Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Percolation and Connectivity in AB Random Geometric Graphs   Given two independent Poisson point processes $\Phi^{(1)},\Phi^{(2)}$ in
$R^d$, the continuum AB percolation model is the graph with points of
$\Phi^{(1)}$ as vertices and with edges between any pair of points for which
the intersection of balls of radius $2r$ centred at these points contains at
least one point of $\Phi^{(2)}$. This is a generalization of the $AB$
percolation model on discrete lattices. We show the existence of percolation
for all $d > 1$ and derive bounds for a critical intensity. We also provide a
characterization for this critical intensity when $d = 2$. To study the
connectivity problem, we consider independent Poisson point processes of
intensities $n$ and $cn$ in the unit cube. The $AB$ random geometric graph is
defined as above but with balls of radius $r$. We derive a weak law result for
the largest nearest neighbour distance and almost sure asymptotic bounds for
the connectivity threshold.

32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