  The neighbourhood of the largest eigenvalue $\lambda_{\rm max}$ in the
Gaussian unitary ensemble (GUE) and Laguerre unitary ensemble (LUE) is referred
to as the soft edge. It is known that there exists a particular centring and
scaling such that the distribution of $\lambda_{\rm max}$ tends to a universal
form, with an error term bounded by $1/N^{2/3}$. We take up the problem of
computing the exact functional form of the leading error term in a large $N$
asymptotic expansion for both the GUE and LUE --- two versions of the LUE are
considered, one with the parameter $a$ fixed, and the other with $a$
proportional to $N$. Both settings in the LUE case allow for an interpretation
in terms of the distribution of a particular weighted path length in a model
involving exponential variables on a rectangular grid, as the grid size gets
large. We give operator theoretic forms of the corrections, which are
corollaries of knowledge of the first two terms in the large $N$ expansion of
the scaled kernel, and are readily computed using a method due to Bornemann. We
also give expressions in terms of the solutions of particular systems of
coupled differential equations, which provide an alternative method of
computation. Both characterisations are well suited to a thinned generalisation
of the original ensemble, whereby each eigenvalue is deleted independently with
probability $(1 - \xi)$. In the final section, we investigate using simulation
the question of whether upon an appropriate centring and scaling a wider class
of complex Hermitian random matrix ensembles have their leading correction to
the distribution of $\lambda_{\rm max}$ proportional to $1/N^{2/3}$.
