More generally let X and Y be Banach spaces and let be bounded linear operators. Then the following commutation formula is true:

(75.1)

in the sense that of = resolvent set of BA, then and

is the resolvent of AB, and vice versa. In particular, we see that (74.1) is true for general bounded operators in Banach spaces. (75.1) is also true, suitably interpreted, for certain classes of unbounded operators (see Deift [D. M. Jour, 1978]).

We now prove (71.2): Let and be given, then

as desired.

The particular case

for a given set of numbers , then the LHS of (71.2) becomes

On the other hand, the RHS of (71.2) gives

Equating these expressions we see that

if

and

we obtain the classical Cauchy-Binet formula

(79.1)

e.g. if

then

Of course if N = M, (79.1) is just the familiar fact that

and for N > M, both LHS and RHS are 0 (why?)

We will also need the following basic definition.

Let be a Borel measure on with all moments finite,

(80.1)

.

Suppose that the support of is infinite, i.e., is not a finite linear combination of delta functions, then by the Gram-Schmidt procedure there exist unique monic polynomials

which are orthogonal with respect to i.e.,

for .

For

,

set

.

The polynomials {} are the orthonormal polynomials with respect to , i.e.,

The s are called the orthonormal polynomials (w.r.t. ) and the s are called the monic orthogonal polynomials (w.r.t. ).

Exercise: what happens if has finite support?

As we will see, the s and s play a central role in RMT.

---

We now show how to compute certain basic statistics for invariant unitary ensembles with probability distributions

where sufficiently rapidly (see 84.x - 85.x below) as .

In particular, we will compute

for f(M) of the form

(82.0)

for bounded functions .

From the Hermitized analog of (68.1) we have

(82.1)

where

(82.2)

(82.3)

VanderMonde =

and

(82.4)

As the integrands in (82.1) and (82.4) are invariant under all permutation

,

we may rewrite (82.1) (82.4) in the following symmetrized form

(83.1)

where

(83.2)

Now define

(83.3)

then

Hence we can apply the generalized Cauchy-Binet formula to (83.1) with measure (the fact that may be a signed measure, as opposed to a positive measure, clearly does not affect the proof of (83.1)).

We obtain

where we note from (82.4) that depends only on N and , but not on g(x).

We have

(84.1)

.

Let be the orthogonal polynomials with respect to the measure i.e.,

.

(Note that does not have finite support; also by "sufficient decay" we are assuming, at least, that has finite moments; note that this implies

so that is a normalized probability measure with finite moments).

Now as we can add rows and columns without changing a determinant, we have (recall )

(85.1)

Set

(85.2)

have

then (85.1) takes the form

(85.3)

where again does not depend on . Setting g = 0, we have f(M) = 1 and so LHS of (85.3) = 1, but RHS = , so =1.

Thus

(86.1)

Now we are primarily interested in the situation where the size N of the matrices become large. We see from (86.1) that <f> is expressed in terms of determinants of larger and larger size. Limits of this kind are generally very difficult to control. Fortunately, we can use (71.1) to reduce (86.1) to the (Fredholm) determinant on a fixed space (see below). Such limits are, generally speaking, easier to control; in particular if is a trace class operator on a (fixed) Hilbert space H, and is trace norm,

then

It is just a matter of continuity of the determinant in the trace norm.

Now let denote the bounded operator

(87.1)

and let denote the bounded operator

(87.2)

then AB maps and for

(87.3)

and we see from (86.1) that

(88.1)

On the other hand, and for

where

(88.2)

is the so-called correlation kernel for the ensemble: K(x, y) plays a central role in RMT.

Thus

(88.3)

where denotes multiplication by g(x) and K denotes the operator with kernel K i.e., .

Assembling the above results we obtain the key formula

(89.1)

Note that as A (and also B!) is finite rank, A is trace class and the Fredholm determinant in (89.1) is well-defined. There are analogous, but more complicated, formulae for and (see Ref (3)).

If is a Borel set in and

then we see from (89.1) that the gap probability considered before is given by

(89.2)

Prob (no eigenvalues in )

, = characteristic function of

Note that if , then RHS = 1, i.e. Prob (no eigenvalues in ) = 1, as it should be. On the other hand for , Prob (no eigenvalues in ) = 0.

On the other hand, for any

Thus 1 is an eigenvalue of K (of multiplicity N > 0) and so det(I - K) = 0, as it should be.

Now take for any , then clearly

Prob (no eigenvalues in ) = Prob ( )

where is the largest eigenvalues of M. Thus

(90.1)

Prob ( ) =

This is a key formula in RMT for unitary ensembles.

Exercise

If is a union of disjoint sets in , for , then we have computed

Prob (0 eigenvalues in , ..., 0 eigenvalues in ) = Prob (no eigenvalues in ).

Derive an explicit formula for

Prob ( eigenvalues in , ..., eigenvalues in )

for any non-negative integers . (see ref (3) pp 86-88).

We now show how to compute correlation functions for unitary ensembles (recall that 2-point correlation functions arose in the study of the non-trivial zeros of the Riemann zeta function in the first lecture).

If is the probability distribution for a system of N identical random particles

symmetric in the s

then the n-point correlation function

for is defined by

(92.1)

Note that and hence is not a probability distribution (more later!).

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