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Our derivation of (2.4)-(2.8) uses a recent result of Nagao and Slevin [8]
to obtain (2.4), and the theory of Tracy and Widom [9] to obtain (2.7). Nagao
and Slevin consider the random matrix ensemble with unitary symmetry
defined by the eigenvalue
p.d.f.
They prove that in the thermodynamic limit, with each scaled
so that the bulk density is , the
corresponding **n**-particle distribution is given by
where is given by (2.5) (for , ,
in the denominator needs to be replaced by , however below
we will only consider the case ).

It follows by substituting (2.10) into the general formula
where denotes the probability of an interval
being free of eigenvalues in the ensemble (2.9), that
Since in the case **b=1** (2.9) is precisely the eigenvalue
probability density function of the GUE with an eigenvalue fixed at the
origin, the result (2.4) follows.

The derivation of (2.7) relies on a set of coupled equations for quantities
associated with the integral operator . To present these equations, the
particular quantities must first be defined.
** Definition 2.1**

Suppose **A** is an integral operator on the interval with kernel
:

We write
to specify that the kernel of **A** is . Denote by **K** an integral
operator of this type with kernel of the form
and write
Also let
where **k=0,1** and **j=1,2**.
The coupled equations which imply (2.7) can now be stated.

** Proposition 2.1**

Consider the kernel (2.5) with on so that in the
setting of Definition 2.1 and . With the notation
, , we have

where the dashes denote differentiation with respect to **t**.
The theory of Tracy and Widom [9] allows equations for the quantities of
Definition 2.1 to be derived which imply the equations of Proposition 2.1.
Before presenting these equations let us show how (2.6) and
(2.7) can be derived from
the equations of Proposition 2.1.
First consider (2.7). We multiply (ii) by **p**, multiply (iii) by **q**, add and use
(v) to obtain

and consequently
Substituting (3.2) and (iv) in (i) gives
which relates **tR** to **pq**. On the other hand, another equation relating these
two quantities is obtained by squaring (iv) and the first equality in (2.12)
and subtracting:
Solving (2.14) for **pq** (it follows from a
small-**t** expansion that the negative square root is to be taken) and ,
substituting in (2.15) and introducing the notation
gives (2.7).
The boundary condition (2.8) follows from the fact that as and the corresponding behaviour of
deduced from (7).
To derive (2.6) we simply substitute (2.16) in (vi) and integrate. The factor
in the upper terminal of (2.6) results from changing
the mean eigenvalue spacing from to .

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