2.2 Derivation

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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

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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