Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Supercritical percolation on graphs of polynomial growth   We consider Bernoulli percolation on transitive graphs of polynomial growth.
In the subcritical regime ($p<p_c$), it is well known that the connection
probabilities decay exponentially fast. In the present paper, we study the
supercritical phase ($p>p_c$) and prove the exponential decay of the truncated
connection probabilities (probabilities that two points are connected by an
open path, but not to infinity). This sharpness result was established by
[CCN87] on $\mathbb Z^d$ and uses the difficult slab result of Grimmett and
Marstrand. However, the techniques used there are very specific to the
hypercubic lattices and do not extend to more general geometries. In this
paper, we develop new robust techniques based on the recent progress in the
theory of sharp thresholds and the sprinkling method of Benjamini and Tassion.
On $\mathbb Z^d$, these methods lead to a completely new proof of the slab
result of Grimmett and Marstrand.

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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: quant-ph