Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exceptional graphs for the random walk   If $\mathcal{W}$ is the simple random walk on the square lattice
$\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any
spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing
$\mathcal{W}$ as a uniformly random infinite word on the alphabet
$\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk
$\mathcal{W}_G$ starts at the origin and follows the directions specified by
$\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk
$\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset
\mathbb{Z}^2$, the walk $\mathcal{W}_G$ is distributed as the simple random
walk on $G$, and hence $\mathcal{W}_G$ is almost surely recurrent in the sense
that $\mathcal{W}_G$ visits every site reachable from the origin in $G$
infinitely often. This fact naturally leads us to ask the following: does
$\mathcal{W}$ almost surely have the property that $\mathcal{W}_G$ is recurrent
for \emph{every} subgraph $G \subset \mathbb{Z}^2$? We answer this question
negatively, demonstrating that exceptional subgraphs exist almost surely. In
fact, we show more to be true: exceptional subgraphs continue to exist almost
surely for a countable collection of independent simple random walks, but on
the other hand, there are almost surely no exceptional subgraphs for a
branching random walk.

10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat