Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp hierarchical upper bounds on the critical two-point function for
  long-range percolation on $\mathbb{Z}^d$   Consider long-range Bernoulli percolation on $\mathbb{Z}^d$ in which we
connect each pair of distinct points $x$ and $y$ by an edge with probability
$1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta\geq
0$ is a parameter. We prove that if $0<\alpha<d$ then the critical two-point
function satisfies \[ \frac{1}{|\Lambda_r|}\sum_{x\in \Lambda_r}
\mathbf{P}_{\beta_c}(0\leftrightarrow x) \preceq r^{-d+\alpha} \] for every
$r\geq 1$, where $\Lambda_r=[-r,r]^d \cap \mathbb{Z}^d$. In other words, the
critical two-point function on $\mathbb{Z}^d$ is always bounded above on
average by the critical two-point function on the hierarchical lattice. This
upper bound is believed to be sharp for values of $\alpha$ strictly below the
crossover value $\alpha_c(d)$, where the values of several critical exponents
for long-range percolation on $\mathbb{Z}^d$ and the hierarchical lattice are
believed to be equal.

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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
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The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph