Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bootstrap percolation on the product of the two-dimensional lattice with
  a Hamming square   Bootstrap percolation on a graph is a deterministic process that iteratively
enlarges a set of occupied sites by adjoining points with at least $\theta$
occupied neighbors. The initially occupied set is random, given by a uniform
product measure with a low density $p$. Our main focus is on this process on
the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph.
We investigate how $p$ scales with $n$ so that a typical site is eventually
occupied. Under critical scaling, the dynamics with even $\theta$ exhibits a
sharp phase transition, while odd $\theta$ yields a gradual percolation
transition. We also establish a gradual transition for bootstrap percolation on
$\mathbb{Z}^2\times K_n$. The main tool is heterogeneous bootstrap percolation
on $\mathbb{Z}^2$.

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The categories of this paper are: bayes-an
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The categories of this paper are: econ
20
The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph