Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological embeddings into random 2-complexes   We consider 2-dimensional random simplicial complexes $Y$ in the
multi-parameter model. We establish the multi-parameter threshold for the
property that every 2-dimensional simplicial complex $S$ admits a topological
embedding into $Y$ asymptotically almost surely. Namely, if in the procedure of
the multi-parameter model, each $i$-dimensional simplex is taken independently
with probability $p_i=p_i(n)$, from a set of $n$ vertices, then the threshold
is $p_0 p_1^3 p_2^2 = \frac{1}{n}$. This threshold happens to coincide with the
previously established thresholds for uniform hyperbolicity and triviality of
the fundamental group.
  Our claim in one direction is in fact slightly stronger, namely, we show that
if $p_0 p_1^3 p_2^2$ is sufficiently larger than $\frac{1}{n}$ then every $S$
has a fixed subdivision $S'$ which admits a simplicial embedding into $Y$
asymptotically almost surely. The main geometric result we prove to this end is
that given $\epsilon>0$, there is a subdivision $S'$ of $S$ such that every
subcomplex $T \subseteq S'$ has $\frac{f_0(T)}{f_1(T)}>\frac{1}{3}-\epsilon$
and $\frac{f_0(T)}{f_2(T)}>\frac{1}{2}-\epsilon$, where $f_i(T)$ denotes the
number of simplices in $T$ of dimension $i$.
  In the other direction we show that if $p_0 p_1^3 p_2^2$ is sufficiently
smaller than $\frac{1}{n}$, then asymptotically almost surely, the torus does
not admit a topological embedding into $Y$. Here we use a result of Z. Gao
which bounds the number of different triangulations of a surface.

19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: physics