Fall 2009 MATH 833 – Random Matrices B.

Valkó

Lecture 1 : Basic random matrix models

Notes prepared by: B. Valkó

Our aim in this course to study the asymptotic behavior of the spectrum of certain random matrices.

Wigner Matrices

Definition 1 (real Wigner matrices). For 1 ≤ i < j < ∞ let Xi,j be i.i.d. (real) random variables
with mean 0 and variance 1 and set Xj,i = Xi,j . Let Xi,i be i.i.d. (real) random variables (with
possibly a different distribution) with mean 0 and variance 1. Then Mn = [Xi,j ]ni,j=1 will be a
random n × n symmetric matrix.
Definition 2 (complex Wigner matrices). For 1 ≤ i < j < ∞ let Xi,j be i.i.d. (complex) random
variables with mean 0, E|Xi,j |2 = 1 and set Xj,i = Xi,j . Let Xi,i be i.i.d. (real) random variables
with mean 0 and variance 1. Then Mn = [Xi,j ]ni,j=1 will be a random n × n hermitian matrix.

In both cases there are n random eigenvalues which we will denote by

λ1 ≤ λ2 ≤ . . . λn .

(We will denote the dependence on n). Fact (which we will prove later): these are continuous
functions of Mn hence they are random variables themselves.

We would like to study the scaling limit of the empirical spectral measure
n
1X
νn∗ = δ λi .
n
i=1

This is a random discrete probability measure which puts n−1 mass to each (random) eigenvalue.
The following picture shows the histogram of eigenvalues for a certain 200 × 200 Wigner matrix.

14

12

10

-20 -10 0 10 20

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The picture suggests that there is a nice deterministic limiting behavior. In order to figure out
the right scaling, we first compute the order of the empirical mean and second moment of the
eigenvalues.
n n
1X 1 1X
λi = TrMn = Xi,i
n n n
i=1 i=1
n n
1X 1 1 X
λi = TrMn2 = 2
Xi,j
n n n
i=1 i,j=1

The first moment converges to 0 by the strong law of large numbers. However the second moment
is of O(n) as we have about n2 /2 independent terms in the sum with a normalization of n1 instead
of n12 . This suggests that in order to see a meaningful limit, we need to scale the eigenvalues (or
the matrix) by √1n .

The following theorem states that in case we indeed have a deterministic limit.
Theorem 3 (Wigner’s semicircle law). Let
n
1X
νn = δ √λi .
n n
i=1

be the normalized empirical spectral measure. Then as n → ∞ we have

νn ⇒ ν a.s.
where ν has density
dν 1 p
= 4 − x2 1{|x| ≤ 2}.
dx 2π
(There willl be some assumptions on the distribution of the random entries of Mn . . . )

Gaussian Ensembles

We also discussed some special Wigner matrix models.

√
Definition 4 (GOE). Consider a real Wigner matrix where Xi,j ∼ N (0, 1) and Xi,i ∼ 2N (0, 1).
The resulting random matrix model is called Gaussian Orthogonal Ensemble (or GOE).
Another construction: let ai,j , i, j ∈ Z be i.i.d. standard normals and An = [ai,j ]ni,j=1 . (Note that
An +AT
this is not a symmetric matrix!). Then the distribution of Mn = √ n
2
is GOE.

It is easy to check the following useful fact: if C ∈ Rn×n is orthogonal (i.e. CC T = I) the C T Mn C
has the same distribution as Mn . (The GOE is invariant to orthogonal conjugation.) It is a bit
harder (we will prove it later) that one can actually compute the joint eigenvalue density which is
given by
1 Y 1 Pn 2
f (λ1 , . . . , λn ) = |λj − λi |e− 4 i=1 λi .
Z1
i<j

Here Z1 is an explicitly computable normalizing constant (which also depends on n).

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Definition 5 (GUE). Consider a complex Wigner matrix where Xi,j is standard complex Gaussian
(i.e. Xi,j ∼ N (0, 12 ) + iN (0, 12 )) and Xi,i ∼ N (0, 1) (real). The resulting random hermitian matrix
model is called Gaussian Unitary Ensemble (or GUE).
Another construction: let ai,j , i, j ∈ Z be i.i.d. standard complex Gaussians and An = [ai,j ]ni,j=1 .
An +A∗n
(Note that this is not a symmetric matrix!). Then the distribution of Mn = √
2
is GUE.

As the name suggests, GUE is invariant under unitary conjugation. If C ∈ Cn×n is unitary
(i.e. CC ∗ = I) the C T Mn C has the same distribution as Mn . (We will later show that the joint
eigenvalue density is given by
1 Y 1 Pn 2
f (λ1 , . . . , λn ) = |λj − λi |2 e− 2 i=1 λi .
Z2
i<j

Here Z2 is an explicitly computable normalizing constant (which also depends on n).

One can see the similarity between the two densities: they are contained in the following one-
parameter family of densities:
1 Y β Pn 2
f (λ1 , . . . , λn ) = |λj − λi |β e− 4 i=1 λi . (1)
Zβ
i<j

For a given β > 0 the resulting distribution (on ordered n-tuples in R) is called Dyson’s β-ensemble.
For β = 1 one gets the eigenvalue density of GOE, for β = 2 we get the GUE. The β = 4 case is also
special: it is related another classical random matrix model, the Gaussian Symplectic Ensemble
(GSE), which can be defined using quaternions.

For other values of β there are no ’nice’ random matrices in the background. (We will see that one
can still build random matrices from which we get the general β-ensemble, but they won’t have
such nice symmetry properties.)

Later in the semester we will show that if one scales the β ensembles properly (’zooming in’ to see
the individual eigenvalues near a point) then one gets a point process limit. The limiting point
process is especially nice in the β = 2 case (GUE). It is conjectured that its distribution appears
among the critical line zeros of the Riemann-ζ function.

Another symmetric random matrix model

Another way of constructing a symmetric random matrix is the following. Let ai,j be i.i.d random
real random variables with mean 0 and variance 1. Let A = [ai,j ]n,m
i,j=1 ∈ R
n×m be a random matrix

with n rows and m columns (with n ≤ m). Then Mn = AAT is a (positive) symmetric random
matrix. We will show that the appropriately normalized empirical spectral measure will converge
to a deterministic limit. (This is the Marchenko-Pastur law). A similar statement holds if we
construct our matrix from i.i.d. complex random variables.