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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. 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