Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A sharp threshold for collapse of the random triangular group   The random triangular group $\Gamma(n,p)$ is the group given by a random
group presentation with $n$ generators in which every relator of length three
is present independently with probability $p$. We show that in the evolution of
$\Gamma(n,p)$ the property of collapsing to the trivial group admits a very
sharp threshold.

3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc