  In this paper we analyze the length-spectrum of rainbows in RNA secondary
structures. A rainbow in a secondary structure is a maximal arc with respect to
the partial order induced by nesting. We show that there is a significant gap
in this length-spectrum. We shall prove that there asymptotically almost surely
exists a unique longest rainbow of length at least $n-O(n^{1/2})$ and that with
high probability any other rainbow has finite length. We show that the
distribution of the length of the longest rainbow converges to a discrete limit
law and that, for finite $k$, the distribution of rainbows of length $k$,
becomes for large $n$ a negative binomial distribution. We then put the results
of this paper into context, comparing the analytical results with those
observed in RNA minimum free energy structures, biological RNA structures and
relate our findings to the sparsification of folding algorithms.
