3.1 Solution of the non-linear equation

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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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