Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Freiheitssatz and phase transition for the density model of random
  groups   Magnus' Freiheitssatz states that if a group is defined by a presentation
with $m$ generators and a single relator containing the last generating letter,
then the first $m-1$ letters freely generate a free subgroup. We study an
analogue of this theorem in the Gromov density model of random groups, showing
a phase transition phenomenon at density $d_r = \min\{\frac{1}{2},
1-\log_{2m-1}(2r-1)\}$ with $1\leq r\leq m-1$: we prove that for a random group
with $m$ generators at density $d$, if $d < d_r$ then the first $r$ letters
freely generate a free subgroup; whereas if $d > d_r$ then the first $r$
letters generate the whole group.

37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol