  We study random knots, which we define as a triple of random periodic
functions (where a random function is a random trigonometric series,
\[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\]
with $a_k, b_k$ are independent gaussian random variables with mean $0$ and
variance $\sigma(k)^2$ - our results will depend on the functional dependence
of $\sigma$ on $k.$ In particular, we show that if $\sigma(k) = k^\alpha,$ with
$\alpha < -3/2,$ then the probability of getting a knot type which admits a
projection with $N$ crossings, decays at least as fast as $1/N.$ The constant
$3/2$ is significant, because having $\alpha < -3/2$ is exactly the condition
for $f(\theta)$ to be a $C^1$ function, so our class is precisely the class of
random \emph{tame} knots. We also find some suprising experimental observations
on the zeros of Alexander polynomials of random knots (with slowly and
non-decaying coefficients), and even more surprising observations on their
coefficients. Our observations persist in other models of random knots, making
it likely that the results are universal.

Geometric Topology;Statistical Mechanics;Probability