w$, the $\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.
