Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Isoperimetric lower bounds for critical exponents for long-range
  percolation   We study independent long-range percolation on $\mathbb{Z}^d$ where the
vertices $x$ and $y$ are connected with probability
$1-e^{-\beta\|x-y\|^{-d-\alpha}}$ for $\alpha > 0$. Provided the critical
exponents $\delta$ and $2-\eta$ defined by $\delta = \lim_{n\to \infty}
\frac{-\log(n)}{\mathbb{P}_{\beta_c}\left(|K_0|\geq n\right)}$ and $2-\eta =
\lim_{x \to \infty} \frac{\log\left(\mathbb{P}_{\beta_c}\left(0\leftrightarrow
x\right)\right)}{\log(\|x\|)} + d$ exist, we show that
  \begin{equation*}
  \delta \geq \frac{d+(\alpha\wedge 1)}{d-(\alpha\wedge 1)} \ \text{ and } \
2-\eta \geq \alpha \wedge 1 \text.
  \end{equation*} The lower bound on $\delta$ is believed to be sharp for $d =
1, \alpha \in \left[\frac{1}{3},1\right)$ and for $d = 2, \alpha \in
\left[\frac{2}{3},1\right]$, whereas the lower bound on $2-\eta$ is sharp for
$d=1, \alpha \in (0,1)$, and for $\alpha \in \left(0,1\right]$ for $d>1$, and
is not believed to be sharp otherwise. Our main tool is a connection between
the critical exponents and the isoperimetry of cubes inside $\mathbb{Z}^d$.

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The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph