Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariance under quasi-isometries of subcritical and supercritical
  behaviour in the Boolean model of percolation   In this work we study the Poisson Boolean model of percolation in locally
compact Polish metric spaces and we prove the invariance of subcritical and
supercritical phases under mm-quasi-isometries. In other words, we prove that
if the Poisson Boolean model of percolation is subcritical or supercritical (or
exhibits phase transition) in a metric space M which is mm-quasi-isometric to a
metric space N, then these phases also exist for the Poisson Boolean model of
percolation in N. Then we apply these results to understand the phenomenon of
phase transition in a large family of metric spaces. Indeed, we study the
Poisson Boolean model of percolation in the context of Riemannian manifolds, in
a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the
existence of a subcritical phase in Gromov spaces with bounded growth at some
scale.

9
The categories of this paper are: nlin
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th