Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the dimensional crossover critical exponent   We consider independent anisotropic bond percolation on $\mathbb{Z}^d\times
\mathbb{Z}^s$ where edges parallel to $\mathbb{Z}^d$ are open with probability
$p<p_c(\mathbb{Z}^d)$ and edges parallel to $\mathbb{Z}^s$ are open with
probability $q$, independently of all others. We prove that percolation occurs
for $q\geq 8d^2(p_c(\mathbb{Z}^d)-p)$. This fact implies that the so-called
Dimensional Crossover critical exponent, if it exists, is greater than 1. In
particular, using known results, we conclude the proof that, for $d\geq 11$,
the crossover critical exponent exists and equals 1.

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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga