Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Slightly supercritical percolation on nonamenable graphs I: The
  distribution of finite clusters   We study the distribution of finite clusters in slightly supercritical ($p
\downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs,
proving in particular that if $G$ is a transitive nonamenable graph satisfying
the $L^2$ boundedness condition ($p_c<p_{2\to 2}$) and $K$ denotes the cluster
of the origin then there exists $\delta>0$ such that $$ \mathbf{P}_p(n \leq |K|
< \infty) \asymp n^{-1/2} \exp\left[ -\Theta \Bigl( |p-p_c|^2 n\Bigr) \right]
$$ and \[ \mathbf{P}_p(r \leq \operatorname{Rad}(K) < \infty) \asymp r^{-1}
\exp\left[ -\Theta \Bigl( |p-p_c| r\Bigr) \right] \] for every $p\in
(p_c-\delta,p_c+\delta)$ and $n,r\geq 1$, where all implicit constants depend
only on $G$. We deduce in particular that the critical exponents $\gamma'$ and
$\Delta'$ describing the rate of growth of the moments of a finite cluster as
$p \downarrow p_c$ take their mean-field values of $1$ and $2$ respectively.
  These results apply in particular to Cayley graphs of nonelementary
hyperbolic groups, to products with trees, and to transitive graphs of spectral
radius $\rho<1/2$. In particular, every finitely generated nonamenable group
has a Cayley graph to which these results apply. They are new for graphs that
are not trees. The corresponding facts are yet to be understood on
$\mathbb{Z}^d$ even for $d$ very large. In a second paper in this series, we
will apply these results to study the geometric and spectral properties of
infinite slightly supercritical clusters in the same setting.

7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int