I have a sequence defined as $\exp((\ln n)^{2})$. As $n \rightarrow \infty$, this sequence diverges.
What can I say about its rate of divergence? It is faster than any polynomial rate in $n$, but is there a better, more informative expression for it?
Similary, if the sequence were $\exp((\ln n)^{1/2})$. This should be slower than any polynomial rate, but again is there more to be said about this sequence?
$\exp((\ln n)^{2})$ is an example of quasipolynomial growth. This grows faster than polynomial but not as fast as exponential.
The general quasipolynomial form is $\exp(f(\ln n))$ for some polynomial $f(x)$.
But for $\exp((\ln n)^{1/2})$ you get sublinear growth because this is $o(n)$. But it's not as good as polylogarithmic growth which takes a form $f(\ln n)$ for a polynomial $f(x)$.
You can read more on the Wikipedia article: https://en.wikipedia.org/wiki/Time_complexity