The diameter of random Cayley digraphs of given degree
We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k...
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| creator | Potočnik, Primož Širáň, Jozef Šiagiová, Jana Lladser, Manuel E Wilson, Mark C |
| description | We consider random Cayley digraphs of order $n$ with uniformly distributed
generating set of size $k$. Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as $n\to\infty$ and
$k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n
\log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the
probability converges exponentially fast to 1. |
| doi_str_mv | 10.48550/arxiv.0706.3539 |
| format | Article |
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generating set of size $k$. Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as $n\to\infty$ and
$k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n
\log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the
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generating set of size $k$. Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as $n\to\infty$ and
$k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n
\log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the
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generating set of size $k$. Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as $n\to\infty$ and
$k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n
\log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the
probability converges exponentially fast to 1.</abstract><doi>10.48550/arxiv.0706.3539</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | The diameter of random Cayley digraphs of given degree |
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