Random matrices, continuous circular systems and the triangular operator
aa r X i v : . [ m a t h . OA ] J un RANDOM MATRICES, CONTINUOUS CIRCULAR SYSTEMSAND THE TRIANGULAR OPERATOR
ROMUALD LENCZEWSKI
Abstract.
Using suitably defined continuous analogs of the matricial circular sys-tems and the direct integral of Hilbert spaces H “ ş ‘ Γ H p γ q dγ , we study the op-erators living in H which give the asymptotic joint *-distributions of complex in-dependent Gaussian random matrices with not necessarily equal variances of theentries. These operators are decomposed in terms of continuous circular systems t ζ p x, y ; u q : x, y P r , s , u P U u acting between the fibers of H , the continuous analogsof matricial circular systems obtained when the Gaussian entries are block-identicallydistributed. In the case of square matrices with i.i.d. entries, we obtain the circularoperators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d.entries, we obtain the triangular operators of Dykema and Haagerup. We apply thisapproach to give a bijective proof of the formula for the moments of T ˚ T , where T is atriangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizterfor alternating ordered rooted trees. Introduction
Independent Gaussian random matrices with suitably normalized complex i.i.d. en-tries are asymptotically free with respect to the normalized trace composed with classi-cal expectation. The limit mixed (*-) moments can be expressed in terms of mixed (*-)moments of free circular operators. This fundamental result was shown by Voiculescu[16], who also found a relation to free group factors [17].Asymptotic freeness of Voiculescu was later generalized in many directions, in par-ticular, to the non-Gaussian random matrices by Dykema [3] and to Gaussian bandmatrices by Shlyakhtenko [13]. In the latter case, where the Gaussian variables arenot assumed to be identically distributed, scalar valued freeness [18] is not sufficient todescribe the asymptotics of matrices and one has to use freeness with amalgamationover some subalgebra, a generalization of freeness, in which a state is replaced by anoperator-valued conditional expectation with values in this subalgebra. This approachwas later used by Benaych-Georges [1] in his study of rectangular block random ma-trices, who introduced a rectangular analog of Voiculescu’s R-transform [18]. Analyticmethods like operator-valued transforms were also applied to the study of blocks of ran-dom matrices (most of these results are mentioned in the recently published monographof Mingo and Speicher [11]).The main point of our approach is that our realizations of the limit joint *-distributionsof Gaussian random matrices are built from operators living in Hilbert spaces ratherthan in Hilbert modules. Our combinatorics also has new features since it is based : 46L54, 60B20, 47C15
Key words and phrases: free probability, matricial freeness, random matrix, triangular operatorThis work is supported by Narodowe Centrum Nauki grant No. 2014/15/B/ST1/00166 on coloring noncrossing pair partitions rather than on using nested evaluations whichare needed when applying the conditional expectation and operator-valued free prob-ability. Although one can translate our approach to that of the operator-valued freeprobability, the new language provides a new realization of limit distributions as wellas connections to some nice results in combinatorics. For instance, the combinatorics ofcolored partitions allows us to find a connection between certain results on moments ofthe triangular operator defined by Dykema and Haagerup [4] with known enumerationresults for alternating ordered rooted trees. The triangular operator got some attentionin the context of the famous invariant subspace problem in the papers of Dykema andHaagerup on the so-called DT-operators and their decompositions [4,5], including thecircular operator of Voiculescu and the triangular operator T . The moments of T ˚ T were also found by Dykema and Haagerup in [4] and the more general case of the mo-ments of T ˚ k T k was treated by ´Sniady [14], who used the so-called generalized circularelements and the combinatorics of nested evaluations.Our original approach to the study of block matrices was related directly to theconnection between free probability and random matrices [7,8,9,10]. Since independent(Gaussian and also more general) random matrices t Y p u, n q : u P U u , are asymptoticallyfree under the expectation of the trace as n Ñ 8 , their blocks t S p,q p u q : 1 ď p, q ď r, u P U u should be asymptotically ’matricially free’ under ‘partial traces’. This idea was actuallyour main motivation to introduce a concept of a ‘matricial’ form of independence, called matricial freeness [7], where a family of subalgebras of a given algebra is replaced byan array of its subalgebras and a state is replaced by a family of states correspondingto the array’s diagonal. Although our construction was entirely based on scalar-valuedstates, it turned out later that this concept was related to freeness with amalgamationand operator-valued conditional expectation, at least in the context of matricial circularsystems studied in [10], where we refer the reader for details on this relation.As far as matricial circular systems are concerned, they describe the asymptoticsof blocks of independent Gaussian random matrices under partial traces. If we aregiven an ensemble of independent non-Hermitian n ˆ n Gaussian random matrices t Y p u, n q : u P U u with suitably normalized independent block-identically distributed( i.b.i.d. ) complex entries for each natural n , then the mixed *-moments of their (ingeneral, rectangular) blocks S p,q p u, n q converge under normalized partial traces to themixed *-moments of certain bounded operators, which we write informallylim n Ñ8 S p,q p u, n q “ ζ p,q p u q , where u P U and 1 ď p, q ď r and the operators ζ p,q p u q are called matricial circularoperators [10]. The arrays of such operators play the role of matricial analogs of circularoperators. If we want to find the counterparts of these operators in the operator-valuedfree probability, we need to take operators of the form F p c p u q F q , where t c p u q : u P U u is a family of circular elements living in the Fock space over Hilbert A -module, where A is the algebra of r ˆ r diagonal matrices with canonical generators F , . . . , F r .In particular, if the matrices Y p u, n q are square and have i.i.d. (normalized in astandard way) Gaussian entries and we divide them into r rectangular blocks S p,q p u q ,whose sizes tend to d p ˆ d q as n Ñ 8 , respectively, where d ` . . . ` d r “
1, then thelimit joint *-distribution of the family t Y p u, n q : u P U u under the expectation of the ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 3 normalized trace is that of the family of standard circular operators t ζ p u q : u P U u andthese are decomposed in terms of matricial circular operators as ζ p u q “ r ÿ p,q “ ζ p,q p u q for any u P U . They are expressed in terms of the canonical matricial creation andannihilation operators as ζ p,q p u q “ ℘ p,q p u q ` ℘ ˚ q,p p u q , which corresponds to the the well-known realization of the circular operator of Voiculescu c “ ℓ ` ℓ ˚ , where ℓ , ℓ are free. Here, we use the indices u , u to encode the fact thatwe have two free copies of creation-annihilation pairs related to the array labeled by u . If matrices have entries which are only i.b.i.d. and not i.i.d., then a similar formulaholds except that operators ζ p u q are no longer circular. In particular, this includes thecase when the matrices Y p u, n q are block strictly upper triangular (it suffices to set thecovariances of all variables lying on and under the diagonal to be equal to zero). In asimilar manner, block lower triangular matrices can be considered.One has to stress that objects like matricial circular operators are not new. Voiculescuintroduced and studied such objects in [17], although perhaps not in the context of theasymptotics of rectangular blocks. Namely, if A is the C ˚ -algebra generated by thesystem of free (generalized) circular operators t c p p, q, u q : 1 ď p, q ď r, u P U u , then ζ p,q p u q “ c p p, q, u q b e p p, q q P M r p A q for any p, q, u . These operators are the generators of free group factors used by Voiculescuin his proof of free group factors isomorphisms [17, Theorem 3.3]. We defined them dif-ferently, as operators on the matricially free Fock space of tracial type, but we showedin [10] that our matricial circular operators were in fact of the same tensor productform.In this paper, we generalize the decomposition of the operators ζ p u q , which give theasymptotic *-joint distributions of matrices Y p u, n q , to the continuous case. Namely,we investigate the limits of the *-mixed moments obtained in the discrete case as r Ñ 8 and show that they agree with the *-mixed moments of operators that can be writtenin the form ζ p g, u q “ ż ‘ Γ g p x, y q ζ p x, y ; u q dxdy, where Γ “ r , s ˆ r , s , g P L p Γ q , and the operators ζ p x, y ; u q are continuousgeneralizations of the matricial circular operators ζ p,q p u q : roughly speaking, p { r Ñ x , q { r Ñ y as r Ñ 8 . The operators ζ p x, y ; u q are continuous analogs of the matricialcircular operators ζ p,q p u q , namely ζ p x, y ; u q “ ℘ p x, y ; u q ` ℘ ˚ p y, x ; u q for any x, y P r , s and u P U , where t ℘ p x, y ; u q : x, y P r , s , u P U Y U u R. LENCZEWSKI is a suitably defined continuous family of creation operators, where U and U denotereplicas of U and the star stands for the adjoint operation.In order to define all these operators, we introduce a continuous analog of the matri-cially free Fock space of tracial type , namely the direct integral of Hilbert spaces H “ ż ‘ Γ H p γ q dγ, where Γ “ Ť n “ Γ n and Γ n “ I n ` , with I “ r , s ( dγ stands for the direct sum ofLebesgue measures on various Γ n ), equipped with the state ϕ p F q “ ż I x F p x q Ω p x q , Ω p x qy dx, where dx is the Lebesgue measure on I and F “ ş ‘ I F p x q dx is a decomposable operatorfield. In order to decompose the operators ζ p g, u q (circular, triangular, or other) asintegrals over Γ (or its subset, if g vanishes outside this subset) one uses variousdecompositions of H into direct integrals.In particular, if g “ χ ∆ , where ∆ “ tp x, y q P Γ : x ă y u , then we obtain the integrals T p u q “ ż ‘ ∆ ζ p x, y ; u q dxdy, which play the role of continuous decompositions of the family of free triangular op-erators of Dykema and Haagerup [4]. We apply this approach to study the mixed*-moments of the triangular operators. In particular, we provide a new bijective proofof the formula for the moments ϕ pp T ˚ T q n q “ n n p n ` q ! , shown by Dykema and Haagerup [4] by a different method, where T is a triangularoperator, using the nice enumeration result of Chauve, Dulucq and Rechnitzer [2] foralternating ordered rooted tress. In this context, let us remark that there is a moregeneral bijective proof of ´Sniady [15], based on a certain algorithm of counting totalorders on directed trees.The paper is organized as follows. In Section 2 we recall the notions related tomatricial circular systems. In Section 3, we introduce the direct integral of Hilbertspaces, on which we define a family of creation and annihilation operators. In Section4, we introduce continuous circular systems as isometries between suitably defined fiberHilbert spaces. Mixed *-moments of matricial circular operators and their convergenceto the mixed *-moments of the operators decomposed in terms of the continuous circularsystems are discussed in Section 5. In Section 6, we apply our approach to the triangularoperators and we provide a bijective proof of the formula for the moments of T ˚ T .We adopt the convention that the stars which indicate adjoints are written closelyto the main symbol, for instance ℘ ˚ p g, u q and ℘ ˚ p x, y ; u q are the adjoints of ℘ p g, u q and ℘ p x, y ; u q , respectively. 2. Matricial circular systems
The matricial circular operators studied in [10] live in a kind of matricially free Fockspace. The original definition given in [7] was slightly generalized in [9,10], where we
ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 5 used the name of the ‘matricially free Fock space of tracial type’ which will be usedbelow.Let r r s : “ t , , . . . , r u and let U be a countable set. To each p p, q q P J Ă r r s ˆ r r s and u P U we associate a Hilbert space H p,q p u q . Using this family of Hilbert spaces, wecan construct the matricially free Fock space of tracial type. Definition 2.1.
By the matricially free Fock space of tracial type we understand thedirect sum of Hilbert spaces M “ r à q “ M q , where each summand is of the form M q “ C Ω q ‘ à m “ à p ,...,pmu ,...,un H p ,p p u q b H p ,p p u q b . . . b H p m ,q p u m q , where Ω q is a unit vector for any q P r r s , endowed with the canonical inner product.We denote by Ψ q the state associated with Ω q .Let us recall a number of basic facts and notions from [8,9,10].(1) In the special case when each H p,q p u q “ C e p,q p u q for any p, q, u , where e p,q p u q isa unit vector, the canonical orthonormal basis of M consists of tensors of theform e p ,p p u q b e p ,p p u q b . . . b e p m ,q p u m q , where p , . . . , p m , q P r r s , u , . . . , u m P U and m P N , and of vacuum vectorsΩ , . . . , Ω r . In the general case, a canonical basis of similar form can be given,except that the basis of each H p,q p u q may contain more than one vector.(2) The matricially free creation operators associated with matrices B p u q “ p b p,q p u qq of non-negative real numbers (covariance matrices) are bounded operators whosenon-trivial action onto the basis vectors is ℘ p,q p u q Ω q “ b b p,q p u q e p,q p u q ,℘ p,q p u qp e q,t p s qq “ b b p,q p u qp e p,q p u q b e q,t p s qq ,℘ p,q p u qp e q,t p s q b w q “ b b p,q p u qp e p,q p u q b e q,t p s q b w q , for any p, q, t P r r s and u, s P U , where e q,t p s q b w is a basis vector. Actiononto the remaining basis vectors gives zero. The corresponding matricially freeannihilation operators are their adjoints denoted ℘ ˚ p,q p u q . If b p,q p u q “
1, we callthe associated operators standard .(3) Matricially free creation operators can be realized as operator-valued matrices[10]. If we are given a C ˚ -probability space p A , φ q and a family of free creationoperators t ℓ p p, q, u q : p, q P r r s , u P U u with covariances b p,q p u q , respectively,which are *-free with respect to φ , and p e p p, q qq is the array of matrix units in M r p C q , then ℘ p,q p u q “ ℓ p p, q, u q b e p p, q q R. LENCZEWSKI for any p, q
P r r s and u P U . This equality holds in the sense that the mixed*-moments of the operators ℘ p,q p u q under the states Ψ j agree with the corre-sponding mixed *-moments of the above matrices under the states Φ j “ φ b ψ j ,where ψ j is the state associated with the canonical basis vector e p j q of C r , where j P r r s .(4) If ℓ ˚ p p, q, u q is the free annihilation operator corresponding to ℓ p p, q, u q , then ℓ ˚ p p, q, u q ℓ p p , q , u q “ δ p,p δ q,q δ u,u b p,q p u q for any p, q, u, p , q , u .(5) The matricially free Gaussian operators (called also matricial semicircular op-erators ) take the form ω p,q p u q “ ℘ p,q p u q ` ℘ ˚ p,q p u q , where p, q P r r s and u P U . We give this definition for completeness since wewill not use these operators in this paper.(6) The matricial circular operators are obtained from a family of arrays of matri-cially free creation operators of the form ζ p,q p u q “ ℘ p,q p u q ` ℘ ˚ q,p p u q , where u P U “ r t s , p, q P r r s and u , u are different copies of u . As we showedin [9], they can be realized as matrices of the form ζ p,q p u q “ c p p, q, u q b e p p, q q , where p, q P r r s , u P U , t c p p, q, u q : p, q P r r s , u P U u is a family of freegeneralized circular operators, i.e. c p p, q, u q “ ℓ p p, q, u q` ℓ ˚ p q, p, u q and p e p p, q qq is an array of matrix units. We use the term ‘generalized’ since, in general, thecovariances of the creation operators are arbitrary. If all creation operators arestandard, this is a family of free circular operators. The above equality holdsin the same sense as in (3). The corresponding family of arrays of operators iscalled the matricial circular system .3. Direct integrals
We would like to construct a continuous analog of the matricially free Fock spaceof tracial type. For that purpose, we will use the formalism of direct integrals (formore on direct integrals, see, for instance, [6]). However, two different direct integraldecompositions of the considered Fock space and of the canonical operator fields actingon this Fock space will be helpful.We begin with a decomposition which is a straightforward generalization of the dis-crete matricially free Fock space of tracial type. For I “ r , s , letΓ “ à n “ Γ n be the direct sum of measure spaces, where Γ n “ I n ` is equipped with the Lebesguemeasure denoted dγ n , and let us denote by dγ the corresponding direct sum of measureson the set Γ. ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 7
Definition 3.1.
By the continuous matricially free Fock space we understand the directintegral of Hilbert spaces of the form H “ ż ‘ Γ H p γ q dγ, where Hilbert spaces are associated to γ P Γ as follows:(1) if γ “ x P Γ “ I , then H p γ q “ C Ω p x q , where Ω p x q is a unit vector,(2) if γ “ p x , x , . . . , x n ` q P Γ n and n P N , then H p γ q “ H p x , x q b H p x , x q b . . . b H p x n , x n ` q , where each H p x, y q is a separable Hilbert space, and each H p γ q is equipped withthe canonical inner product,(3) the canonical inner product in H is then given by x F, G y “ ż Γ x F p γ q , G p γ qy dγ, where F “ ş ‘ Γ F p γ q dγ, G “ ş ‘ Γ G p γ q dγ P H are measurable square integrablefields with the natural assumption that F p γ q , G p γ q P H p γ q . Remark 3.1.
Let us collect certain basic facts about the Hilbert spaces defined above.(1) The continuous family of unit vectors t Ω p x q : x P I u replaces the finite set ofvacuum vectors t Ω , . . . , Ω r u used in the discrete case. The corresponding directintegral H : “ ż ‘ I H p x q dx – L p I q will be called the vacuum space . In this paper, we will be mainly concerned withthe function on I which is constantly equal to one since it corresponds to thecanonical trace on the algebra of random matrices. However, weighted traceswill, in general, lead to different elements of L p I q .(2) In the case when H p x, y q – G for any p x, y q P Γ , where G is a separable Hilbertspace (with an orthonormal basis indexed by U ), we also have isomorphisms forhigher order integrals H n : “ ż ‘ Γ n H p γ q dγ n – L p Γ n , G b n q , where n ě L p Γ n , G b n q denotes the Hilbert space of square integrable G b n -valued functions over the set Γ n with respect to dγ n ( dγ restricted to Γ n ).Thus, in this particular case, we have the isomorphism H – L p I q ‘ à n “ L p Γ n , G b n q . (3) Fields F “ ş ‘ Γ F p γ q dγ, G “ ş ‘ Γ G p γ q dγ P H have direct sum decompositions F “ ÿ n “ F n and G “ ÿ n “ G n , R. LENCZEWSKI where F n , G n P ş Γ n H p γ q dγ in the natural sense. Under the above isomorphismassumptions, F , G P L p I q and F n , G n P L p Γ n , G b n q for n ě
1. In mostcomputations, it is enough to consider these to be of the form F n p γ q “ f p x , x q b . . . b f n p x n , x n ` q ,G n p γ q “ g p x , x q b . . . b g n p x n , x n ` q , for γ “ p x , . . . , x n ` q and n ě
1, with f i p x i , x i ` q , g i p x i , x i ` q P G for any i .(4) The canonical inner product in H decomposes as x F, G y “ ÿ n “ ż Γ n x F n p γ q , G n p γ qy dγ n for any F, G P H , and an analogous equation holds for squared norms.This setting is suitable for introducing continuous analogs of sums of matricially freecreation operators ℘ p u q “ r ÿ p,q “ ℘ p,q p u q , where the covariance of each ℘ p,q p u q is assumed to be b p,q p u q ě u P U . In particular,if b p,q p u q “ d p for any p, q, u , where d ` ¨ ¨ ¨ ` d r “
1, then t ℘ p u q : u P U u is a familyof standard free creation operators. We would like to find a continuous analog of thesedecompositions, using direct integrals.The continuous analogs of the matricially free creation operators will be denoted ℘ p f q ,where f is an essentially bounded G -valued function on Γ , namely f P L p Γ , G q , wherethe square Γ is equipped with the two-dimensional Lebesgue measure. Definition 3.2.
For given f P L p Γ , G q , let us define bounded linear operators ℘ p f q on H by ℘ p f q ˆż ‘ I F p x q dx ˙ “ ż ‘ Γ f p x, x q F p x q dxdx for any F P L p I q , and ℘ p f q ˆż ‘ Γ n F n p x , . . . , x n ` q dx . . . dx n ` ˙ “ ż ‘ Γ n ` f p x, x q b F n p x , . . . , x n ` q dxdx . . . dx n ` for any F n P L p Γ n , G b n q , where n P N . In the special case, when f “ g b e p u q , where e p u q is some basis unit vector of G , under the identification L p Γ , G q – L p Γ q b G ,we will write ℘ p g, u q instead of ℘ p f q .In order to give formulas for the adjoints of ℘ p f q , we first need to define boundedoperators which multiply each F p γ q by an essentially bounded function g of the firstcoordinate of γ . The explicit definition is given below. Definition 3.3.
For k P L p I q , define bounded linear operators M p k, γ q : H p γ q Ñ H p γ q ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 9 for any γ “ p x , . . . , x n ` q P Γ n and n ě M p k, γ q F n p γ q “ k p x q F n p γ q , and the associated decomposable operator in the direct integral form M p k q : “ ż ‘ Γ M p k, γ q dγ, which is a bounded linear operator on H .The operator M p k q reminds the gauge operator on the free Fock space associatedwith the multiplication operator by k , but one important difference is that M p k q isnon trivial on the vacuum space unless k vanishes outside of the set of measure zero.Moreover, we will use the shorthand notations F n p γ q “ f p x , x q b . . . b f n p x n , x n ` q ,F n ´ p γ q “ f p x , x q b . . . b f n p x n , x n ` q , where γ “ p x , . . . , x n ` q P Γ n , γ “ p x , . . . , x n ` q P Γ n ´ and each f p x i , x i ` q is anelement of the Hilbert space G . Proposition 3.1.
The adjoints of the operators ℘ p f q are given by ℘ ˚ p f q ż ‘ I F p γ q dγ “ ℘ ˚ p f q ż ‘ Γ n F n p γ q dγ n “ ż ‘ Γ n ´ M p k, γ q F n ´ p γ q dγ n ´ where k p x q “ ż x f p x , x q , f p x , x qy dx , and x ., . y is the canonical inner product in G .Proof. The first formula is obvious since the range of ℘ p f q is contained in the orthog-onal complement of L p I q . To prove the second formula, we can take F n p γ q and G n p γ q to be simple tensors of the form F n p γ q “ f p x , x q b . . . b f n p x n , x n ` q ,G n p γ q “ g p x , x q b . . . b g n p x n , x n ` q , where γ “ p x , . . . , x n ` q . Then x ℘ p f q ż ‘ Γ n ´ G n ´ p γ q dγ n ´ , ż ‘ Γ n F n p γ q dγ n y“ x ż ‘ Γ n f p x , x q b G n ´ p γ q dγ n ´ , ż ‘ Γ n F n p γ q dγ n y“ ż Γ n x f p x , x q , f p x , x qyx G n ´ p γ q , F n ´ p γ qy dx . . . dx n ` “ ż Γ n ´ ˆż I x f p x , x q , f p x , x qy dx ˙ x G n ´ p γ q , F n ´ p γ qy dx . . . dx n ` “ x ż ‘ Γ n ´ G n ´ p γ q dγ n ´ , ℘ ˚ p f q ż ‘ Γ n F n p γ q dγ n p γ qy , where γ “ p x , . . . , x n ` q and γ “ p x , . . . , x n ` q . The proof is completed. (cid:4) Corollary 3.1.
For any f, f P L p Γ , G q , it holds that ℘ ˚ p f q ℘ p f q “ M p k q , where k is of the same form as in Proposition 3.1. Remark 3.2.
Let us consider some special cases and one property of the operatorsstudied above.(1) It is easy to see that if the functions f, f do not depend on the second coordinate,i.e. f p x , x q “ r f p x q and f p x , x q “ r f p x q , then k p x q “ ż x r f p x q , r f p x qy dx , for any x and thus M p k q reduces to the multiplication by a constant and thuswe can write the relation ℘ ˚ p f q ℘ p f q “ x f , f y “ x r f , r f y , and thus the operators ℘ p f q , ℘ ˚ p f q reduce to free creation and annihilation oper-ators, respectively, with the natural inner product for square integrable G -valuedfunctions.(2) In the above case, if we take two functions of the form: f “ χ Γ b e p u q and f “ χ Γ b e p u q , where χ Γ is the characteristic function of the square, anddenote the associated creation operators by ℘ p u q , ℘ p u q , respectively, then it iseasy to see that t ζ p u q : u P U u , where ζ p u q “ ℘ p u q ` ℘ ˚ p u q and u ‰ u , viewed as two ‘copies’ of u , is a family of free circular operators(in other words, instead of the set U we have to consider a twice bigger set ofindices).(3) If we use characteristic functions of the triangle and take f “ χ ∆ b e p u q and f “ χ ∆ b e p u q , where ∆ “ tp x, y q : 0 ď x ă y ď u and e p u q , e p u q areorthonormal basis vectors in G , then k p x q “ δ u,u ż x dx “ δ u,u x , and thus M p k q reduces to the multiplication by x times the Kronecker deltarelated to basis vectors, and thus the relation between the creation and annihi-lation operators becomes ℘ ˚ p f q ℘ p f q “ δ u,u M p id q , which corresponds to the case when we deal with strictly upper triangular Gauss-ian random matrices and the operatorial limit is the triangular operator. ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 11 (4) For simplicity, we will assume from now on that f “ g b e p u q and that g does not depend on u . Let us observe that if p g n q is a sequence of func-tions from L p Γ q which converges in norm to g P L p Γ q , then the cor-responding sequences of operators considered above converge strongly on H ,namely s ´ lim n Ñ8 ℘ p g n , u q “ ℘ p g, u q , s ´ lim n Ñ8 ℘ ˚ p g n , u q “ ℘ ˚ p g, u q and s ´ lim n Ñ8 M p g n q “ M p g q .4. Continuous circular systems
Other decompositions of H are also relevant since they give useful decompositionsof the operators of interest. We will introduce decompositions in which the sets offibers are relatively small (indexed by I ), but the fibers themselves are ‘long’. Thesedecompositions allow us to introduce continuous analogs of matricial circular systemsand interpret the operators of interest as integrals of two-dimensional ‘densities’. Definition 4.1.
For each x P r , s , let us define the associated fiber Hilbert space thatbegins with x : N p x q : “ ż ‘ Γ p y q N p γ q d r γ – C Ω p x q ‘ ż ‘ I H p x, y q dy ‘ ż ‘ Γ H p x, y q b H p y, z q dydz ‘ . . . , where Γ p x q “ tp x, γ q : γ P Γ u for any fixed x P I , with r γ pt x uq “ r γ pt x uˆ A q “ λ p A q (the Lebesgue measure of A ) for any A Ă Γ, and let H “ ż ‘ I N p x q dx, be the associated direct integral decomposition. All Hilbert spaces involved are equippedwith canonical inner products. Definition 4.2.
In a similar fashion, for all p x, y q P Γ , define Hilbert spaces N p x, y q : “ H p x, y q ‘ ż ‘ Γ H p x, y q b H p y, z q dz ‘ . . . , equipped with the canonical inner products and let H a H “ ż ‘ Γ N p x, y q dxdy, be the associated direct integral decomposition, where Hilbert spaces involved areequipped with canonical inner products. Definition 4.3.
Let us suppose that t e p y, z ; u q : u P U u is a countable orthonormalbasis of H p y, z q for each p y, z q P Γ . For any given x, y P I and u P U , define isometries ℘ p x, y ; u q : N p y q Ñ N p x, y q by the direct integral extension of ℘ p x, y ; u q Ω p y q “ e p x, y ; u q ,℘ p x, y ; u q e p y, z ; s q “ e p x, y ; u q b e p y, z ; s q ,℘ p x, y ; u qp e p y, z ; s q b w q “ e p x, y ; u q b e p y, z ; s q b w, for any x, y, z P I and u, s P U , where e p y, z ; s q b w is a basis vector of some tensorproduct H p y, z q b H p z, z q b . . . b H p z n ´ , z n q . Remark 4.1.
Equivalently, we could act with ℘ p x, y ; u q onto direct integrals in the lasttwo equations. For instance, the second equation would then take the form ℘ p x, y ; u q ż ‘ I g p y, z q e p y, z ; s q dz “ ż ‘ I g p y, z qp e p x, y ; u q b e p y, z ; s qq dz, but it is more convenient to completely decompose the considered fibers since we getsimpler formulas which are in correspondence with the discrete case. As far as thiscorrespondence is concerned, in contrast to the discrete case, we do not include scalarsin the definition of ℘ p x, y ; u q in order to avoid lengthy formulas. These scalars, playingthe role of covariances, are included in the function g when we deal with ℘ p g, u q tothe effect that | g p x, y q| is the continuous analog of b p,q p u q (as we mentioned earlier, weshall assume for simplicity that these covariances do not depend on u ). Proposition 4.1.
For any x, y P I and u P U , let ℘ ˚ p x, y ; u q : N p x, y q Ñ N p y q be thebounded operator defined by the direct integral extension of the formal formulas ℘ ˚ p x, y ; u q e p x, y ; u q “ Ω p y q ,℘ ˚ p x, y ; u qp e p x, y ; u q b w q “ w, for any x, y P I and u P U and w as above, and setting them to be zero on the remainingbasis vectors. Then the operator ℘ ˚ p x, y ; u q is the adjoint of ℘ p x, y ; u q for any x, y, u .Proof. These formulas are obtained by straightforward computations. (cid:4)
Definition 4.4.
