Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Anisotropic bootstrap percolation in three dimensions   Consider a $p$-random subset $A$ of initially infected vertices in the
discrete cube $[L]^3$, and assume that the neighbourhood of each vertex
consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each
$i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy
vertex $x\in [L]^3$ already having $a_3+1$ infected neighbours, and that
infected sites remain infected forever. In this paper we determine the critical
length for percolation up to a constant factor in the exponent, for all triples
$(a_1,a_2,a_3)$. To do so, we introduce a new algorithm called the beams
process and prove an exponential decay property for a family of subcritical
two-dimensional bootstrap processes.

20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol