Correlation functions are useful whenever we want to focus on n < N particles and "average out" the remaining particles. Indeed, suppose is a symmetric function of . Then

Thus

(93.1)

where

(93.2)

is the symmetric extension of to N variables.

Suppose and let be small. Let be the characteristic function of the (disjoint) sets . Let

,

clearly F is symmetric.

Then we have from (93.2)

Remark 94+ (where )

Now as , . Hence

Prob (exactly 1 eigenvalue in each of the intervals )

---

Remark 94+

Note: In the case of RMT, only has physical meaning, even though is symmetric in the s, only when . Indeed, remember that the map always specifies the eigenvalues in some order, in particular, .

When we compute the expectation for some quantity which is symmetric in , we have

(94+.1)

.

However as a computational convenience, we observe that

(94+.2)

.

Although (94+.2) is easier to manipulate, when we want to understand the meaning of the statistic , we must refer to (94+.1).

---

Thus

is the density of the probability that is one eigenvalue at each of the points .

Note the following:

If , the characteristic function of

Thus by (93.1)

(95.1)

Bearing (94+) in mind, we also have for random matrix ensembles,

(95.2)

.

Also if are two disjoint sets in and

,

then

Thus

(96.1)

(# {pairs ( ), : either and or and })

Remark 96+

Exercise: Show how (96.1) changes if .

We now show how to compute using

(89.1)

where and K is given in (88.2).

---

Remark 96+

Again bearing (94+) in mind, we have for random matrix ensembles

Now suppose for definiteness that lies to the left of

then for

= # { ordered pairs of eigenvalues, such that }.

Hence

(96+.1)

{ # { ordered pairs of eigenvalues, , such that }}

.

---

More explicitly for any ,

(97.0)

where

Choose g such that

for some k, where are the characteristic functions of disjoint Borel sets , in , such that .

Here .

Clearly,

(97.1)

.

For any , let

(97.2)

denote the elementary symmetric function and set .

We have

(98.1)

thus

(by symmetry)

Remarks 98+, 98++

Substituting (98++.2) for we find (exercise: see ref (3) p. 87)

(98.2)

---

Remark 98+

Now

(98+.1)

where

(98+.2)

Consider, for example, the case where k=5 and j=6 and and with arranged as follows

Now clearly if and only if .

---

Remark 98++

In particular only one term in (98+.2) contributes. We conclude that

(98++.1)

Thus

(98++.2)

---

where .

On the other hand, by the Fredholm expansion of a determinant,

.

Here we have used the fact that if j > N (why?).

Again expanding out g(x) using (98++.2), we find as above

(99.1)

.

Equating (98.2) and (99.1), and comparing coefficients, we find in particular for

(100.1)

where

As is symmetric, (100.1)

(100.2)

.

Let be disjoint intervals ordered from the left i.e.,

and inserting these s into (100.2) and letting , we obtain

for all and hence for all by symmetry. We conclude that for ,

(100.3)

Exercise: Use the above calculations to rederive (95.1).

Remark: For other proofs of this result see ref 3 p. 96-98 and also ref 2 p. 103-108 (this calculation is taken from [Meh]).

Remark 101+

The above calculations show that in order to evaluate key eigenvalue statistics for Unitary Ensembles we must understand the asymptotic behavior of the correlation kernel

where , and the s are orthonormal w.r.t. the weight ,

.

Thus the problem of the asymptotics of eigenstatistics reduces, for Unitary Ensembles to the classical problem of the asymptotics of orthogonal polynomials (OPs).

---

Remark 101+:

We now compute

Prob { eigenvalues in , ..., eigenvalues in }

where again the s are disjoint and .

Set and set .

Again letting be the characteristic function of , we have, using (98++.1)

Prob { eigenvalues in , ..., eigenvalues in }

now for

we have for

where we have used (98+.1) with j = N

.

Thus

Prob { eigenvalues in , ..., eigenvalues in }

Setting

and hence at ,

where .

It follows now from (89.1)

where K is the correlation kernel in (88.2)

.

Thus, finally,

(101+++.1)

Prob { eigenvalues in , ..., eigenvalues in }

.

---

(Ref: , "Orthogonal Polynomials").

For the next couple of lectures we will consider this problem. A key object that controls the asymptotics of OPs is the so-called equilibrium measure (see Ref 2, Chap 6; see also Saff and Totik "Logarithmic potentials with external fields" for the general theory).

We will see eventually that this quantity is intimately related to the density of states for RMT and also to the one-point correlation function . The calculations below are taken from ref (2), which in turn are based on work of K. Johansson (see ref (2)).

In the calculations that follow we will always assume that the probability density varies with N in the following way

(102.1)

.

As all our calculations so far have assumed that N is given and fixed, they all remain valid: we must just set

.

After integrating out the eigenvectors we obtain as before a probability measure on the eigenvalues

where (after symmetrizing)

(103.1)

where .

Let

(103.2)

be the normalized counting measure for the eigenvalues, and note that can be expressed as follows:

(104.1)

Note that the scaling in the potential

is chosen so that the terms in (104.1) are balanced.

Intuitively, the leading contribution to the partition function (103.1) as comes from those s for which is a minimum. Thus we are led to consider the following energy minimization problem

(104.2)

where and = { is a Borel measure on }.

We will show eventually that (103.2) has a unique minimizer , the equilibrium measure mentioned above. From the definition of , has an electrostatic interpretation: it is the equilibrium configuration for electrons with logarithmic electrostatic repulsion

in an external field . As already indicated, is intimately related to a variety of problems in RMT, and also in analysis. The existence and uniqueness of the solution of the variational problem (104.2) relies ultimately on the fact that we are dealing with a (constrained) convex minimization problem.

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