Problem Set
Math 8660: Random Matrices
April 14, 2014
1. Let X be an n × n matrix with i.i.d. entries coming from a distribution µ. We assume that the
distribution has mean 0 and variance 1, and it is compactly supported.
(a) Show that for each fixed k ∈ N, as n → ∞,
i
h
1
T k
→ Ck ,
E
tr(XX
)
nk+1
2k
1
th
where Ck = k+1
Catalan number.
k is the k
(b) Conclude that the expected ESD L̄n := E[Ln ] of the matrix n−1 XX T converges in distribution to
some probability measure ν. Identify ν. Hint: ν is a simple transformation of Wigner’s semicircle
law.
2. Let Mn be a random matrix of size n whose entries are i.i.d. symmetric ±1 Bernoulli random variables.
In this exercise, we will give a relatively simple proof that P(det(Mn ) = 0) = o(1). Let X1 , X2 , . . . , Xn
be the columns of Mn .
(a) Bound
P(det(Mn ) = 0) ≤
n
X
P(Xk ∈ Hk ),
(1)
k=1
where Hk is the subspace spanned by the vectors X1 , X2 , . . . , Xk−1 .
(b) Let X be chosen uniformly at random from {−1, 1}n . Show that (done in class) for any fixed
subspace W ⊂ Rn of dimension d, P(X ∈ W ) ≤ 2−(n−d) . Use this to bound all but the last
O(log log n) terms in the sum in (1).
(c) Call that a set S ⊆ {−1, 1}n is `-universal if for any set of ` different indices 1 ≤ i1 < i2 < . . . <
i` ≤ n, we have {πI (v) : v ∈ S} = {−1, 1}` . Here πI : {−1, 1}n → {−1, 1}k is the usual projection
onto the coordinates i1 , i2 , . . . , i` , that is, πI (v) = (vi1 , . . . , vi` ).
Prove that for k ≥ n/2, with probability at least 1 − n−1 , Hk is `-universal with ` = log n/10.
(d) From (c) and Erdös-Littlewood-Offord lemma, deduce that for k ≥ n − O(log log n),
P(Xk ∈ Hk ) ≤ n−1 + O(log−1/2 n).
(e) Conclude that P(det(Mn ) = 0) = O(log−1/3 n).
3. Use Laplaces method to prove : as s → ∞ along the positive real axis,
Z ∞
√
Γ(s) =
xs−1 e−x dx ∼ 2πe−s ss−1/2 .
0
This recovers, in particular, the Stirling’s formula.
4. Let µ be a probability measure on R with infinite support. Suppose that p0 , p1 , p2 , . . . are orthogonal
polynomials in L2 (µ) such that pk (x) = ck xk + . . . with ck > 0. Prove that
n−1
X
k=0
pk (x)pk (y) =
cn−1 pn (x)pn−1 (y) − pn−1 (x)pn (y)
·
.
cn
x−y
1
5. Let G be an n×n matrix with i.i.d. standard complex Gaussian entries. The eigenvalues (put in random
exchangeable order) of G have joint density
!
n
X
Y
1
2
Q
exp
−
|z
|
|zi − zj |2 ,
k
n
π n i=1 k!
i<j
k=1
n
with respect to the Lebesgue measure on C . Take this fact as granted.
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(a) Show that the eigenvalues of G form a determinantal point process on (C, π −1 e−|z| dz) with kernel
Kn (z, w) =
n−1
X
zk
φk (z) = √ .
k!
φk (z)φk (z),
k=0
(b) Let Ln and L̄n be the ESD and the expected ESD of n−1/2 G respectively. Write down the density
d
of L̄n with respect to Lebesgue measure on C. Show that L̄n → π −1 1{|z|≤1} dz. The limiting
distribution is known as circular law.
(c) Prove that for each continuous bounded function f ,
Z
E
4
Z
f dLn (z) −
f dL̄n (z)
= O(n−2 ).
(Note that to compute the LHS above, you need to know m-correlation functions of the eigenvalues
of G for m ≤ 4.) Deduce that almost surely,
d
Ln → π −1 1{|z|≤1} dz.
6. Let G ∼ G(n, p) be an Erdös-Rényi random graph. Assume that 0 < p = p(n) ≤ 1/2. Let A be
A. Imitate the proof of Wigner’s semicircle
its adjacency matrix and let Ln be the ESD of √ 1
np(1−p)
law using Stieltjes transform method to show that almost surely, Ln converges in distribution to the
semicircle law if p = p(n) satisfies the condition np → ∞.
7. Let G ∼ G(n, 1/2) be an Erdös-Rényi random graph and let A be the corresponding adjacency
√ matrix.
Set Ā = A − E[A]. The goal of this exercise to show that kĀk := supx:kxk2 =1 |xT Āx| = O( n) with
high probability.
(a) Use Azuma-Hoeffding inequality to show that for a fixed unit vector x ∈ Rn ,
2
PĀ |xT Āx| > t ≤ 2e−t
for any t > 0.
(b) Let B be a symmetric matrix of size n and let v be a unit eigenvector
of B whose eigenvalue has
√
absolute value kBk. Then for any unit vector x such that xT v ≥ 3/2, we have
xT Bx ≥
1
kBk.
2
(c) Let x be chosen uniformly from S n−1 . Then for any fixed unit vector v,
P x xT v ≥
√
1
3/2 ≥ √
.
πn2n−1
(d) Let x be chosen uniformly from S n−1 , independent of Ā. Fix t > 0. Use part (a) and (b) to derive
that
1
1
PĀ kĀk ≥ t .
PĀ,x |xT Āx| ≥ kĀk, kĀk ≥ t ≥ √
n−1
2
πn2
On the other hand, use part (a) to show that
2
1
PĀ,x |xT Āx| ≥ kĀk, kĀk ≥ t ≤ 2e−t /4 .
2
2
(e) Deduce that for any t > 0,
√
2
PĀ kĀk ≥ t ≤ πn2n e−t /4 ,
√
and hence conclude that kĀk = O( n) except for an exponentially small probability.
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