Download Asymptotic Liberation in Free Probability Josu´ e Daniel V´ azquez Becerra

Asymptotic Liberation in Free Probability
by
Josué Daniel Vázquez Becerra
A thesis submitted to the
Department of Mathematics and Statistics
in conformity with the requirements for
the degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
December 2014
c Josué Daniel Vázquez Becerra, 2014
Copyright Abstract
Recently G. Anderson and B. Farrel presented the notion of asymptotic liberation on sequences of families of random unitary matrices. They showed that asymptotically liberating
sequences of families of random unitary matrices, when used for conjugation, delivers asymptotic freeness, a fundamental concept in free probability theory. Furthermore, applying the
Fibonacci-Whitle inequality together with some combinatorial manipulations, they established sufficient conditions on a sequence of families of random unitary matrices in order to
be asymptotically liberating.
On the other hand, a theorem by J. Mingo and R. Speicher states that given a graph
G = (E, V, s, r) there exists an optimal rational number rG , depending only on the structure
of G, such that for any collection of n × n complex matrices {Ae = (Ae (is(e) , ir(e) )) | e ∈ E}
we have
X
n
iv ,...ivm =1
1
!
Y
kAe k
Ae (is(e) , ir(e) ) ≤ nrG
e∈E
e∈E
Y
where V = {v1 , . . . , vm } and k·k denotes the operator norm.
In this report we show how to use the latter inequality to prove the same result as G.
Anderson and B. Farrell regarding sufficient conditions for asymptotic liberation.
i
Acknowledgments
First and foremost, I would like to thank my supervisor James A. Mingo for his continuous
support, patience and time. Thank you James for your guidance throughout this project.
I extend my gratitude to all members and staff in the Department of Mathematics and
Statistics at Queens University who have directly or indirectly supported and helped me
over the past year.
I would also like to thank all of my friends and colleagues for their friendship, comprehension and encouragement.
Finally, I am grateful to my family for your love and incitement to pursue my aspirations.
ii
Contents
Abstract
i
Acknowledgments
ii
Contents
Chapter 1:
iii
Introduction
1
Chapter 2:
Asymptotic Liberation in free probability
2.1 Basic concepts from free probability . . . . . . . . . . . . . . . . . . . . . .
2.2 Random matrices from the point of view of free probability . . . . . . . . .
2.3 Asymptotically liberating sequences . . . . . . . . . . . . . . . . . . . . . .
5
5
10
14
Chapter 3:
A theorem on sufficient conditions for asymptotic liberation
3.1 Main statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Completion of proof via the Fibonacci-Whitle inequality . . . . . . . . . . .
3.3 Completion of proof via sharp bounds for graph sums . . . . . . . . . . . .
19
19
25
39
Bibliography
47
iii
1
Chapter 1
Introduction
Free probability theory was introduced by D. Voiculescu around 1985 motivated by his
research on von Neumann algebras of finitely generated free groups. While he was working
on the classification up to isomorphism of such algebras, he realized that some operator
algebra problems could be addressed by imitating some classical probability theory. Two
of his central observations were that such algebras could be equipped with tracial states,
certain unital linear functionals which resembled expectations of classical probability theory,
and by replacing tensor product with free product we get a new notion of independence.
He named this new notion of independence free independence, or simply freeness, and its
study is now known as free probability.
It should be mentioned that freenes is a notion of independence incorporating both the
probabilistic idea of non-correlation of random variables and the algebraic idea of absence
of algebraic relations. Furthermore, many concepts taken from classical probability theory
such as the notions of distribution, convergence in distribution, central limit theorem, Brownian motion, entropy, and more have natural analogues in suitable free probability spaces.
Therefore, free probability is a non-commutative probability theory in which the concept of
independence of classical probability is replaced by that of freeness and the corresponding
notion of independent random variables is then that of free random variables.
Now, although free probability had its origins in the operator algebras framework, one
2
of its main virtues is its relation with random matrix theory. Recall that random matrices
are simply matrix-valued random variables, and therefore a natural question arises: what
can we say about their eigenvalues? It turns out that free probability helps analyze through
expectations of normalized traces the asymptotic behavior of distributions of eigenvalues of
random matrices.
1
For instance, the following result, originally proved by E. P. Wigner [6]
and considered the starting point of random matrix theory, was showed to be a consequence
of the central limit theorem for free random variables by D. Voiculescu [4].
Wigner’s semicircular law. Let {bi,j |1 ≤ i < j}i,j∈N and {ci }i∈N be two independent
families of independent and identically distributed real-valued random variables with zero
mean such that Var (b1,2 ) = 1 and E |b1,2 |k , E |c1 |k < ∞ for all positive integers k. If
for each n ∈ N we consider the symmetric n-by-n random matrix An with entries given by
An (i, j) = An (j, i) =
then



√1 bi,j
n
if i < j


√1 ci
n
if i = j
Z2 p
i
1
1 h
tk 4 − t2 dt
lim E tr(Akn ) =
n→∞ n
2π
for k ∈ N
−2
where tr(Akn ) denotes the trace of Akn .
Let us now assume we are given a finite non-empty set M and for each positive integer
n ∈ N a family {An,m }m∈M of (possibly random) n-by-n complex matrices. From the
point of view of free probability, one of the most desirable properties, regarding asymptotic
behavior of expectations of normalized traces, that the sequence {{Am }m∈M }n∈N can have
is that of asymptotic freeness. Indeed, if the sequence {{Am }m∈M }n∈N is asymptotically
free with respect to expectations of normalized traces, then it suffices to know the values
1
Let us suppose we are given a n-by-n random complex matrix A defined on a probability space (Ω, F, µ).
Then, n1 tr◦A : Ω → C is a random variable with the peculiarity that for each ω ∈ Ω the value ( n1 tr◦A)(ω) is
the
of
eigenvalues of A(ω) counting multiplicities. Therefore, we might expect that the sequence
the
average
∞
E n1 tr(Ak ) k=1 provides us some information about the distribution of the eigenvalues of A.
3
limn→∞ n1 E tr(Akn,m ) for all m ∈ M and k ∈ N to compute, via free probability relations,
1
l
. . . Akn,m
) where k1 , . . . , kl ∈ N and m1 , . . . , ml ∈ M with
all limits limn→∞ n1 E tr(Akn,m
1
l
m1 6= m2 , . . . , ml−1 6= ml . For example, the limit
i
1 h
k2
k3
k4
1
E tr(Akn,m
A
A
A
)
n,m2 n,m1 n,m2
1
n→∞ n
lim
for k1 , k2 , k3 , k4 ∈ N and m1 , m2 ∈ M with m1 6= m2 would be given by
i
i
i
1 h
1 h
1 h
k2
k4
k3
1
E tr(Akn,m
E
tr(A
E
tr(A
A
)
lim
)
)
lim
n,m
n,m
n,m
1
2
2
1
n→∞ n
n→∞ n
n→∞ n
h
h
h
i
i
i
1
1
1
k3
k2
k4
1
E
tr(A
E
tr(A
)
lim
)
lim
A
lim E tr(Akn,m
)
n,m1
n,m2 n,m2
1
n→∞ n
n→∞ n
n→∞ n
h
h
h
i
i
i
i
1
1
1 h
1
k3
k2
k4
1
E
tr(A
E
tr(A
E
tr(A
)
lim
)
lim
)
lim
)
.
lim E tr(Akn,m
n,m1
n,m2
n,m2
1
n→∞ n
n→∞ n
n→∞ n
n→∞ n
lim
+
−
Hence, we may ask: what sort of sequences of families of random matrices are asymptotically
free? In particular, how can we construct asymptotically free sequences of families of random
matrices? In this respect, D. Voiculescu [4] showed that sequences of families of independent
Haar-distributed random unitary matrices are asymptotically free and lead to asymptotic
freeness when used for conjugation. To be more precise, he proved the following theorem.
Theorem. Fix a finite non-empty set M . Suppose that for each n ∈ N we are given a family
{Un,m }m∈M of independent Haar-distributed random unitary matrices. Then the sequence
{{Un,m }m∈M }n∈N is asymptotically free. Moreover, if for every n ∈ N we have a family
{An,m }m∈M of constant n-by-n Hermitian matrices such that sup max kAn,m k < ∞ and
n∈N m∈M
1
the limit lim tr(Akn,m ) exists for all indexes m ∈ M and positive integers k ∈ N, then
n→∞ n
∗ A U
{{Un,m
m n,m }m∈M }n∈N is asymptotically free.
Recently G. Anderson and B. Farrel [1] presented the notion of asymptotically liberating
sequences of families of random unitary matrices. They showed that such sequences can be
used to the same end as the sequence of families of independent Haar-distributed random
4
unitary matrices in the latter conclusion of the previous theorem. Thus, asymptotically
liberating sequences of families of random unitary matrices also lead to asymptotic freeness
when used for conjugation, and therefore provide a way of constructing asymptotically free
sequences of families of random matrices.
The aim of this project is to give a new proof of the main theorem of [1] which establishes sufficient conditions on a sequence of families of random unitary matrices in order
to be asymptotically liberating. Its original proof strongly relies on the Fibonacci-Whitle
inequality and some trivial but tedious combinatorial manipulations. Our proof is based on
a theorem by J. Mingo and R. Speicher [3] which provides optimal bounds for the absolute
value of the sum of products of entries of matrices.
The rest of this report is organized as follows. In Chapter 2, we give basic definitions
from free probability, present random matrices as non-commutative probability spaces, and
establish the concept of asymptotic liberating sequences of random unitary matrices. Then,
in Chapter 3, we state the main theorem of [1], present its original proof, and give an
alternative proof based on [3].
5
Chapter 2
Asymptotic Liberation in free probability
In this chapter, we aim to motivate and present the definition of asymptotically liberating
sequence of families of random unitary matrices. For this, we first give some of the most
fundamental concepts from free probability such as non-commutative probability space, free
independence, and asymptotic freeness. Then, we show how the space of n-by-n random
matrices whose entries have finite moments of all orders is viewed as a non-commutative
probability space. Finally, we present the definition of asymptotically liberating sequence of
families of random unitary matrices motivated by a characterization of asymptotic freeness
in terms of joint distributions.
2.1
Basic concepts from free probability
In classical probability theory, the set of complex random variables on a fixed probability
space is naturally endowed with the structure of a complex unital algebra with point-wise
operations. On the other hand, the expected value function is a unital linear functional that,
in many cases, determines the random variables or at least their distributions. Thus, from an
algebraic point of view, we might think of a probability space merely as a pair consisting of
a (possibly non-commutative) complex unital algebra and a unital linear functional defined
on such algebra. So, we have the following definition.
2.1. BASIC CONCEPTS FROM FREE PROBABILITY
6
Definition 1. A non-commutative probability space is a pair (A, φ) consisting of a unital
algebra A over C, with multiplicative identity 1A , and a linear functional φ : A → C such
that φ(1A ) = 1. The elements of A are called non-commutative random variables, or simply
random variables.
To differentiate non-commutative random variables from random variables defined on
measurable probability spaces, we will refer to the latter as classical random variables. Suppose that a is a random variable in a non-commutative probability space (A, φ). Since the
linear functional φ plays now the role of the expected value function in classical probability
theory, we define the kth moment of a as the value φ(ak ) for each positive integer k. Recall
that, in many cases, the distribution of a classical random variable is determined by its
moments, and hence we might be tempted to define the distribution of a as the sequence
{φ(ak )}∞
k=1 . However, it is more convenient to define the distribution of a non-commutative
random variable as follows, where C[X] denotes the polynomial algebra in the variable X
over the complex numbers C.
Definition 2. Let a be a random variable in a non-commutative probability space (A, φ).
The distribution of a is the linear functional ψa : C[X] → C given by ψa (p) = φ(p(a)) for
every polynomial p ∈ C[X] where p(a) ∈ A is defined by substituting X = a.
One advantage of the previous definition is that it can be easily adjusted to define
the (joint) distribution of a family of non-commutative random variables. To do this,
given a non-empty set M we denote by C h{Xm }m∈M i the non-commutative polynomial
algebra over C generated by the family {Xm }m∈M of non-commutative variables. In other
words, C h{Xm }m∈M i is the unital non-commutative algebra over C with Hamel basis given
by the set of all monomials Xm1 Xm2 . . . Xml with l ∈ {0, 1, . . .} and m1 , . . . , ml ∈ M
and multiplication on basis elements acting by juxtaposition, the multiplicative identity
1Ch{Xm }m∈M i is identified with the empty monomial. Recall that if m1 belongs to the
set M , then we canonically identify C[Xm1 ] as a sub-algebra of C h{Xm }m∈M i. Given a
2.1. BASIC CONCEPTS FROM FREE PROBABILITY
7
polynomial p ∈ C h{Xm }m∈M i and a family {am }m∈M of elements of a unital algebra A,
the evaluation p({am }m∈M ) ∈ A is defined by substituting Xm = am for all m ∈ M .
Definition 3. Let {am }m∈M be a family of random variables in a non-commutative probability space (A, φ). The joint distribution of the family {am }m∈M is the linear functional ψ : C h{Xm }m∈M i → C given by ψ(p) = φ(p({am }m∈M )) for every polynomial
p ∈ C h{Xm }m∈M i .
Note that if {am }m∈M is a family of random variables in a non-commutative probability
space (A, φ) with joint distribution ψ : C h{Xm }m∈M i → C, then for every m1 ∈ M
the restriction of ψ to the sub-algebra C[Xm1 ] is just the distribution of am1 , and hence
Definition 3 generalizes Definition 2.
We now present the notion of independence in non-commutative probability spaces
whose conception by D. Voiculescu constituted the beginning of free probability.
Definition 4. Let (A, φ) be a non-commutative probability space. A family {Am }m∈M of
unital sub-algebras of A is freely independent, or simply free, with respect to φ if for all
positive integers l ∈ N, indexes m1 , m2 , . . . , ml ∈ M with m1 6= m2 , . . . , ml−1 6= ml , and
random variables ak ∈ Amk with φ(ak ) = 0 for 1 ≤ k ≤ l we have φ(a1 a2 . . . al ) = 0.
If we are given a family {am }m∈M of random variables in a non-commutative probability
space (A, φ), we say that the random variables {am }m∈M are free if the unital sub-algebras
Am = {q(am ) ∈ A
| q ∈ C[X]} of A generated by the elements am are free in the sense
just defined.
As we previously mentioned, the aim of this report is to provide an alternative proof
of a theorem regarding asymptotically liberating sequences of families of random unitary
matrices. Such sequences, when used for conjugation, lead to asymptotically free sequences
of families of random matrices under standard conditions. We still need to show how
the space of n-by-n random matrices whose entries have finite moments of all orders can
2.1. BASIC CONCEPTS FROM FREE PROBABILITY
8
be viewed as a non-commutative probability space, however, we must first establish what
asymptotically free means.
Definition 5. Let M be a non-empty set. Assume for each n ∈ N we have a family
{an,m }m∈M of random variables in a non-commutative probability space (An , φn ). We say
that the sequence {{an,m }m∈M }n∈N of families of random variables is asymptotically free if
there exists a family {am }m∈M of free random variables in some non-commutative probability
space (A, φ) such that for all positive integers l ∈ N and indexes m1 , . . . , ml ∈ M we have
lim φn (an,m1 . . . an,ml ) = φ(am1 . . . aml ) .
n→∞
So, a sequence {{an,m }m∈M }n∈N of families of random variables is asymptotically free
if the random variables {an,m }m∈M behave as some {am }m∈M free random variables as n
goes to ∞.
To end this section, we prove a result which characterizes asymptotic freeness in terms of
joint distributions, and, as we will show later, it motivates the definition of asymptotically
liberating sequences of random unitary matrices.
Proposition 6. Let M be a non-empty set. Suppose that for each n ∈ N we are given
a family {an,m }m∈M of random variables in a non-commutative probability space (An , φn )
with joint distribution ψn : C h{Xm }m∈M i → C. Then the sequence {{an,m }m∈M }n∈N of
families of random variables is asymptotically free if the following two conditions hold:
(i) the limit lim ψn (p) exists for every index m ∈ M and every polynomial p ∈ C[Xm ],
n→∞
(ii) the limit lim ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) equals 0 for all integers l ≥ 2, indexes
n→∞
m1 6= m2 ,. . .,ml−1 6= ml in M , and polynomials p1 ∈ C[Xm1 ], . . . , pl ∈ C[Xml ].
Proof. Let φ : C h{Xm }m∈M i → C be defined by φ(p) = lim ψn (p) for every p ∈
n→∞
C h{Xm }m∈M i. We must show that φ is well-defined. To this end, it suffices to show
by induction on l that the following holds:
2.1. BASIC CONCEPTS FROM FREE PROBABILITY
9
(iii) the limit lim ψn (p1 . . . pl ) exists for all positive integers l, indexes m1 6=
n→∞
m2 ,. . .,ml−1 6= ml in M , and polynomials p1 ∈ C[Xm1 ], . . . , pl ∈ C[Xml ] .
Clearly, (iii) holds for l = 1 by hypothesis (i). Suppose (iii) holds for 1, 2, . . . , l−1. Take
m1 6= m2 ,. . .,ml−1 6= ml in M , and polynomials p1 ∈ C[Xm1 ], . . . , pl ∈ C[Xml ]. For each
subset I of {1, 2, . . . , l} =: [l], let us take pI as the product of p1 , p2 , . . . , pl after removing
Y
from the latter list the entries indexed by i ∈ I, and αn,I as the product
ψn (pi ) for each
i∈I
n ∈ N. Note that for every non-empty subset I of [l] the limits lim ψn (pI ) and lim αn,I
n→∞
n→∞
exist by induction hypothesis and hypothesis (i), respectively. Hence, the limit
lim
X
n→∞
(−1)|I| αn,I ψn (pI ) =
I⊂[l]
∅6=I
X
(−1)|I| lim αn,I ψn (pI )
n→∞
I⊂[l]
∅6=I
also exists. Now, for every n ∈ N we have
ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) = ψn (p1 . . . pl ) +
X
(−1)|I| αn,I ψn (pI )
I⊂[l]
∅6=I
and by hypothesis (ii) we have lim ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) = 0. Thus, the limit
n→∞
lim ψn (p1 p2 . . . pl ) = − lim
n→∞
X
n→∞
(−1)|I| αn,I ψn (pI )
(1)
I⊂[l]
∅6=I
exists. Therefore, (iii) holds for every positive integer l and hence φ is well-defined.
Put now A = C h{Xm }m∈M i. Consider the non-commutative probability space (A, φ)
and take am = Xm ∈ A for every m ∈ M . By definition of ψn and φ we have
lim φn (an,m1 . . . an,ml ) = lim ψn (Xm1 . . . Xml ) = φ(am1 . . . aml )
n→∞
n→∞
for all positive integers l ∈ N and indexes m1 , . . . , ml ∈ M . So, to prove that {{an,m }m∈M }n∈N
2.2. RANDOM MATRICES FROM THE POINT OF VIEW OF FREE
PROBABILITY
10
is asymptotically free, it only remains to show that {am }m∈M is free in (A, φ).
Let Am denote the unital sub-algebra of A generated by am , i.e., Am = C[Xm ], for
every m ∈ M . Suppose that l ∈ N, m1 , . . . , ml ∈ M with m1 6= m2 , . . . , ml−1 6= ml , and
ak := pk ∈ Amk with φ(ak ) = 0 for 1 ≤ k ≤ l. For each subset I of [l] and each n ∈ N, let
us take pI and αn,I as before. Note that for every non-empty subset I of [l] we have
lim αn,I = lim
n→∞
n→∞
Y
ψn (pi ) = 0
i∈I
since lim ψn (pk ) =: φ(pk ) = φ(ak ) = 0 for 1 ≤ k ≤ l. Thus, by equation (1) we have
n→∞
φ(a1 a2 . . . al ) := lim ψn (p1 p2 . . . pl ) = − lim
n→∞
X
n→∞
(−1)|I| αn,I ψn (pI ) = 0
I⊂[l]
∅6=I
Therefore, the sequence {{an,m }m∈M }n∈N of families of random variables is asymptotically
free.
2.2
Random matrices from the point of view of free probability
In this section, we give some elementary definitions from random matrix theory, and provide
some related notation which will be used through this report. We also establish the usual
setting under which we identify the space of random matrices whose entries have finite
moments of all orders as a non-commutative probability space.
We denote by Mn (C) the set of n-by-n matrices with complex entries for each positive
integer n ∈ N. For a matrix A ∈ Mn (C), we denote by A∗ its conjugate transpose, and by
A(i, j) its entry in the ith row and j th column. Recall that Mn (C) is a unital algebra over
C under the usual matrix operations. The identity matrix in Mn (C) is denoted by In . A
matrix U ∈ Mn (C) is called unitary if U U ∗ = U ∗ U = In .
The operator norm k·k : Mn (C) → [0, ∞) and the Hilbert-Schmidt norm k·k2 : Mn (C) →
2.2. RANDOM MATRICES FROM THE POINT OF VIEW OF FREE
PROBABILITY
11
[0, ∞) and are the norms on Mn (C) defined for each matrix A ∈ Mn (C) by
kAk = sup{|Ax| : x ∈ Cn , |x| = 1}
and
v
uX
u n
kAk2 = t
|A(i, j)|2 .
i,j=1
where | · | denotes the euclidean norm in Cn . Recall that for all matrices A, B ∈ Mn (C) we
have the inequalities
kAk ≤ kAk2 ≤
√
nkAk
and
kABk ≤ kAkkBk .
(2.1)
Hence, the Hilbert-Schmidt norm and the operator norm induce the same topology on
Mn (C). We always assume that Mn (C) is endowed with such topology.
The trace of a matrix A ∈ Mn (C), denoted tr(A), is defined to be the sum of its entries
on the main diagonal, i.e.,
tr(A) :=
n
X
A(i, i).
i=1
A matrix A ∈ Mn (C) is of trace zero if its trace equals 0. Notice that for all matrices
A, B ∈ Mn (C), and complex numbers λ ∈ C we have
tr(λA + B) = λtr(A) + tr(B),
tr(AB) = tr(BA),
and
1
|tr(A)| ≤ kAk .
n
(2.2)
Suppose we are given a classical probability space (Ω, F, µ), i.e., Ω is a non-empty set,
F is a σ-algebra of subsets of Ω, and µ : F → [0, 1] is a probability measure. A function
X : Ω → C is a classical random variable on (Ω, F, µ) if for every open set U of the complex
numbers C we have X −1 (U) = {ω ∈ Ω : X(ω) ∈ U} ∈ F. For each positive integer k ∈ N,
the kth moment of a classical random variable X on (Ω, F, µ) is defined to be the integral
Z
Ω
[X(ω)]k dµ(ω)
2.2. RANDOM MATRICES FROM THE POINT OF VIEW OF FREE
PROBABILITY
12
if such integral exists, and it is not defined otherwise. The first moment of X, denoted
E[X], is also called the expected value of X, and it is well-defined if and only if the integral
Z
|X(ω)| dµ(ω)
Ω
exists. The set of classical random variables on (Ω, F, µ) which have moments of all orders
is denoted by L−∞ (Ω, F, µ). The set of complex numbers C is identified as the subset of
L−∞ (Ω, F, µ) consisting of constant classical random variables. From classical probability
theory we know that L−∞ (Ω, F, µ) is a unital algebra over the complex numbers under pointwise operations, and the expected value E[·] : L−∞ (Ω, F, µ) → C defines a linear functional.
Moreover, for each real number p ∈ [1, ∞) the p-norm k · kp : L−∞ (Ω, F, µ) → [0, ∞) is the
semi-norm on L−∞ (Ω, F, µ) defined for each X ∈ L−∞ (Ω, F, µ) by
Z
kXkp =
1
p
.
|X(ω)| dµ(ω)
p
Ω
Hölder’s inequality tells us that if p1 , p2 , . . . , pl ∈ (1, ∞) are such that
1
p1
+ p12 + . . . + p1l = 1
then for all X1 , X2 , . . . , Xl ∈ L−∞ (Ω, F, µ) we have
kX1 X2 . . . Xl k1 ≤ kX1 kp1 kX2 kp2 . . . kXl kpl .
Now, in analogy to classical random variables, we have the following definition.
Definition 7. Let (Ω, F, µ) be a classical probability space. A function X : Ω → Mn (C) is
a n-by-n random matrix on (Ω, F, µ) if for every open set U of Mn (C) we have
X −1 (U) = {ω ∈ Ω : X(ω) ∈ U} ∈ F .
For a classical probability space (Ω, F, µ) we denote by Mn (Ω, F, µ) the set of n-byn random matrices on (Ω, F, µ). Notice that a function X : Ω → Mn (C) belongs to
2.2. RANDOM MATRICES FROM THE POINT OF VIEW OF FREE
PROBABILITY
13
Mn (Ω, F, µ) if and only if there are classical random variables X(i, j) : Ω → C on (Ω, F, µ)
for i, j = 1, 2, . . . , n such that for every ω ∈ Ω we have
X(ω)(i, j) = X(i, j)(ω) .
(2.3)
Thus, a random matrix on (Ω, F, µ) is a matrix whose entries are classical random variables
on (Ω, F, µ). For a random matrix X ∈ Mn (Ω, F, µ) and a pair 1 ≤ i, j ≤ n we denote by
X(i, j) the unique classical random variable on (Ω, F, µ) which satisfies (2.3), note that this
notation agrees with our notation for matrices in Mn (C). The set of n-by-n random matrices
on (Ω, F, µ) whose entries have finite moments of all orders is denoted by Mn (L−∞ (Ω, F, µ)),
or simply Mn (L−∞ ), i.e.,
Mn (L−∞ (Ω, F, µ)) = X ∈ Mn (Ω, F, µ) : X(i, j) ∈ L−∞ (Ω, F, µ) for 1 ≤ i, j ≤ n .
We identify Mn (C) as the subset of Mn (L−∞ ) consisting of constant random matrices
and extend the matrix operations of Mn (C) to Mn (L−∞ ) in the obvious way. Thus, for all
random matrices X, Y ∈ Mn (L−∞ ), complex numbers λ ∈ C, and integers i, j = 1, 2, . . . , n
we have
(X + Y )(i, j) = X(i, j) + Y (i, j), (XY )(i, j) =
n
X
X(i, k)Y (k, j), and (λY )(i, j) = λY (i, j).
k=1
Hence, we think of Mn (L−∞ ) as a unital algebra over the complex numbers with multiplicative identity In .
For a random matrix X ∈ Mn (L−∞ (Ω, F, µ)), we denote by X ∗ its conjugate transpose.
In other words, X ∗ is the unique random matrix in Mn (L−∞ (Ω, F, µ)) such that X ∗ (i, j) =
X(j, i) for all 1 ≤ i, j ≤ n. Here, X(i, j)(ω) := X(i, j)(ω) for every ω ∈ Ω and for all α ∈ C
we denote by α its complex conjugate. A random matrix U ∈ Mn (L−∞ ) is called unitary if
U ∗ U = U U ∗ = In .
2.3. ASYMPTOTICALLY LIBERATING SEQUENCES
14
Now, extending the case for matrices in Mn (C), the trace of a random matrix X, denoted
tr(X), is defined as the sum of its entries on the main diagonal. Note that if X ∈ Mn (L−∞ )
Pn
then tr(X) =
i=1 X(i, i) is a classical random variable, and hence, E[tr(X)] is welldefined. Moreover, for all random matrices X, Y ∈ Mn (L−∞ ) and complex numbers λ ∈ C
we have
tr(λX + Y ) = λtr(X) + tr(Y )
and tr(XY ) = tr(Y X) .
Hence, the normalized expected value of the trace
linear functional such that
1
n E[tr(In )]
1
n E[tr(·)]
: Mn (L−∞ ) → C defines a
= 1.
Therefore, for a probability space (Ω, F, µ) the pair Mn (L−∞ (Ω, F, µ)) , n1 E[tr(·)] is
a non-commutative probability space. This is the usual setting under which we identify
the space of random matrices whose entries have finite moments of all orders as a noncommutative probability space.
In this report, each family {Xm }m∈M of n-by-n random matrices is assumed to be
contained in Mn (L−∞ (Ω, F, µ)) for some classical probability space (Ω, F, µ); moreover, its
joint distribution, in the sense of Definition 3, is understood to be defined with respect to
the linear functional
2.3
1
n E[tr(·)]
.
Asymptotically liberating sequences
For this section, let M be a fixed non-empty set and assume for each n ∈ N we have a family
{Un,m }m∈M of n-by-n random unitary matrices defined in the same classical probability
space. Let us suppose that for each n ∈ N we are given a family {Bn,m }m∈M of n-by-n
complex matrices such that sup sup kBn,m k < ∞. Under these assumptions, we would like
n∈N m∈M
to find a condition for {{Un,m }m∈M }n∈N based on Proposition 6 which allows us to conclude
∗ B
that the sequence {{Un,m
n,m Un,m }m∈M }n∈N is asymptotically free.
∗ B
Let ψn denote the joint distribution of {Un,m
n,m Un,m }m∈M for each n ∈ N. Suppose
we are given an integer l ≥ 2, indexes m1 , . . . , ml ∈ M with m1 6= m2 ,. . .,ml−1 6= ml , and
2.3. ASYMPTOTICALLY LIBERATING SEQUENCES
15
polynomials pk ∈ C[Xmk ] ⊂ C h{Xm }m∈M i for 1 ≤ k ≤ l. By Proposition 6 we just need a
condition on {{Un,m }m∈M }n∈N under which
lim ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) = 0 .
n→∞
Let us take qn,k = pk − ψn (pk ) ∈ C[Xmk ] and An,k = qn,k (Bn,mk ) for every n ∈ N and
k = 1, 2, . . . , l. Notice that
ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) =
1 ∗
∗
E tr Un,m
A
U
.
.
.
U
A
U
n,1
n,m
n,m
n,l
n,m
1
l
1
l
n
since ψn ((p1 − ψn (p1 )) . . . (pl − ψn (pl ))) = ψn (qn,1 . . . qn,l ) and
1 ∗
∗
E tr qn,1 (Un,m
Bn,m1 Un,m1 ) . . . qn,l (Un,m
B
U
)
n,m
n,m
l
l
1
l
n
1 ∗
∗
q (Bn,m1 )Un,m1 . . . Un,m
q (Bn,ml )Un,ml .
= E tr Un,m
1 n,1
l n,l
n
ψn (qn,1 . . . qn,l ) =
So, we are actually looking for a condition on {{Un,m }m∈M }n∈N such that
1 ∗
∗
E tr Un,m
An,1 Un,m1 . . . Un,m
An,l Un,ml = 0 .
1
l
n→∞ n
lim
(2.4)
Let us now note that the matrices An,k are of trace zero and uniformly bounded with respect
to the operator norm. Indeed, by definition of the matrices An,k we have
An,k = qn,k (Bn,mk ) = pk (Bn,mk ) −
1
tr(pk (Bn,mk ))In
n
since
ψn (pk ) =
=
1
nE
∗
tr(pk (Un,m
Bn,mk Un,mk )) =
k
1
n E [tr(pk (Bn,mk ))]
=
1
nE
∗
tr(Un,m
p (Bn,mk )Un,mk )
k k
1
n tr (pk (Bn,mk ))
.
2.3. ASYMPTOTICALLY LIBERATING SEQUENCES
16
Hence, for every n ∈ N and 1 ≤ k ≤ l we get
tr(An,k ) = 0 .
(2.5)
Furthermore, taking C = sup sup kBn,m k < ∞, for every n ∈ N and 1 ≤ k ≤ l we get
n∈N m∈M
kAn,k k ≤ kpk (Bn,mk )k + k n1 tr(pk (Bn,mk ))In k = kpk (Bn,mk )k + n1 |tr(pk (Bn,mk ))|
≤
where we take |pk | =
2k(pk (Bn,mk ))k
Pnk
λ=0 |αλ |X
λ
≤
∈ C[X] for pk (X) =
∞
2|pk |(C)
Pnk
λ=0 αλ X
λ
∈ C[X], and thus,
l
sup maxkAn,k k < ∞ .
(2.6)
n=1 k=1
Therefore, in light of (2.4), (2.5), and (2.6), we have the following definition.
Definition 8. The sequence {{Un,m }m∈M }n∈N of families of random unitary matrices is
asymptotically liberating if for every integer l ≥ 2 and indexes m1 , . . . , ml ∈ M with m1 6=
m2 , . . . , ml−1 6= ml there exists a constant C(m1 , . . . , ml ), depending only on l, m1 , . . ., and
ml , such that
l
Y
∗
∗
kAn,k k
E tr(Un,m1 An,1 Un,m1 . . . Un,ml An,l Un,ml ) ≤ C(m1 , . . . , ml )
(2.7)
k=1
for all positive integers n ∈ N and n-by-n complex matrices An,1 , . . . , An,l of trace zero.
We finish this chapter with a convenient reformulation of the previous definition granted
by the following result.
Proposition 9. The sequence {Un,m }n∈N,m∈M is asymptotically liberating (if and only)
if for every integer l ≥ 2 and indexes m1 , . . . , ml ∈ M with m1 6= m2 ,. . .,ml−1 6= ml ,
ml 6= m1 there exists a constant C(m1 , . . . , ml ), depending only on l, m1 , . . ., and ml ,
2.3. ASYMPTOTICALLY LIBERATING SEQUENCES
17
such that inequality (2.7) holds for all positive integers n ∈ N and n-by-n complex matrices
An,1 , . . . , An,l of trace zero.
Proof. Let us assume we are given a positive integer l ≥ 2 and indexes m1 6= m2 , . . . , ml−1 6=
ml in M . Suppose that for each positive integer n ∈ N we have matrices An,1 , . . . , An,l ∈
Mn (C) of trace zero. We must show that there exists a constant constant C(m1 , . . . , ml ),
depending only on l, m1 , . . ., and ml , such that inequality (2.7) holds for all positive integers n ∈ N. By hypothesis, such constant exists whenever ml 6= m1 , in particular, if l = 2.
Thus, we may assume l ≥ 3 and m1 = ml .
Fix a positive integer n ∈ N. To simplify notation, let us put I = In , Uk = Un,mk and
Ak = An,k for k = 1, 2, . . . , l. Consider the constant matrix A = Al A1 − n1 tr(Al A1 ) of trace
∗ . Then,
zero and the random matrix X = U2 A2 U2∗ . . . Ul−1 Al−1 Ul−1
E tr(U1 A1 U1∗ U2 A2 U2∗ . . . Ul Al Ul∗ ) = E tr(U1 A1 U1∗ XUl Al Ul∗ ) ,
but, since U1 = Ul and Ul∗ U1 = I because m1 = ml , we have
E tr(U1 A1 U1∗ XUl Al Ul∗ ) = E tr(Ul Al Ul∗ U1 A1 U1∗ X) = E tr(U1 Al A1 U1∗ X) .
Moreover, by linearity we get
1
E tr(U1 Al A1 U1∗ X) = E tr(U1 AU1∗ X) + tr(Al A1 )E tr(X) ,
n
and thus,
1
E tr(U1 A1 U1∗ . . . Ul Al Ul∗ ) = E tr(U1 AU1∗ X) + tr(Al A1 )E tr(X) .
n
(1)
By induction on l, we may assume there are C(m1 , . . . , ml−1 ), C(m2 , . . . , ml−1 ) ≥ 0,
2.3. ASYMPTOTICALLY LIBERATING SEQUENCES
18
depending only on l − 1, m1 , m2 , . . . , and ml−1 , such that
l−1
Y
E tr(U1 AU1∗ U2 A2 U2∗ . . . Ul−1 Al−1 U ∗ ) ≤ C(m1 , . . . , ml−1 )kAk
kAk k
l−1
(2)
k=2
and
l−1
Y
E tr(U2 A2 U2∗ . . . Ul−1 Al−1 U ∗ ) ≤ C(m2 , . . . , ml−1 )
kAk k .
l−1
(3)
k=2
Therefore, taking C(m1 , . . . , ml ) = 2C(m1 , . . . , ml−1 ) + C(m2 , . . . , ml−1 ), by inequalities
(1), (2), (3), and since
1
tr(Al A1 ) ≤ kAl A1 k ≤ kAl kkA1 k
n
and
1
1
kAk ≤ kAl A1 k + k tr(Al A1 )Ik ≤ kAl kkA1 k + tr(Al A1 ) ≤ 2kAl kkA1 k,
n
n
we conclude
l
Y
E tr(U1 A1 U1∗ . . . Ul Al U ∗ ) ≤ C(m1 , . . . , ml )
kAk k .
l
k=1
19
Chapter 3
A theorem on sufficient conditions for asymptotic liberation
This chapter contains the contribution of this project, namely, an alternative proof for the
main theorem in [1] which is a theorem on sufficient conditions for a sequence of families
of random unitary matrices in order to be asymptotically liberating. First, we state such
a theorem and present its original proof which is primarly based on the Fibonacci-Whitle
Inequality. Then, we recall the principal result of [3], providing optimal bounds for the
absolute value of the sum of products of entries of matrices. Finally, we show how to apply
the latter result to alternatively prove the main theorem of [1].
3.1
Main statement
Let us first introduce some useful notation then recall some terminology for matrices. For
each positive integer n ∈ N we denote by [n] set of the first n positive integers {1, 2, . . . , n}.
Additionally, for a non-empty set I and a positive integer l ∈ N we denote by I l the set of
l-tuples with entries in I, i.e., I l = {(i1 , i2 , . . . , il ) : ik ∈ I for every k ∈ [l]}.
Two families {Xm }m∈M and {Ym }m∈M of n-by-n random matrices defined in the same
classical probability space are said to be equal in distribution if for all positive integers
l ∈ N, and indexes m1 , m2 , . . . , ml ∈ M we have
E[Xm1 (i1 , i2 ) . . . Xml (i2l−1 , i2l )] = E[Ym1 (i1 , i2 ) . . . Yml (i2l−1 , i2l )]
3.1. MAIN STATEMENT
20
for every (i1 , i2 , . . . , i2l ) ∈ [n]2l . A matrix W ∈ Mn (C) is a signed permutation matrix if
there exists 1 , . . . , n ∈ {−1, 1} and a permutation σ ∈ Sn = {f : [n] → [n] : f is bijective}
such that
W (i, j) = i δi,σ(j) =


 i
if i = σ(j)

 0
otherwise
.
Note that if V is a n-by-n random matrix then for i, j ∈ [n] we have
(W ∗ V W )(i, j) = σ(i) σ(j) V (σ(i), σ(j))
We now state the main theorem of [1].
Theorem 10. Let M be a non-empty set. Assume we have a family {Un,m }m∈M of n-by-n
random unitary matrices defined on a probability space (Ωn , Fn , µn ) for each n ∈ N. Suppose
{Un,m }m∈M n∈N satisfies the following two conditions:
(I) for every n ∈ N and every signed permutation matrix Wn ∈ Mn (C) we have that the
∗
∗ U
Un,m0 m,m0 ∈M and W ∗ Un,m
families Un,m
n,m0 W m,m0 ∈M are equal in distribution,
m6=m0
m6=m0
(II) for every l ∈ N there exist a constant Kl < ∞ such that
√ ∗ U
n Un,m
n,m0 (i, j) l < Kl
for all m, m0 ∈ M with m 6= m0 , n ∈ N, and 1 ≤ i, j ≤ n.
Then {Un,m }m∈M n∈N is asymptotically liberating.
For now, we prove Theorem 10 modulo any of Theorems 21 and 27. Proof. Let
us assume we are given a positive integer l ≥ 2 and indexes m1 , . . . , ml ∈ M such that
m1 6= m2 , . . ., ml−1 6= ml , and ml 6= m1 . Suppose that for each n ∈ N we have matrices
An,1 , An,2 , . . . , An,l ∈ Mn (C) of trace zero.
By Proposition 9, to prove {Un,m }m∈M n∈N is asymptotically liberating, it suffices to
show that there exists a constant 0 ≤ Cl < ∞ depending only on l such that
l
Y
∗
∗
kAn,k k
E tr(Un,m1 An,1 Un,m1 . . . Un,ml An,l Un,ml ) ≤ Cl
k=1
3.1. MAIN STATEMENT
21
for all positive integers n ∈ N. Fix a positive integer n ∈ N and consider the functions
A, F : [n]2l → C defined for i = (i1 , . . . , i2l ) ∈ [n]2l by
A(i) = An,1 (i1 , i2 ) . . . An,l (i2l−1 , i2l )
F(i) = E [Vn,1 (i2 , i3 ) . . . Vn,l−1 (i2l−2 , i2l−1 )Vn,l (i2l , i1 )]
∗
for k = 1, 2, . . . l, and ml+1 := m1 . Then, we have the equality
where Vn,k = Un,m
U
k n,mk+1
X
n
X
A(i)F(i) :=
An,1 (i1 , i2 ) . . . An,l (i2l−1 , i2l )E Vn,1 (i2 , i3 ) . . . Vn,l (i2l , i1 )
i1 ,...,i2l =1
i∈[n]2l
n
X
=
E An,1 (i1 , i2 )Vn,1 (i2 , i3 ) . . . An,l (i2l−1 , i2l )Vn,l (i2l , i1 )
i1 ,...,i2l =1
n
h X
=E
An,1 (i1 , i2 )Vn,1 (i2 , i3 ) . . . An,l (i2l−1 , i2l )Vn,l (i2l , i1 )
i
i1 ,...,i2l =1
= E tr(An,1 Vn,1 . . . An,l Vn,l )
∗
∗
= E tr(An,1 Un,m
U
. . . An,l Un,m
U
)
1 n,m2
l n,m1
∗
∗
= E tr(Un,m1 An,1 Un,m
.
.
.
U
A
U
)
n,m
n,l
n,m
l
1
l
(3.1)
by linearity of the expectation E[·] and the trace tr(·).
By hypothesis (II) there exists a constant 0 ≤ Kl < ∞ depending only on l such that
for all 2l-tuples (j1 , . . . , j2l ) ∈ [n]2l we have
√
√ ∗
n Un,mk Un,mk+1 (j2k−1 , j2k )l = n kVn,k (j2k−1 , j2k )kl < Kl ,
and hence, by Hölder’s inequality we get
kVn,1 (j1 , j2 ) . . . Vn,l (j2l−1 , j2l )k1 ≤ kVn,1 (j1 , j2 )kl . . . kVn,l (j2l−1 , j2l )kl <
K
√l
n
l
.
3.1. MAIN STATEMENT
22
Thus, we have the following inequality
max |F(i)| ≤
i∈[n]2l
max
i∈[n]2l
Vn,1 (i2 , i3 ) . . . Vn,l−1 (i2l−2 , j2i−1 )Vn,l (i2l , i1 ) ≤ (Kl )l n −l
2 . (3.2)
1
i=(i1 ,...,i2l )
Note then that to conclude {Un,m }m∈M n∈N is asymptotically liberating we only need
to show that there exists a constant 0 ≤ κl < ∞ depending only on l such that
l
Y
X
≤ κl n 2l max |F(i)|
kAn,k k .
F(i)A(i)
i∈[n]2l
i∈[n]2l
k=1
(3.3)
Indeed, if κl satisfies (3.3) and we take Cl = (Kl )l κl , then
l
Y
∗
∗
E tr(Un,m An,1 Un,m
. . . Un,ml An,l Un,ml ) ≤ Cl
kAn,k k
1
1
k=1
by (3.1), (3.2), and (3.3).
Suppose now we take an arbitrary permutation σ ∈ Sn and integers 1 , . . . , n ∈ {−1, 1}.
Consider the signed permutation matrix W ∈ Mn (C) with entries W (i, j) = i δi,σ(j) . Then,
by hypothesis (I) for every 2l-tuple (i1 , i2 , . . . , i2l ) ∈ [n]2l we have
F((i1 , i2 , . . . , i2l )) = σ(i1 ) σ(i2 ) . . . σ(i2l ) F((σ(i1 ), σ(i2 ), . . . , σ(i2l ))) .
(3.4)
Note that the last equation implies the following:
i) F(i) = 0 for every i = (i1 , i2 , . . . , i2l ) ∈ [n]2l for which there exists a k ∈ [2l] such that
ik 6= ik0 for every k 0 ∈ [2l] \ {k},
ii) F(i) = F(j) for all i = (i1 , i2 , . . . , i2l ), j = (j1 , j2 , . . . , j2l ) ∈ [n]2l for which there exists
a permutation σ ∈ Sn such that jk = σ(ik ) for every k ∈ [2l].
Therefore, F satisfies hypothesis of Theorems 21 and 27, and hence, the constant κl is
3.1. MAIN STATEMENT
23
provided by any of such Theorems.
The final two sections in this chapter are devoted to state and prove Theorems 21 and
27. In [1] G. Anderson and B. Farrell proved Theorem 21, and, as we will see, its proof relies
on the Fibonacci-Whitle inequality. Theorem 27 and its proof is the principal contribution
in this report. Both Theorems 21 and 27 deal with functions of the χχ-class which are
defined at the end of this section.
Before moving foward, we need to introduce some terminology and notation regarding
set partitions. Given a positive integer l ∈ N a partition of [l] is a set of non-empty pair-wise
disjoint subsets of [l] whose union equals [l] . In other words, a set π is a partition of [l] if
S
∅=
6 B ⊂ [l] for every B ∈ π, B ∩ B 0 6= ∅ implies B = B 0 for all B, B 0 ∈ π, and [l] = B∈π B.
The elements of a partition are called its blocks and if we write π = {B1 , B2 , . . . , Bp } for a
partition π, it is always assumed that two block Bk and Bk0 are the same only if k = k 0 .
The set of partitions of [l] is denoted by P (l). Additionally, there are two special cases
of set partitions which will be extensively used,
Pχ (l) = {π ∈ P (l) : {k} ∈
/ π for k ∈ [l]}, and
Pχχ (l) = {π ∈ Pχ (2l) : {2k − 1, 2k} ∈
/ π for k ∈ [l]} .
So, Pχ (l) denotes the set of partitions of [l] lacking of singletons and Pχχ (2l) denotes the
set of partitions of [2l] lacking of singletons and blocks of the form {2k − 1, 2k}.
The set of partitions P (l) is endowed with the partial order ≤ such that for partitions
π1 , π2 ∈ P (l) we write π1 ≤ π2 if for every block B1 ∈ π1 there exists a second block B2 ∈ π2
such that B1 is contained in B2 . We denote by 0l the minimal partition in P (l), i.e., 0l :=
{{k} : k ∈ [l]}. Thus, for instance, π1 = {{1, 2}, {3, 4}, {5}} and π2 = {{1, 2, 5}, {3, 4}} are
both partitions of [5] = {1, 2, . . . , 5}, 05 = {{1}, {2}, . . . , {5}} ∈ P (5), and 05 ≤ π1 ≤ π2 .
If we are given a l-tuple i = (i1 , i2 , . . . , il ) ∈ [n]l for some positive integer n ∈ N, the
3.1. MAIN STATEMENT
24
partition of [l] generated by i, denoted Π(i), is defined by
Π(i) = {i−1 (λ) 6= ∅ : λ = 1, 2, . . . , n}
where i−1 (λ) = {k ∈ [l] : ik = λ} for λ = 1, 2, . . . , n. For example, for i = (1, 2, 3, 1, 2) ∈ [3]5
and j = (2, 2, 1, 1, 2) ∈ [3]5 we have Π(i) = {{1, 4}, {2, 5}, {3}} and Π(j) = {{1, 2, 5}, {3, 4}}.
Remark 11. For all i = (i1 , . . . , il ), j = (j1 , . . . , jl ) ∈ [n]l we have that Π(i) = Π(j) if and
only if there is a permutation σ ∈ Sn such that jk = σ(ik ) for k = 1, 2, . . . , l.
Let us now define the functions of the χ-class and the χχ-class.
Definition 12. A function f : [n]l → C belongs to the χ-class if f (i) = 0 whenever i ∈ [n]l
is such that Π(i) contains a singleton.
Definition 13. A function f : [n]2l → C belongs to the χχ-class if it belongs to the χ-class
and f (i) = f (j) whenever i = (i1 , . . . , i2l ), j = (ij , . . . , j2l ) ∈ [n]l are such that {2k − 1, 2k}
is a block of both Π(i) and Π(j) for some k ∈ [l] and iλ = jλ for λ = 1, 2, . . . , 2k − 2, 2k +
1, . . . , 2l.
The clump of a partition π ∈ P (2l), denoted Clump(π), is defined as the union of
all blocks of π except for those which have the form {2k − 1, 2k}.
For instance, for
π1 = {{1, 2}, {3, 4}, {5}}, π2 = {{1, 2, 5}, {3, 4}}, and π3 = {{1, 3}, {2, 4}, {5}} in P (5)
we have Clump(π1 ) = {5}, Clump(π2 ) = {1, 2, 5}, and Clump(π3 ) = {1, 2, 3, 4, 5}. Given
i = (i1 , . . . , i2l ) ∈ [n]2l and j = (j1 , . . . , j2l ) ∈ [n]2l we say that i and j are clump-equivalent
and write i ∼ j if Π(i) = Π(j) and iλ = jλ for λ ∈ Clump(Π(i))
Proposition 14. Let f : [n]2l → C be a function of the χχ-class. If n ≥ 6l then f is
constant on clump-equivalence classes.
Proof. Let i = (i1 , i2 , . . . , i2l ) ∈ [n]2l and put π = Π(i). Suppose π contains a singleton.
Then since f belongs to the χ-class we have f (j) = 0 for all j ∈ [n]2l such that Π(j) = π.
So, we may assume π lacks singletons, i.e., π ∈ Pχ (2l).
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
25
Suppose now i ∼ j for some j = (j1 , . . . , j2l ) ∈ [n]2l . Note that if π ∈ Pχ (2l) but
π ∈
/ Pχχ (2l) then Clump(π) = [2l] and hence i ∼ j implies i = j. So, let us assume
π ∈ Pχχ (2l). By symmetry, without loss of generality we may additionally assume
π = θ ∪ {{2s + 1, 2s + 2}, . . . , {2l − 1, 2l}}
for some s = l − t ∈ {0, 1, . . . , l} and θ = if s = 0 and θ ∈ Pχχ (2s) otherwise.
By hypothesis n ≥ 6l, and thus there exists distinct k1 , . . . , kt ∈ [n]\{i1 , . . . , i2l , j1 , . . . , j2l }.
For each λ = 0, 1 . . . , t let us consider iλ , jλ ∈ [n]2l given by
jλ = (j1 , j2 , . . . , j2s , k1 , k1 , . . . , kλ , kλ , j2s+2λ , j2s+2λ+1 , . . . , j2l )
iλ = (i1 , i2 , . . . , i2s , k1 , k1 , . . . , kλ , kλ , i2s+2λ , i2s+2λ+1 , . . . , i2l ) .
Note that jt = it since ik = jk for all k ∈ [2s] because Clump(π) = [2s]. Moreover, we have
i0 = i ∼ i1 ∼ . . . ∼ it , and thus f (i) = f (it ) since f belongs to the χχ-class. Similarly,
f (j) = f (jt ). Therefore, f (j) = f (i).
3.2
Completion of proof via the Fibonacci-Whitle inequality
In this section, we state and prove Theorem 21 to complete the proof of Theorem 10. Here
is a brief description of the results of this section. Theorem 21 is a consequence of Lemmas
17 and 20. Proposition 18 and Corollary 19 provide the main combinatorial results and
are applied to prove Lemma 20 from Lemma 17. Finally, the proof of Lemma 17 relies on
Propositions 15 and 16 which involves Fibonacci random variables and Whitle’s inequality.
The following is the main result in [5].
Whitle’s inequality. Let X1 , . . . , Xn be real independent random variables with zero mean.
Suppose there exists a real number l ∈ [2, ∞) such that kXi k2l < ∞ for i = 1, . . . , n. Then
there exists a constant 0 ≤ Cl < ∞ depending only on l such that for all matrices A ∈ Mn (C)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
26
with real entries we have

1
2
n
n
X
X
2
2
2

A(i, j)(Xi Xj − E[Xi Xj ])
|A(i, j)| kXi k2l kXj k2l  .
≤ Cl
i,j=1
i,j=1
l
A X classical random variable defined on a probability space (Ω, F, µ) is a Fibonacci
random variable if X 2 = X + 1 almost surely and E[X] = 0. Note that if X is a Fibonacci
random variable then for k = 0, 1, . . . we have X k+2 = X k+1 + X k almost surely, and hence
E[X k+2 ] = E[X k+1 ] + E[X k ]. Therefore,
E[X] = 0, E[X 2 ] = 1, E[X 3 ] = 1, E[X 4 ] = 2, E[X 5 ] = 3, E[X 6 ] = 5, . . .
Proposition 15. Let X1 , . . . , Xn be independent identically distributed Fibonacci random
variables. Then for every real number l ∈ [2, ∞) there exists a constant 0 ≤ Kl < ∞
depending only on l such that for all matrices A ∈ Mn (C) we have
n
X
|A(i, j)|(Xi Xj − δi,j )
≤ Kl kAk2 .
i,j=1
l
We refer to this inequality as the Fibonacci-Whittle inequality.
Proof. Let l ∈ [2, ∞). Then, since X1 , . . . , Xn independent identically distributed Fibonacci random variables, for i, j = 1, 2 . . . , n we have
E[Xi Xj ] = δi,j =


 1, if i = j
and
kX1 k2l = kXi k2l < ∞ .

 0, otherwise
Thus, by Whitle’s inequality we get

1
2
n
n
X
X
2
2
2
2


=
C
kX
k
|A(i,
j)|(X
X
−
δ
)
≤
C
|A(i,
j)|
kX
k
kX
k
1 2l kAk2
i j
i,j i 2l
j 2l
l
l
i,j=1
i,j=1
l
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
27
for some constant 0 ≤ Cl < ∞ depending only on l.
Proposition 16. Suppose X1 , . . . , Xn are independent identically distributed Fibonacci random variables. Then for every for every l-tuple i = (i1 , . . . , i2l ) ∈ [n]2l we have
"
E
l
Y
#
(Xi2k−1 Xi2k − δi2k−1 ,i2k ) ≥


 1, if Π(i) ∈ P χχ(2l)
.

 0, otherwise
k=1
Proof. Suppose X1 , . . . , Xn are independent identically distributed Fibonacci random variables. Let i = (i1 , . . . , i2l ) ∈ [n]2l . Note that for 1 ≤ i, j ≤ n (for i, j ∈ [n]) we have
Xi Xj − δi,j = Xi Xj (1 − δi,j ) − Xi δi,j =


 Xi Xj
, if i 6= j
(1)

 Xi2 − 1 = Xi , otherwise
since X1 , . . . , Xn are Fibonacci. Thus,
l
Y
(Xi2k−1 Xi2k − δi2k−1 ,i2k ) =
k=1
n
Y
Xλαλ
(2)
λ=1
for some α1 , . . . , αn ∈ {0, 1, 2, . . .}, and hence
"
E
l
Y
#
(Xi2k−1 Xi2k − δi2k−1 ,i2k ) =
k=1
l
Y
E Xλαλ ≥ 0
λ=1
since X1 , . . . , Xn are independent Fibonacci.
Suppose now αλ = 1 for some λ ∈ {1, 2, . . . , n}. Then, by (1) and (2) there is one and
only one k ∈ {1, . . . , l} such that Xλ is a factor of Xi2k−1 Xi2k − δi2k−1 ,i2k . So, λ belongs to
{i2k−1 , i2k } but does not belong to {i2k0 −1 , i2k0 } for k 0 ∈ {1, . . . , l} \ {k}. Thus, either Π(i)
contains a singleton if i2k−1 6= i2k , or Π(i) contains a the block {2k − 1, 2k} if i2k−1 = i2k .
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
28
Therefore, if Π(i) ∈ Pχχ (2l) then αλ 6= 1 for λ = 1, . . . , n, and thus
"
E
l
Y
#
(Xi2k−1 Xi2k − δi2k−1 ,i2k ) =
l
Y
E Xλαλ ≥ 1
λ=1
k=1
Lemma 17. Suppose we have matrices A1 , . . . , Al ∈ Mn (C). Let A : [n]2l → C defined
l
Q
for every i = (i1 , . . . , i2l ) ∈ [n]2l by A(i) =
Ak (i2k−1 , i2k ). Then there exists a constant
k=1
0 ≤ Cl < ∞ depending only on l such that
X
|A(i)| ≤ Cl
i∈[n]2l
l
Y
kAk k2 .
k=1
Π(i)∈Pχχ (2l)
Proof. We may assume l ≥ 2 since otherwise there is nothing to prove because Pχχ (2l) = ∅.
Let X1 , . . . , Xl be independent identically distributed Fibonacci random variables. Consider
G : [n]2l → C defined for each i = (i1 , . . . , i2l ) ∈ [n]2l by
G(i) =
l
Y
(Xi2k−1 Xi2k − δi2k−1 ,i2k ) .
k=1
By Proposition 16 and linearity of E[·] we have

X
i∈[n]2l
Π(i)∈Pχχ (2l)
|A(i)| ≤
X
i∈[n]2l
|A(i)| E [G(i)] = E 

X
i∈[n]2l
|A(i)| G(i)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
29
Now, note that
X
|A(i)|G(i) =
i∈[n]2l
X
l
Y
i∈[n]2l
k=1
|Ak (i2k−1 , i2k )|(Xi2k−1 Xi2k − δi2k−1 ,i2k )
i=(i1 ,...,i2 )
l
Y
=


n X
n
X

|Ak (i, j)|(Xi Xj − δi,j )
k=1
i=1 j=1
and hence, by Hölder’s inequality we get
l X
n
l
Y
Y
n X
l
E
|A(i)|G(i) ≤
|A
(i,
j)|(X
X
−
δ
)
kAk k2
≤
(K
)
i j
i,j k
l
2l
i=1
j=1
k=1
k=1
i∈[n]


X
l
where the constant Kl is given by Proposition 15.
Proposition 18. Suppose we have functions f1 , . . . , fl : [n] → C and let f : [n]l → C be
defined for i = (i1 , . . . , il ) ∈ [n]l by f (i) = f1 (i1 ) . . . fl (il ). Then for every subset I of [n] we
have
X
f (i1 , . . . , il ) =
distinct
i1 ,...,il ∈I
X
µ(0l , π)
X
f (i)
(3.5)
i∈I l
π∈P (l)
π≤Π(i)
where 0l := {{k} : k ∈ [l]} ∈ P (l) and µ : P (l) × P (l) → C is the Möbius inversion function
for the poset P (l); additionally, for every partition π ∈ P (l) we have
X
i∈I l
f (i) =
YXY
fb (i).
(3.6)
B∈π i∈I b∈B
π≤Π(i)
If J is a subset of [n] and
we have
Pn
i=1 fk (i)
= 0 for k = 1, 2, . . . , l, then for every partition π ∈ P (l)
Y X Y
X
|f (i)|
fb (i) ≤
B∈π i∈[n]\J b∈B
i∈[n]l
χ,J
where [n]lχ,J := i = (i1 , . . . , i2l ) ∈ [n]l : {k} ∈ Π(i) ⇒ ik ∈ J .
(3.7)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
30
Proof. We first prove equality (3.5). Let I ⊂ [n]. Since Π(i) = {i−1 (λ) 6= ∅ : λ =
1, 2, . . . , n} where i−1 (λ) = {k ∈ [l] : ik = λ} for λ = 1, 2, . . . , n, we have that
X
X
f (i1 , . . . , il ) =
f (i).
i∈I l
distinct
i1 ,...,il ∈I
Π(i)=0l
Now, recall that the Möbius inversion function for P (l) is the unique function µ : P (l) ×
P (l) → C such that for σ, θ ∈ P (l) we have
X
µ(σ, π)ζ(π, θ) =
π∈P (l)
σ≤π≤θ


 1
if σ = θ

 0
otherwise
(3.8)
where ζ : P (l) × P (l) → C is defined for π, θ ∈ P (l) by
ζ(π, θ) =


 1
if π ≤ θ

 0
otherwise
.
Hence, for θ ∈ P (l) we have
X
π∈P (l)
π≤θ
µ(0l , π) =
X
µ(0l , π) =
π∈P (l)
0l ≤π≤θ
X
µ(0l , π)ζ(π, θ) =
π∈P (l)
0l ≤π≤θ
where 0l := {{k}|k = 1, . . . , l} ∈ P (l). Thus, we get
X
i∈I l
Π(i)=0l
f (i) =
X
i∈I l
f (i)
X
π∈P (l)
π≤Π(i)
µ(0l , π)


 1
if 0l = θ

 0
otherwise
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
31
but
X
X
f (i)
i∈I l
X X
µ(0l , π) =
f (i)µ(0l , π)
i∈I l
π∈P (l)
π≤Π(i)
X
X
π∈P (l)
i∈I l
π∈P (l)
π≤Π(i)
=
X
=
f (i)µ(0l , π)
i∈I l ,π∈P (l)
π≤Π(i)
f (i)µ(0l , π) =
X
X
µ(0l , π)
f (i).
i∈I l
π∈P (l)
π≤Π(i)
π≤Π(i)
and therefore,
X
f (i1 , . . . , il ) =
X
f (i) =
i∈I l
distinct
i1 ,...,il ∈I
X
µ(0l , π)
X
f (i).
i∈I l
π∈P (l)
Π(i)=0l
π≤Π(i)
We now prove equality (3.6). Let π ∈ P (l) and write π = {B1 , B2 , . . . , Bm } and Bk =
{bk,1 , bk,2 , . . . , bk,pk } for k = 1, . . . , m with Bk = Bk0 only if k = k 0 and bk,t = bk0 ,t0 only if
k = k 0 , t = t0 . Then, since π is a partition of [l], for every i = (i1 , . . . , il ) ∈ I l we have
f (i) = f1 (i1 ) . . . fl (il ) =
Y Y
fb (ib ) =
pk
m Y
Y
fbk,t (ibk,t ).
k=1 t=1
B∈π b∈B
Now, note that for each i = (i1 , . . . , il ) ∈ I l we have π ≤ Π(i) if and olny if ibk,1 = ibk,2 =
. . . = ibk,pk for k = 1, 2, . . . , m. Thus, we get
X
f (i) =
i=∈I l
π≤Π(i)
pk
m Y
Y
X
)∈I l
i=(i1 ,...,il
π≤Π(i)
X
fbk,t (ibk,t ) =
k=1 t=1
pk
m Y
Y
fbk,t (ik ) ,
i1 ,...,im ∈I k=1 t=1
and therefore,
X
i=∈I l
f (i) =
X
pk
m Y
Y
i1 ,...,im ∈I k=1 t=1
fbk,t (ik ) =
pk
m XY
Y
k=1 i∈I t=1
fbk,t (i) =
YXY
fb (i).
B∈π i∈I b∈B
π≤Π(i)
Finally, we prove equality (3.7). Let J be a subset of [n] and suppose
Pn
i=1 fk (i)
=0
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
32
for k = 1, 2, . . . , l. After relabeling if necessary, we may assume r ∈ {0, . . . , m} is such that
B1 , . . . , Br are singletons whereas Br+1 , . . . , Bm are not. Take Bk = {bk } for 1 ≤ k ≤ r.
Note that
r
Y
X
Y
r
Y
X
r
fb (i) = (−1)
fbk (ik ) = (−1)
i1 ,...,ir ∈J k=1
k=1 i∈[n]\J b∈Bk
X
r
r Y
Y
fb (ik )
i1 ,...,ir ∈J k=1 b∈Bk
since by hypothesis for 1 ≤ k ≤ r we have
X
Y
X
fb (i) =
i∈[n]\J b∈Bk
fbk (i) = −
X
fbk (i) = −
i∈J
i∈[n]\J
X Y
fb (i) ,
i∈J b∈Bk
and hence,
r
Y
X
Y
fb (i) =
k=1 i∈[n]\J b∈Bk
r
Y

−

X Y
fb (i) = (−1)r
fb (ik ).
i1 ,...,ir ∈J k=1 b∈Bk
i∈J b∈Bk
k=1
r Y
Y
X
Note also that
m
Y
X
Y
fb (i) =
k=r+1 i∈[n]\J b∈Bk
m
Y
X
Y
fb (ik ).
ir+1 ,...,im ∈[n]\J k=r+1 b∈Bk
Thus, since

Y X Y
fb (i) = 
B∈π i∈[n]\J b∈B
r
Y
X

Y
fb (i) 
k=1 i∈[n]\J b∈Bk
m
Y

X
Y
fb (i) ,
k=r+1 i∈[n]\J b∈Bk
it follows

Y X Y
B∈π i∈[n]\J b∈B
fb (i) = (−1)r
X
m Y
Y

i1 ,...,ir ∈J
ir+1 ,...,im ∈[n]\J
k=1 b∈Bk

fb (ik ) .
(1)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
33
Consider now the subset [n]l,J of [n]l given by
[n]l,J



ib1 , . . . , ibr ∈
J
:= (i1 , . . . , il ) ∈ [n]l 

ibk,1 = . . . = ibk,pk ∈ [n] \ J



.

for r + 1 ≤ k ≤ m 
Observe that

X
f (i) =
X

X
fb (ib ) =

(i1 ,...,il )∈[n]l,J
i∈[n]l,J

m Y
Y

i1 ,...,ir ∈J
ir+1 ,...,im ∈[n]\J
k=1 b∈Bk
m Y
Y

fb (ik ) .
(2)
k=1 b∈Bk
Thus, by (1) and (2) it only remains to show that [n]l,J is contained in [n]lχ,J to prove (3.7).
Suppose i = (i1 , . . . , il ) ∈ [n]l,J and {k 0 } ∈ Π(i) for some k 0 ∈ {1, . . . , l}. Then, ik0 = ik
if and only if k = k and by definition of [n]l,J we have ibk,1 = . . . = ibk,pk with 2 ≤ pk for
r + 1 ≤ k ≤ m. Hence, k 0 ∈ {b1 , . . . , br }, and so ik0 ∈ J. It follows i ∈ [n]lχ,J . Thus, [n]l,J is
a subset of [n]lχ,J .
Corollary 19. Suppose we have functions f1 , . . . , fl : [n] → C and let f : I l → C be
defined for each i = (i1 , . . . , il ) ∈ [n]l by f (i) = f1 (i1 ) . . . fl (il ) . If J is a subset of [n] and
Pn
i=1 fk (i) = 0 for k = 1, . . . , n, then
X
X
f (i1 , . . . , il ) ≤ l2l
|f (i)|
distinct
l
i∈[n]χ,J
i1 ,...,il ∈[n]\J
(3.9)
where [n]lχ,J := i = (i1 , . . . , il ) ∈ [n]l : {k} ∈ Π(i) ⇒ ik ∈ J .
Proof. Let J be a subset of [n]. Then, by (3.5) and (3.6), we have
X
distinct
i1 ,...,il ∈[n]\J
f (i1 , . . . , il ) =
X
π∈P (l)
µ(0l , π)
X
i∈([n]\J)l
π≤Π(i)
f (i) =
X
π∈P (l)
µ(0l , π)
Y X Y
B∈π i∈[n]\J b∈B
fb (i)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
34
and hence, by (3.7), we get
X
X
X
≤
|f (i)| .
|µ(0
,
π)|
f
(i
,
.
.
.
,
i
)
1
l
l
π∈P (l)
distinct
l
i∈[n]
i1 ,...,il ∈[n]\J
χ,J
The result follows since
X
|µ(0l , π)|
π∈P (l)
X
i∈[n]lχ,J
|f (i)| ≤
X
π∈P (l)
ll
X
|f (i)| ≤ l2l
i∈[n]lχ,J
X
|f (i)| .
i∈[n]lχ,J
because |P (l)| ≤ ll and |µ(0l , π)| ≤ ll for every π ∈ P (l).
Lemma 20. Suppose A1 , . . . , Al ∈ Mn (C) are of trace zero and let F : [n]2l → C be a
function of the χχ-class. Then there exists a constant 0 ≤ cl < ∞ depending only on l such
that for all partitions π ∈ Pχ (2l) \ Pχχ (2l) we have
l
X
Y
A(i)F(i) ≤ cl max |F(i)|
kAk k2
i∈[n]2l
2l
k=1
i∈[n]
Π(i)=π
where A(i) =
l
Q
Ak (i2k−1 , i2k ) for each i = (i1 , . . . , i2l ) ∈ [n]2l .
k=1
Proof. Let π ∈ Pχ (2l) \ Pχχ (2l). Then, π lacks of singletons, but it does contain some
blocks of the form {2k − 1, 2k}. By symmetry, we may assume that for some s ∈ {0, . . . , l}
we have
π = θ ∪ {{2s + 1, 2s + 2}, . . . , {2l − 1, 2l}}
where θ is the empty set if s = 0 and otherwise belongs to Pχχ (2s). Put t = l − s
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
35
Let us note we may assume 6l ≤ n. Indeed, by Cauchy-Schwarz inequality we have
2
2
X
X
X
X

|A(i)|2
12 = n2l
|A(i)|2
|A(i)| · 1 ≤
|A(i)| = i∈[n]2l
i∈[n]2l
i∈[n]2l
i∈[n]2l
i∈[n]2l

X
but
n
X
i∈[n]2l
|A(i)|2 =
n
X
l
Y
|Ak (i2k−1 , i2k )|2 =
i1 ,...,i2l =1 k=1
l X
n
Y
|Ak (i, j)|2 =
k=1 i,j=1
l
Y
kAk k22 .
k=1
Hence,
X
|A(i)| ≤
i∈[n]2l
X
|A(i)| ≤ n
i∈[n]2l
l
l
Y
kAk k2 ,
k=1
Π(i)=π
and therefore, for n < 6l we have
l
X
X
Y
l
A(i)F(i) ≤ max |F(i)|
kAk k2 .
|A(i)| ≤ (6l) max |F(i)|
i∈[n]2l
i∈[n]2l
k=1
i∈[n]2l
i∈[n]2l
Π(i)=π
Π(i)=π
Assume 6l ≤ n. By Proposition 14 we have
F(i1 , . . . , i2s , j1 , j1 , . . . , jt , jt ) = F(i1 , . . . , i2s , j10 , j10 , . . . , jt0 , jt0 )
for i1 , . . . , i2s ∈ [n] and j1 , j10 , . . . , jt , jt0 ∈ [n] \ {i1 , . . . , i2s } such that jk = jk0 only if k = k 0
and jk0 = jk0 0 only if k = k 0 . Thus, the function F : [n]2s → C defined for i = (i1 , . . . , i2s ) ∈
[n]2s by
F(i) = F(i1 , . . . , i2s , j1 , j1 , . . . , jt , jt )
where we take any distinct j1 , . . . , jt ∈ [n] \ {i1 , . . . , i2s }, is well-defined.
Now, to abbreviate notation, for i = (i1 , . . . , i2s ) ∈ [n]2s and j = (j1 , . . . , jt ) ∈ [n]t we
take {i} = {i1 , . . . , i2s } subset of [n], and i ? j2 = (i1 , . . . , i2s , j1 , j1 , . . . , jt , jt ) element of
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
36
[n]2l .
b : [n]t → C given by
Consider the functions A : [n]2s → C and A
A(i) = A1 (i1 , i2 ) . . . As (i2s−1 , i2s )
and
b
A(j)
= As+1 (j1 , j1 ) . . . Al (jt , jt )
for i = (i1 , . . . , i2s ) ∈ [n]2s and j = (j1 , . . . , jt ) ∈ [n]t . By Corollary 19 we have
X X X
X b b
A(i) ≤
A(i) t2t
A(j)
A(j
,
.
.
.
,
j
)
1
t
distinct
i∈[n]2s
j∈[n]tχ,{i}
i∈[n]2s
j1 ,...,jt ∈[n]\{i}
Π(i)=θ
Π(i)=θ
where [n]tχ,{i} = j = (j1 , . . . , jt ) ∈ [n]t
|
{k} ∈ Π(j) ⇒ jk ∈ {i} . Hence, it follows
X X
X
X b
A(i ? j2 )
≤ l2l
A(i) A(j
,
.
.
.
,
j
)
1
t
distinct
i∈[n]2s j∈[n]tχ,{i}
i∈[n]2s
j1 ,...,jt ∈[n]\{i}
Π(i)=θ
Π(i)=θ
b
since t2t ≤ l2l and A(i ? j2 ) = A(i)A(j)
for i ∈ [n]2s and j ∈ [n]t .
Note that for i = (i1 , . . . , i2l ) ∈ [n]2l we have Π(i) = π if and only if Π((i1 , . . . , i2s )) = θ,
i2s+1 = i2s+2 , . . . , i2l−1 = i2l , i2s+2 , . . . , i2l ∈ [n] \ {i1 , . . . , i2s }, and i2s+2 , i2s+4 , . . . , i2l are
all distinct. Hence,
X
F(i)A(i) =
2l
i∈[n]
Π(i)=π
=
X
X
i∈[n]2s
j∈([n]\{i})t
Π(i)=θ
Π(j)=0t
X
i∈[n]2s
Π(i)=θ
F(i)A(i)
F(i ? j2 )A(i ? j2 )
X
j∈([n]\{i})t
Π(j)=0t
b
A(j).
=
X
X
i∈[n]2s
j∈([n]\{i})t
Π(i)=θ
Π(j)=0t
b
F(i)A(i)A(j)
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
37
Thus, since
X
X
X
X
b ≤ max |F(i)|
b ,
A(i) F(i)A(i)
A(j)
A(j)
i∈[n]2l
i∈[n]2s
j∈([n]\{i})t
j∈([n]\{i})t
i∈[n]2s
Π(i)=θ
Π(j)=0t
Π(j)=0t
Π(i)=θ
it follows
X
X
X A(i ? j2 )
F(i)A(i) ≤ l2l max |F(i)|
i∈[n]2l
i∈[n]2l
i∈[n]2s j∈[n]tχ,{i}
Π(i)=π
Π(i)=θ
Observe now that Π(i ? j2 ) belongs to Pχχ (2l) whenever i ∈ [n]2s with Π(i) = θ and
j ∈ [n]tχ,{i} . Hence, we have
X
X
i∈[n]2s
j∈[n]tχ,{i}
Π(i)=θ
X
A(i ? j2 ) ≤
|A(i)|
i∈[n]2l
Π(i)∈Pχχ (2l)
and therefore, by Lemma 17 we get
l
X
X
Y
2l
2l
F(i)A(i) ≤ l max |F(i)|
|A(i)| ≤ l Cl max |F(i)|
kAk k2
i∈[n]2l
i∈[n]2l
2l
k=1
i∈[n]2l
i∈[n]
Π(i)=π
Π(i)∈Pχχ (2l)
for some constant 0 ≤ Cl < ∞ depending only on l.
Theorem 21. Suppose A1 , . . . , Al ∈ Mn (C) are of trace zero and let F : [n]2l → C be a
function of the χχ-class. Then there exists a constant 0 ≤ κl < ∞ depending only on l such
that
X
l
Y
≤ κl n 2l max |F(i)|
F(i)A(i)
kAk k
i∈[n]2l
i∈[n]2l
k=1
where A(i) := A1 (i1 , i2 ) . . . Al (i2l−1 , i2l ) for each i = (i1 , . . . , i2l ) ∈ [n]2l .
Proof. Since P (2l) is the disjoint union of P (2l) \ Pχ (2l), Pχ (2l) \ Pχχ (2l), and Pχχ (2l),
3.2. COMPLETION OF PROOF VIA THE FIBONACCI-WHITLE
INEQUALITY
38
we have
X
i∈[n]
F(i)A(i) =
X
F(i)A(i)+
i∈[n]2l
2l
X
X
F(i)A(i)+
2l
i∈[n]
Π(i)∈Pχχ (2l)
Π(i)∈P (2l)\Pχ (2l)
F(i)A(i).
i∈[n]2l
Π(i)∈Pχ (2l)\Pχχ (2l)
But, F(i) = 0 whenever Π(i) ∈ P (2l) \ Pχ (2l) since F belongs to th χ-class, and thus we get
X
i∈[n]
X
F(i)A(i) =
2l
2l
i∈[n]
Π(i)∈Pχχ (2l)
X
F(i)A(i) +
X
F(i)A(i).
2l
π∈Pχ (2l)\Pχχ (2l) i∈[n]
Π(i)=π
Now, by Lemmas 17 and 20 there is a constant cl such that
X
2l
i∈[n]
Π(i)∈Pχχ (2l)
and
|F(i)A(i)| ≤ cl max |F(i)|
i∈[n]2l
l
Y
kAk k2 ,
k=1
l
X
X
X
Y
F(i)A(i) ≤
cl max |F(i)|
kAk k2 .
i∈[n]2l
2l
k=1
π∈Pχ (2l)\Pχχ (2l) i∈[n]
π∈Pχ (2l)\Pχχ (2l)
Π(i)=π
Thus, taking κl = cl (1 + 2l2l ) ≥ cl (1 + |Pχ (2l) \ Pχχ (2l)|), we have
X
l
Y
≤ κl max |F(i)|
F(i)A(i)
kAk k2
i∈[n]2l
i∈[n]2l
k=1
Therefore, since kAk2 ≤
√
nkAk for all matrices A ∈ Mn (C), we conclude
l
Y
X
l
2
F(i)A(i) ≤ κl n max |F(i)|
kAk k
i∈[n]2l
i∈[n]2l
k=1
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
3.3
39
Completion of proof via sharp bounds for graph sums
In this section, we state and prove Theorem 27. For this, we first introduce some terminology
and notation from graph theory. Then, we recall and use the main result of [3] to prove
Lemma 25. Finally, we show how Theorem 27 is a consequence of Lemma 25 and Corollary
26.
A graph G is an ordered pair (V, E) consisting of a non-empty set V of vertices and a
(possibly empty) set E of edges together with an incidence function Ψ : E → V ∗ where
V ∗ := {{v, v 0 } : v, v 0 ∈ V }. A graph is said to be finite if its vertex and edge sets are both
finite. Unless otherwise specified, given a graph G, we write VG for its vertex set, EG for
its edge set, and ΨG for its incidence function.
If e is and edge and u and v are vertices in a graph G such that ΨG (e) = {u, v}, then e
is said to join u and v, e is said to be incident with u and v, and the vertices u and v are
called the ends of e. An edge with identical ends is called a loop. The usual way to picture
a graph is by drawing a dot for each vertex and joining two of these dots by a line for each
edge whose ends are represented exactly by these two dots.
A graph H is called a subgraph of a graph G (also H is contained in G) if VH ⊂ VG ,
EH ⊂ EG , and ΨH (e) = ΨG (e) for every edge e ∈ EH . Two subgraphs are disjoint(or
disconnected ) if they have no common vertex. Note that two subgraphs H and H 0 are
disjoint if and only if EH ∩ EH 0 = ∅ and VH ∩ VH 0 = ∅ since EH ∩ EH 0 6= ∅ implies
VH ∩VH 0 6= ∅. If H1 , . . . , Hm are pair-wise disjoint subgraphs of G such that EG = ∪m
k=1 EHk
and VG = ∪m
k=1 VHk , we say that G is the disjoint union of H1 , . . . , Hm .
A walk W in a graph G is a non-empty alternating sequence v0 , e1 , v1 , . . . , vl−1 , el , vl of
vertices and edges in G such that vk−1 and vk are the ends of ek for k = 1, 2, . . . , l. The
vertices v0 and vl called the end of W ; moreover, v0 and vl are said to be connected by W .
If in addition v1 , . . . , vl are all distinct, W is called a path. A graph G is connected if every
two distinct vertices of G are connected by a path in G. A connected component of G is a
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
40
subgraph which is connected and is not properly contained in a larger connected subgraph.
Proposition 22. Every finite graph is the disjoint union of its connected components.
A cutting edge of a finite graph is an edge whose removal would increase the number
of connected components. In particular, an edge e in a connected graph G is a cutting
edge if and only if the graph with edge set EG \ {e}, vertex set VG , and incidence function
defined as ΨG restricted to EG \ {e} is not connected, i.e., it has two connected components.
A graph is two-edge connected if it is connected and does not contain a cutting edge. A
two-edge connected component of a graph is a subgraph which is two-edge connected and is
not properly contained in a larger two-edge connected subgraph.
Proposition 23. Suppose G is a finite graph. Then there exists a positive integer m ∈ N
and D1 , . . . , Dm pair-wise disjoint subgraphs of G such that
(i) D1 , . . . , Dm are two edge-connected components of G,
(ii) every vertex of G belongs to some subgraph Dk , i.e., VG = ∪m
k=1 VDk , and
(iii) an edge e ∈ EG is a cutting edge if and only if it does belong to EG \ ∪m
k=1 EDk
Thus, if we remove all cutting edges of a graph, what we get is a subgraph whose
connected components are precisely the two-edge connected components of the initial graph.
To each finite graph G, we associate a new graph whose vertices are the two-edge connected
components of G, its edges are the cutting edges of G, and two vertices of the new graph
are joined by an edge if there is a cutting edge between vertices from the two corresponding
components in G. To be precise, we set the following.
Notation. Suppose G is a finite graph and let m and D1 , . . . , Dm be as in the previous
proposition. We denote by F(G) the graph with vertex set VF(G) = {D1 , . . . , Dm }, edge set
∗
EF(G) = EG \ ∪m
k=1 EDk , and incidence function ΨF(G) : EF(G) → VF(G) defined for e ∈ EF(G)
by ΨF(G) (e) = {Di , Dj } if there are vertices v ∈ VDi and v 0 ∈ VDj such that ΨG (e) = {v, v 0 }.
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
41
A path in a graph G with identical ends is called a cycle, and a graph without cycles
is called forest. Notice then that F(G) is a forest for every finite graph G. We called F(G)
the forest of two-edge connected components of G.
Recall that the degree of a vertex v in a graph G, denoted by degG (v), is the number of
edges of G incident with v, each loop counting as two edges. Moreover, if H is a subgraph
of G, we define the degree of H (with respect to G), denoted degG (H), as the sum of the
degrees of all vertices in H, i.e.,
degG (H) :=
X
degG (v) .
v∈VH
Now, a leaf in a forest is a vertex whose degree is at most 1. If we are given a forest F, we
associate to F the rational number
r(F) =
X
rl
l∈LF
where LF denotes the set of leaves in F and for each leaf l ∈ LF we take rl =
1
2
if degF (l) = 1
and rl = 1 otherwise. Note that if F is connected then r(F) equals max{1, |L2F | }. Furthermore, if G is the disjoint union of its subgraphs H1 , . . . , Hm and we take F = F(G) and
Fk = F(Hk ) for k = 1, 2, . . . , m, then we have the relation
r(F) = r(F1 ) + r(F2 ) + . . . + r(Fm ) .
Notation. Given a partition π = {B1 , . . . , Bp } ∈ P (2l), we denote by Gπ the graph with
vertex set Vπ = {B1 , . . . , Bp }, edge set Eπ = {{2k − 1, 2k} : 1 ≤ k ≤ l}, and incidence
function Ψ : Eπ → Vπ∗ defined by Ψ({2k − 1, 2k}) = {Bi , Bj } if 2k − 1 ∈ Bi and 2k ∈ Bj
for k = 1, . . . , l.
We should mention that the following result and its proof have been taken from [2].
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
42
Proposition 24. Suppose π ∈ Pχχ (2l). Then any connected component of Gπ has at least
two edges and any two-edge connected component of Gπ that becomes a leaf in F(Gπ ) has
at least degree three.
Proof. Let H be a connected component of Gπ and assume v is a vertex in H. By the
construction of Gπ , the vertex v is in a block of π, say B. Thus, since π lacks singletons,
because it belongs to Pχχ (2l), there are b1 , b2 ∈ [2l] with b1 6= b2 such that {b1 , b2 } is
contained in B.
Note now that if {b1 , b2 } = {2k − 1, 2k} for some k ∈ [l], then there exists b3 ∈ B \
{b1 , b2 } since {2k − 1, 2k} can not be a block of π because π belongs to Pχχ (2l). Hence, by
connectedness, H contains the edge e = {2k − 1, 2k} and the edge e0 6= e containing b3 .
On the other hand, if {b1 , b2 } =
6 {2k −1, 2k} for all k ∈ [l], then the edge e1 containing b1
is distinct from the edge e2 containing b2 and both e1 and e2 belong to H by connectedness.
Let D be a two-edge connected component of Gπ . Note that every vertex of Gπ has at
least degree two since π does not contain singletons. Thus, D has degree at least four if it
contains two distinct vertices. So, let us assume D has only one vertex, say v. Then, the
degree of D is exactly the degree of v. Moreover, if the degree of v is two, then D does not
become a leaf in F(Gπ ).
Therefore, any two-edge connected component of Gπ that becomes a leaf in F(Gπ ) has
at least degree three.
The following is the main result of [3] and represents the cornerstone of Theorem 27.
Sharp bound. Suppose π is a partition of [2l] and take F = F(Gπ ). Then for all positive
integer n ∈ N and matrices A1 , . . . , Al ∈ Mn (C) we have
l
X
Y
A(i) ≤ nr(F)
kAk k
k=1
i∈[n]2l
π≤Π(i)
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
43
where A(i) := A1 (i1 , i2 ) . . . Al (i2l−1 , i2l ) for i = (i1 , . . . , i2l ) ∈ [n]2l .
Lemma 25. Let A1 , . . . , Al ∈ Mn (C). Then for every partition π ∈ Pχχ (2l) we have
l
X
l Y
A(i) ≤ n 2
kAk k
2l
k=1
i∈[n]
π≤Π(i)
where A(i) =
l
Q
Ak (i2k−1 , i2k ) for each i = (i1 , . . . , i2l ) ∈ [n]2l .
k=1
Proof. Let π ∈ Pχχ (2l) and put F = F(Gπ ). Assume G1 , . . . , Gm are the connected components of Gπ and let F1 , . . . , Fm be their respective trees of two-edge connected components.
For each k = 1, . . . , m, let us take pk as the number of edges in Gk , qk as the number of
leaves in Fk , and denote by Dk,1 , . . . , Dk,qk the two-edge connected components of Gk that
become a leaf in Fk .
By Proposition 24, we have pk ≥ 2 and degGk (Dk,λ ) ≥ 3 for k = 1, 2, . . . , m and
λ = 1, 2, . . . , qk . Then, we get
n q o
pk
k
≥ max 1,
2
2
q
q
k
k
1
1X
1X
degGk (Dk,λ ) ≥
3 ≥ qk .
deg(Gk ) ≥
2
2
2
λ=1
λ=1
Now, since Gk is connected we have r(Fk ) = max 1, q2k for k = 1, 2, . . . , m. Moreover,
since
pk
2
≥ 1 and pk =
since Gπ is the disjoint union of G1 , . . . , Gm we get
r(F) =
m
X
k=1
r(Fk ) ≤
m
X
pk
k=1
2
=
l
2
Therefore, the result follows from Sharp bound.
Corollary 26. Suppose A1 , . . . , Al ∈ Mn (C) are of trace zero. Then there exists a constant
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
44
0 ≤ cl < ∞ depending only on l such that for all partitions π ∈ Pχ (2l) we have
l
X
l Y
2
A(i)
≤
c
n
kAk k
l
2l
k=1
i∈[n]
Π(i)=π
where A(i) =
l
Q
Ak (i2k−1 , i2k ) for each i = (i1 , . . . , i2l ) ∈ [n]2l .
k=1
Proof. Let us consider the functions G, H : P (2l) → C given for π ∈ P (2l) by
G(π) =
X
A(i)
and
X
H(π) =
i∈[n]2l
π=Π(i)
A(i) .
i∈[n]2l
π≤Π(i)
Note that for every θ ∈ P (2l) we have
H(θ) =
X
A(i) =
i∈[n]2l
X
X
A(i) =
i∈[n]2l
π∈P (2l)
θ≤π π=Π(i)
θ≤Π(i)
X
G(π).
π∈P (2l)
θ≤π
Thus, by the Möbius inversion formula, there is a function µ : P (2l) × P (2l) → C such that
G(π) =
X
µ(π, θ)H(θ)
θ∈P (2l)
π≤θ
for every π ∈ P (2l).
Now, fix π ∈ Pχ (2l). Note that θ ∈ P (2l) and π ≤ θ implies θ ∈ Pχ (2l). Then, we have
G(π) =
X
θ∈Pχ (2l)
π≤θ
µ(π, θ)H(θ) =
X
θ∈Pχχ (2l)
π≤θ
µ(π, θ)H(θ) +
X
µ(π, θ)H(θ)
θ∈Pχ (2l)\Pχχ (2l)
π≤θ
since Pχ (2l) is the disjoint union of Pχχ (2l) and Pχ (2l) \ Pχχ (2l).
Suppose θ ∈ Pχ (2l) \ Pχχ (2l). Then θ does not contain singletons but it does contain
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
45
a block of the form {2k − 1, 2k}. By symmetry, we may assume θ = ϑ ∪ {{2l − 1, 2l}}
for some ϑ ∈ P (2l − 2). Hence, taking B(j) = B1 (j1 , j2 ) . . . Bl−1 (j2l−3 , j2l−1 ) for each
j = (j1 , . . . , j2l−2 ) ∈ [n]2l−2 , we have
H(θ) =
X
n
X
i∈[n]2l
j∈[n]2l−2
i=1
θ≤Π(i)
ϑ≤Π(j)
X
A(i) =
B(j)Al (i, i) =
X
B(j)tr(Al ) .
j∈[n]2l−2
ϑ≤Π(j)
Thus, H(θ) = 0 since by hypothesis Al is of trace zero. Consequently,
X
A(i) = G(π) =
i∈[n]2l
π=Π(i)
X
µ(π, θ)H(θ)
θ∈Pχχ (2l)
π≤θ
Therefore, by Lemma 25 we have


l
X
X
 X
 l Y


2
kAk k
A(i) ≤
|µ(π, θ)| |H(θ)| ≤ 
|µ(π, θ)| n
k=1
i∈[n]2l
θ∈Pχχ (2l)
θ∈Pχχ (2l)
π=Π(i)
π≤θ
π≤θ
Theorem 27. Suppose A1 , . . . , Al ∈ Mn (C) are of trace zero. Let F : [n]2l → C be a
function of the χ-class such that F(i) = F(j) for all i, j ∈ [n]2l for which Π(i) = Π(j). Then
there exists a constant 0 ≤ κl < 0 depending only on l such that
X
l
Y
≤ κl n 2l max |F(i)|
F(i)A(i)
kAk k
i∈[n]2l
i∈[n]2l
k=1
where A(i) := A1 (i1 , i2 ) . . . Al (i2l−1 , i2l ) for each i = (i1 , . . . , i2l ) ∈ [n]2l .
3.3. COMPLETION OF PROOF VIA SHARP BOUNDS FOR GRAPH
SUMS
46
Proof. Note first that
X
i∈[n]
F(i)A(i)
X
=
F(i)A(i)
2l
2l
X
+
F(i)A(i) .
2l
i∈[n]
Π(i)∈P (2l)\Pχ (2l)
i∈[n]
Π(i)∈Pχ (2l)
Thus, since F belongs to the χ-class, we get
X
i∈[n]
F(i)A(i)
=
X
F(i)A(i) .
2l
2l
i∈[n]
Π(i)∈Pχ (2l)
Now, by hypothesis we have Π(j) = Π(j0 ) implies F(j) = F(j0 ), and hence
X
2l
i∈[n]
Π(i)∈Pχ (2l)
F(i)A(i) =
X
X
F(i)A(i) =
π∈Pχ (2l) i∈[n]2l
Π(i)=π
X
π∈Pχ (2l)
F(π)
X
A(i)
i∈[n]2l
Π(i)=π
where F(π) = F(i) for some i ∈ [n]2l such that Π(i) = π. Therefore, by Corollary 26 we get


X
l
X
X X
l Y
≤
F(π) 

2
A(i)
≤
max
|F(i)|
F(i)A(i)
c
n
kAk k
l
i∈[n]2l
i∈[n]2l
π∈Pχ (2l)
2l
k=1
i∈[n]
π∈Pχ (2l)
Π(i)=π
for some constant 0 ≤ cl < ∞ depending only on l.
BIBLIOGRAPHY
47
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