Using the continuous family t ℘ p x, y ; u q : x, y P I, u P U u and thefamily of their adjoints, one then defines the continuous analogs of the matricial circularoperators as ζ p x, y ; u q “ ℘ p x, y ; u q ` ℘ ˚ p y, x, u q , for p x, y q P Γ and u P U and u , u are copies of u , as in Remark 3.2. This definition isin agreement with the definition of matricial circular systems and therefore the family t ζ p x, y ; u q : x, y P I, u P U u will be called the continuous circular system . Proposition 4.2. If f p x, y q “ g p x, y q b e p u q , the matrix elements of operators ℘ p f q “ ℘ p g, u q and their adjoints of the form x ℘ p g, u q h , h y “ ż Γ g p x, y qx ℘ p x, y ; u q h p y q , h p x, y qy dxdy, x h , ℘ ˚ p g, u q h y “ ż Γ g p x, y qx h p y q , ℘ ˚ p x, y ; u q h p x, y qy dxdy, where g P L p Γ q and u P U , are well defined for any h “ ş ‘ I h p y q dy and h “ ş ‘ Γ h p x, y q dxdy according to the decompositions of H and H a H , in Definitions 4.1and 4.2, respectively.Proof. Observe that the integrals on the RHS are well defined since h and h havesquare integrable norms by assumption, each ℘ p x, y ; u q is an isometry from N p y q to N p x, y q and g is essentially bounded on Γ . By Definition 3.2 and Proposition 3.1, theintegrals on the RHS give the desired matrix elements. This completes the proof. (cid:4) ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 13
In the above situation, we can write a decomposition of the creation operators ℘ p g, u q in the direct integral form ℘ p g, u q “ ż ‘ Γ g p x, y q ℘ p x, y ; u q dxdy, and an analogous formula for the annihilation operators ℘ ˚ p g, u q , namely ℘ ˚ p g, u q “ ż ‘ Γ g p x, y q ℘ ˚ p x, y ; u q dxdy. We use the symbol ‘ with a slight abuse of notation since the considered families ofintegrands are ‘almost decomposable’ with respect to the direct integral decompositionof Definition 5.2. The operators ℘ p x, y ; u q ( ℘ ˚ p x, y ; u q ) can be interpreted as operatorscreating (annihilating) vector e p u q of color x ‘under condition y ’. The ‘condition’ y refersto the color of the vector onto which the operator ℘ p x, y ; u q acts. If the given pairing isa block in the mixed *-moment of creation and annihilation operators, integration over x of the associated | g p x, y q| gives the contribution of the given pairing to the mixed*-moment.It is also natural to define the corresponding ’circular operator’ ζ p g, u q “ ż ‘ Γ p g p x, y q ℘ p x, y ; u q ` g p y, x q ℘ ˚ p y, x ; u qq dxdy which becomes a circular operator ζ p u q if g “ χ Γ . Similarly, if we set g “ χ Γ , weobtain canonical creation and annihilation operators associated with the basis vector e p u q , denoted ℘ p u q and ℘ ˚ p u q . 5. Mixed *-moments
We would like to discuss the combinatorics of the mixed *-moments of the creationoperators and of certain operators obtained from them. A very interesting exampleis that of the triangular operator obtained as the limit realization of sctrictly uppertriangular Gaussian random matrices.In our previous works we have studied the combinatorics of *-moments of various op-erators (creation, semicircular, circular, etc.) in the matricial (discrete) case [7,8,9,10].It was based on the class of colored labeled noncrossing partitions (if only one label isused, we just have colored noncrossing partitions). In the case of *-moments of creation,semicircular or circular operators, it suffices to consider pair partitions (see, for instance[12]). The general idea of block coloring is very straightforward. We color the blockswith natural numbers from the finite set r r s , where r is related to the block structureof the considered block random matrices in the sense that these matrices are assumedto have r blocks. At the same time r is the number of summands in the direct sumdecomposition of our Fock space (with r vacuum vectors). The color of each block isrelated in a matricial way to the color of its nearest outer block [7], as shown in Fig. 1.In addition to colors, we equip blocks with labels, if necessary (namely, if we deal withmore than one matrix), but labels are rather easy to deal with since they just have tomatch within a block.Roughly speaking, each creation-annihilation pair of operators indexed by the pair p p, q q and labeled by u produces the covariance b p,q p u q , with q depending on the vectorin the space M onto which the creation operator acts. If we associate a block with a ijk i jk k ijl i jklb i,j b j,k b i,k b j,k b i,j b j,l b k,l b i,k b j,k b k,l Figure 1.
Examples of colored non-crossing pair partitions. To eachpartition we assign the product of covariances which depend on the colorsof block and its nearest outer blocks. Labels are assumed to be the samefor all blocks and are omitted.given creation-annihilation pair, this vector corresponds to the nearest outer block ofthat block and must be colored by q if the action is to be non-trivial. That is whywe color blocks with colors from the set r r s and thus we can think that the covariance b p,q p u q is assigned to each block and its nearest outer block. Examples of such partitionsare shown in Fig. 1, where, for simplicity, we assume that all labels are the same andcan be omitted. One has to remark that we extend each partition by the imaginaryblock of some color q which corresponds to the vacuum vector Ω q . Now, the new ideain this paper is that in the limit r Ñ 8 the combinatorics is still described by coloredlabeled noncrossing pair partitions, but the discrete set of colors r r s is replaced by theinterval r , s .The basic definitions and notations are given below. If π is a non-crossing pair-partition of the set r m s , where m is an even positive integer, which is denoted π P N C m ,the set B p π q “ t V , . . . , V s u is the set of its blocks, where m “ s . If V i “ t l p i q , r p i qu and V j “ t l p j q , r p j qu are twoblocks of π with left legs l p i q and l p j q and right legs r p i q and r p j q , respectively, then V i is inner with respect to V j if l p j q ă l p i q ă r p i q ă r p j q . In that case V j is outer withrespect to V i . It is the nearest outer block of V i if there is no block V k “ t l p k q , r p k qu such that l p j q ă l p k q ă l p i q ă r p i q ă r p k q ă r p j q . It is easy to see that the nearestouter block, if it exists, is unique, and we write in this case V j “ o p V i q . If V i does nothave an outer block, we set o p V i q “ V , where V “ t , m ` u is the additional blockcalled imaginary . The partition of the set t , , . . . , m ` u consisting of the blocks of π and of the imaginary block will be denoted by p π .Let us recall how we assigned colors to blocks in the discrete case [7,8,9,10], where wecolored blocks with natural numbers from r r s . We used the set C r p π q of all mappings c : B p π q Ñ r r s called colorings . By a colored noncrossing pair partition we understooda pair p π, c q , where π P N C m and c P C r p π q . Then, the set of pairs B p π, f q “ tp V , c q , . . . , p V s , c qu played the role of the set of colored blocks. We assumed that also the imaginary blockwas colored by a number from the set r r s and thus we could speak of a coloring of p π .Then we assigned to blocks matricial elements associated with the covariance matrices ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 15 B p u q “ p b p,q p u qq P M r p R q , where u P U . For any π P N C m and c P C r p π q , let b q p π, c q “ s ź k “ b q p V k , c q , where b q p V k , c q “ b s,t p u q whenever V k “ t i, j u is colored by s , its nearest outer block o p V k q is colored by t and u i “ u j “ u and we assume that the imaginary block is colored by q P r r s , and otherwisewe set b q p V k , c q “
0. Examples of colored pair partitions with assigned weights are shownin Fig. 1. Let us just say that in the present paper we will repeat this procedure, exceptthat the set of colors will be x , . . . , x s ` lying in the interval r , s and the weights willbe products of g p x i , x o p i q q , whenever V i is a block and o p V i q is its nearest outer block.In order to go from the discrete case to the continuous one, we will concentrate on thecreation and annihilation operators. If a tuple pp ǫ , u q , . . . , p ǫ m , u m qq , where ǫ j P t , ˚u and u j P U for any j , is given, where m is even, we will say that π P N C m is adapted to it if u i “ u j whenever t i, j u is a block and p ǫ i , ǫ j q “ p˚ , q whenever t i, j u is a blockand i ă j . This notion (of a ‘noncrossing partition adapted to stars and labels’) isconvenient when speaking of *-moments of free creation operators. Clearly, if a tuple isgiven, it may have exactly one noncrossing pair partition adapted to it or none at all.If such a partition exists, then it means that the considered mixed *-moment of freecreation operators gives a non-zero contribution.We need to define a continuous analog of the state Ψ considered in the discrete case.For simplicity, we can take d q “ { r for any q , which corresponds to the decompositionof the random matrices into blocks which are asymptotically square and of equal sizes.We will use the state ϕ : B p H q Ñ C of the form ϕ “ ż ‘ I ϕ p γ q dγ, where ϕ p γ q “ ϕ p x q is the vacuum state associated with Ω p x q , namely ϕ p F q “ ż I x F p x q Ω p x q , Ω p x qy dx, where F “ ş ‘ I F p x q dx P B p H q according to the decomposition of H into fibers that endwith x P I , which is a natural continuous analog of the state Ψ when d q “ { r for all q obtained by taking the limit r Ñ 8 .Computations of mixed *-moments of interest always reduce to the mixed *-momentsof the creation operators. Therefore, let us first establish a connection on this level withthe use of the operators ℘ p g, u q introduced in Section 3. We consider the case when b p,q p u q “ d p for any p, q, u . For further simplicity, one can even assume that d p “ { r for all p , but the result given below holds for any asymptotic dimensions. Lemma 5.1.
For any p , q , . . . , p m , q m P r r s , u , . . . , u m P U , ǫ , . . . , ǫ m P t , ˚u andany m P N , it holds that Ψ p ℘ ǫ p ,q p u q . . . ℘ ǫ m p m ,q m p u m qq “ ϕ p ℘ ǫ p f q . . . ℘ ǫ m p f m qq , where f k “ g k b e p u k q for k P r m s and g k is the characteristic function of the rectangle I p k ˆ I q k Ă Γ for any ď k ď m , where I “ I Y . . . Y I r is the partition of I intodisjoint non-empty intervals with natural ordering. Proof.
Let us observe that for any fixed r P N an isometric embedding θ : M Ñ H is given by θ p Ω q q “ a d q ż ‘ I q Ω p x q dx,θ p e p ,p p u q b ¨ ¨ ¨ b e p m ,p n ` p u n qq “ a d p ¨ ¨ ¨ d p n ` ż ‘ I p ˆ¨¨¨ˆ I pn ` e p x , x , u q b ¨ ¨ ¨ b e p x n , x n ` , u n q dx . . . dx n ` , for any q, p , . . . , p n ` P r r s and u , . . . , u n P U . It is then easy to check directly that themixed *-moments of the operators ℘ p,q p u q in the state Ψ agree with the correspondingmixed *-moments of the operators ℘ p g, u q , where g is the characteristic function of I p,q ,respectively. This completes the proof. (cid:4) The combinatorics of mixed *-moments of matricially free creation operators can beexpressed in terms of noncrossing pair partitions adapted to stars and labels. It is nothard to see that they also describe the combinatorics of the mixed *-moments of themuch more general family of operators ℘ p f q , where f “ g b e p u q , in which matriciallyfree creation operators are included if one takes characteristic functions of rectangles asabove. The main reason is that they encode two main facts: blocks must correspondsto pairings of creation and annihilation operators which have the same label, but thecontribution of each partition depends on the inner products. Proposition 5.1.
