I have not yet been able to find a direct wave-mechanical explanation (which would bypass the above semiclassical estimate) for the quasi-orthogonality of dilation. However a clue is given by Berry and Wilkinson's proof [28] that degenerate () off-diagonal elements are exactly zero. An explanation for was first attempted by VS [195,194]. They tried to establish quasi-orthogonality using the identity , with . This identity can be proved in a simple fashion (Eq.(H.25)). However, they then made the assumption (for ) by claiming that eigenstates are uncorrelated across the volume of [194]. (In this would give variables whose variances add linearly). This would give a power law for the off-diagonal growth of .

This is in error for two reasons.
Firstly, we know that the number of degrees of freedom in a constant-wavenumber
function is actually
scaling like the boundary
(see Section 5.3.1), which would imply the naive random-wave estimate
.
Secondly, a random wave estimate of an overlap over a large fraction of the
region is generally very bad (Section 3.3.2).
In fact, comparison to our band profile results (analytical and numerical)
shows that
, a result which cannot
be guessed by random wave assumptions.
This result has also since been verified by Vergini ^{6.1}.

The conclusion is that the original authors' quasi-orthogonality estimate
(which they have used in [195,194,175,196,203])
was in fact pessimistic:
the true power law actually gives *much* smaller
elements close to the diagonal.