Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometry of Gaussian free field sign clusters and random interlacements   For a large class of amenable transient weighted graphs $G$, we prove that
the sign clusters of the Gaussian free field on $G$ fall into a regime of
strong supercriticality, in which two infinite sign clusters dominate (one for
each sign), and finite sign clusters are necessarily tiny, with overwhelming
probability. Examples of graphs belonging to this class include regular
lattices like $\mathbb{Z}^d$, for $d \geqslant 3$, but also more intricate
geometries, such as Cayley graphs of suitably growing (finitely generated)
non-Abelian groups, and cases in which random walks exhibit anomalous diffusive
behavior, for instance various fractal graphs. As a consequence, we also show
that the vacant set of random interlacements on these objects, introduced by
Sznitman in arXiv:0704.2560, and which is intimately linked to the free field,
contains an infinite connected component at small intensities. In particular,
this result settles an open problem from arXiv:1010.1490.

26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom