Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Connectedness of the Free Uniform Spanning Forest as a function of edge
  weights   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.

25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con