Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Clustering, percolation and directionally convex ordering of point
  processes   Heuristics indicate that point processes exhibiting clustering of points have
larger critical radius $r_c$ for the percolation of their continuum percolation
models than spatially homogeneous point processes. It has already been shown,
and we reaffirm it in this paper, that the $dcx$ ordering of point processes is
suitable to compare their clustering tendencies. Hence, it was tempting to
conjecture that $r_c$ is increasing in $dcx$ order. Some numerical evidences
support this conjecture for a special class of point processes, called
perturbed lattices, which are "toy models" for determinantal and permanental
point processes. However, the conjecture is not true in full generality, since
one can construct a Cox point process with degenerate critical radius $r_c=0$,
that is $dcx$ larger than a given homogeneous Poisson point process.
Nevertheless, we are able to compare some nonstandard critical radii related,
respectively, to the finiteness of the expected number of void circuits around
the origin and asymptotic of the expected number of long occupied paths from
the origin in suitable discrete approximations of the continuum model. These
new critical radii sandwich the "true" one. Surprisingly, the inequalities for
them go in opposite directions, which gives uniform lower and upper bounds on
$r_c$ for all processes $dcx$ smaller than some given process. In fact, the
above results hold under weaker assumptions on the ordering of void
probabilities or factorial moment measures only. Examples of point processes
comparable to Poisson processes in this weaker sense include determinantal and
permanental processes. More generally, we show that point processes $dcx$
smaller than homogeneous Poisson processes exhibit phase transitions in certain
percolation models based on the level-sets of additive shot-noise fields, as
e.g. $k$-percolation and SINR-percolation.

31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex