Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Every planar graph with the Liouville property is amenable   We introduce a strengthening of the notion of transience for planar maps in
order to relax the standard condition of bounded degree appearing in various
results, in particular, the existence of Dirichlet harmonic functions proved by
Benjamini and Schramm. As a corollary we obtain that every planar non-amenable
graph admits Dirichlet harmonic functions.

34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th