Threshold functions

There's no strict order on growth rates,
but we can say the following:
in probabilistic applications, a function $\rho$ is a -threshold-
function (is this a good letter??) iff
when $f \ll \rho$, the probability goes to $0$
when $f \gg \rho$, the probability goes to $1$

Note:
when $f$ is -equivalent- to $\rho$, the behaviour can be interesting;
when -incomensurable-, often nothing you can say (duh: could be junk)
...or sometimes can say something interesting, presumably

a priori not obvious that there should be a -unique- threshold
function... (at least to me).

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One big source of this is:
the -expected- number of (whatever) often follows a -Poisson-
distribution: asymptotically, you achieve (a.s.) the expected value.
Hence if the expected value depends on the function
(and goes to 0 or infinity (or even > 1), below/above threshold),
you get a threshold effect.

Behaviour around/at the threshold can be very complicated,
of course.

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For example, there's threshold behaviour in "randomness of card
shuffling" (Diaconis)
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Incidentally, I did an REU in asymptotic methods in probability -- I
have a soft spot in my heart for this stuff.
[b/c it -is- cool.]
