Many biological and biochemical measurements, e.g. the ``fitness'' of a
particular genome, or the binding affinity to a particular substrate,
can be treated as a ``fitness landscape'', an assignment of numerical
values to points in sequence space (or some other configuration space).
As an alternative to the enormous amount of data required to completely
describe such a landscape, we propose a statistical characterization,
based on the properties of a random walk through the landscape, and,
more specifically, its autocorrelation function. Under assumptions
roughly satisfied by two classes of simple model landscapes (the $N$-$k$
model and the $p$-spin model) and by the landscape of estimated free
energies of RNA secondary structures, this autocorrelation function,
along with the mean and variance of individual points and the size of
the landscape, completely characterize it. Having noted that these and
other landscapes of estimated replication and degradation rates all
have a well defined correlation length, we propose a classification of
landscapes depending on how the correlation length scales with the
diameter of the landscape.
The landscapes of some of the kinetic parameters of RNA molecules scale
similarly to the model landscapes introduced into
evolutionary studies from other fields, such as quadratic spin glasses and the
traveling salesman problem, but the correlation length of RNA landscapes
are considerably smaller.
Nevertheless, both the model and some of the RNA landscapes satisfy a
test of self-similarity proposed by Sorkin (1989).