Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local percolative properties of the vacant set of random interlacements
  with small intensity   Random interlacements at level u is a one parameter family of connected
random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the
vacant set at level u, exhibits a non-trivial percolation phase transition in
u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected
component, when it exists, is almost surely unique, see arXiv:0805.4106.
  In this paper we study local percolative properties of the vacant set of
random interlacements at level u for all dimensions d>=3 and small intensity
parameter u>0. We give a stretched exponential bound on the probability that a
large (hyper)cube contains two distinct macroscopic components of the vacant
set at level u. Our results imply that finite connected components of the
vacant set at level u are unlikely to be large. These results were proved in
arXiv:1002.4995 for d>=5. Our approach is different from that of
arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the
proof is a certain conditional independence property of the random
interlacements, which is interesting in its own right.

18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol