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We have computed many terms of the power series expansion of (2.7) about
**s=0** with **b=1** and subject to (2.8). Substitution of the first seven into
(2.4) gives
Comparison with the analogous expansion of (1.3a) (see e.g. [2]) shows that
, which in qualitative terms says that
very small spacings between consecutive eigenvalues will most likely be
nearest neighbour spacings (the factor of 1/2 accounts for the fact that the
nearest neighbour occurs with equal probability to the left or the right).

For the large-**t** expansion, we follow the corresponding analysis of
(see e.g. [7]) and seek a solution of the form
. Indeed (2.7) has a unique solution of this
form, which when integrated according to (2.6) implies
As is well known (see e.g. [7]) this leaves the overall multiplicative constant
**A** unspecified. The large-**t** expansion of is obtained by substituting
(3.2) in (2.4).
The solution of (2.7) with **b=1** was computed numerically by first calculating
the power series expansion of up to and using the
corresponding values of and as initial data in
the
Mathematica routine NDSolve.
The d.e. (2.7) was rewritten so that occured to
the first power (the negative square root is to be taken) and it was found
necessary to use a high precision setting (AccuracyGoal and PrecisionGoal = 20) to
get a stable solution in the interval of interest (**s < 13**).
Although the results of Section 2 are only exact in the limit , it is well known that can be accurately approximated by
considering matrices which give the so called Wigner surmise
(see e.g. [2]). This suggests that may also be insensitive to the
precise dimension of the GUE matrices. Assuming this, to test our exact
expression we have compared as calculated from (2.6) with
as determined empirically from 10,000 numerically generalted
matrices from the GUE (see
Figure 1
). The latter calculation was done using
Mathematica. For each matrix the eigenvalues were calculated
and the nearest neighbour spacing of the middle (8th) eigenvalue was
calculated. After scaling the spacings were tested to count how many fell
into the intervals , and the corresponding
empirical value of plotted at the points .

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