Let f k “ g k b e p u k q and ǫ k P t , ˚u , where k P r m s and m “ s ,be such that there exists a unique non-crossing pair partition π P N C m adapted to pp ǫ , u q . . . , p ǫ m , u m qq . Then ϕ p ℘ ǫ p f q . . . ℘ ǫ m p f m qq “ ż Γ s s ź k “ x f r p k q p x k , x o p k q q , f l p k q p x k , x o p k q qy dx dx . . . dx s , where V k “ t l p k q , r p k qu , k “ , . . . , s , are the blocks of π with l p k q ă r p k q , with x assigned to the imaginary block of π and x , ., . y is the canonical inner product in G .Proof. Each pairing of a creation and annihilation operator produces a function g ofone argument in the operator M p k q . Here, we just compute the inner products in G which appear in the definition of such g for all pairings, which gives x f r p k q p x k , x o p k q q , f l p k q p x k , x o p k q qy “ g r p k q p x k , x o p k q q g l p k q p x k , x o p k q q for each pairing, and then integrate the product of over all the variables x , . . . , x s ,which gives the desired formula. (cid:4) Proposition 5.2.
Under the above assumptions, if g l p k q “ g r p k q “ χ k for all k P r s s ,where χ , . . . , χ s are characteristic functions of some measurable subsets of Γ , then ϕ p ℘ ǫ p f q . . . ℘ ǫ m p f m qq “ V ol p π q , where V ol p π q is the volume of the region V p π q Ď Γ s defined by x , x , . . . , x s for which χ k p x k , x o p k q q “ , for all k “ , . . . , s , with x assigned to the imaginary block. ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 17
Proof.
If we set the functions associated with the left and right legs of the block π k to be equal to χ k , then in the proof of Proposition 5.1 we have g r p k q p x k , x o p k q q g l p k q p x k , x o p k q q “ χ k p x k , x o p k q q , which gives a condition on two variables, x k and x o p k q , from among s ` s ` s . Each inner product in the formula of Lemma5.2 leads to a similar condition, which completes the proof. (cid:4) Example 5.1.
Consider the mixed *-moment associated with the pair partition π “tt , u , t , u , t , uu , where it is natural to assume that f “ f , f “ f and f “ f .Then ϕ p ℘ ˚ p f q ℘ ˚ p f q ℘ ˚ p f q ℘ p f q ℘ p f q ℘ p f qq“ ż Γ k f p y, x q k k f p z, y q k k f p w, z q k dwdzdydx. In the case when f j “ g j b e p u j q for j “ , ,
3, we can replace the norms of f , f , f by the absolute values of g , g , g , respectively. In particular, when these numericalvalued functions are the characteristic functions of the triangle ∆ “ tp x, y q : x ď y u ,this integral is equal to V ol p π q “ ż dx ż x dy ż y dz ż z dw “ . Example 5.2.
Consider the mixed *-moment associated with the pair partition π “tt , u , t , u , t , uu , where it is natural to assume that f “ f , f “ f and f “ f .Then ϕ p ℘ ˚ p f q ℘ ˚ p f q ℘ p f q ℘ ˚ p f q ℘ p f q ℘ p f qq“ ż Γ k f p y, x q k k f p z, y q k k f p w, y q k dwdzdydx. In the case when f j “ g j b e p u j q for j “ , , V ol p π q “ ż dx ż x dy ż y dz ż y dw “ . It is not a coincidence that this volume is twice bigger than that in Example 5.1 (inthe former case we had one 4-dimensional simplex, here we have the union of two suchsimplices).Finally, let us return to the asymptotic *-distributions of Gaussian random matriceswith i.b.i.d. entries. We assume that we have r blocks for each r P N . Later we willgo with r to infinity. Therefore, at this point it seems appropriate to include r in thesymbols denoting random matrices as well as limit operators. Proposition 5.3.
For any r P N , let t Y p u, n, r q : u P U u be a family of square n ˆ n independent complex Gaussian random matrices with i.b.i.d. entries for any natural n and any u P U . Then, lim n Ñ8 τ p n qp Y ǫ p u , n, r q . . . Y ǫ m p u m , n, r qq “ Ψ p ζ ǫ p u , r q . . . ζ ǫ m p u m , r qq for any u , . . . , u m P U and ǫ p q , . . . , ǫ p m q P t , ˚u , where ζ p u, r q “ r ÿ p,q “ ζ p,q p u, r q for any u P U and the operators ζ p,q p u, r q are discrete (generalized) matricial circularoperators corresponding to given r .Proof. The proof of this result was given in [10]. (cid:4)
The next step consists in taking the limit of the *-moments on the RHS as r Ñ 8 .We assume that d p “ { r for any p and any r and, for simplicity, we assume that theblock covariances b p,q p u, r q , defined for all p, q P r r s and all u, r , do not depend on u .From these block covariances we built a sequence of simple functions b r p x, y q “ r ÿ p,q “ b p,q p u, r q χ I p ,I q p x, y q and assume that it converges to some g P L p Γ q as r Ñ 8 . Then we compute thelimit of the mixed *-moments expressed as liner combinations of *-moments of the typegiven by Lemma 5.1.
Theorem 5.1.
Let t Y p u, n, r q : u P U , r P N u be a family of independent n ˆ n randommatrices for any n P N , such that (1) each Y p u, n, r q consists of r blocks of equal size with i.b.i.d. complex Gaussianentries, (2) the sequence of simple functions p b r q converges to g in L p Γ q as r Ñ 8 .Then lim r Ñ8 lim n Ñ8 τ p n qp Y ǫ p u , n, r q . . . Y ǫ m p u m , n, r qq “ ϕ p η ǫ p g, u q . . . η ǫ m p g, u m qq , where η p g, u j q “ ℘ p g, u j q ` ℘ ˚ p g t , u j q , with g t p x, y q “ g p y, x q and all labels u j , u j , j Pr m s , different, and where ϕ “ ş ‘ I ϕ p γ q dγ .Proof. The second limit ( n Ñ 8 ) was computed in Proposition 5.4. It is easy to seethat the moments obtained there, namely Ψ p ζ ǫ p u , r q . . . ζ ǫ m p u m , r qq , can be writtenas linear combinations of mixed *-moments of creation and annihilation operators ofcontinuous type in the state ϕ , namely such as those given in Lemma 5.1, since ζ p u, r q “ r ÿ p,q “ ζ p,q p u, r q “ r ÿ p,q p ℘ p,q p u , r q ` ℘ ˚ q,p p u , r qq , where the matricial creation and annihilation operators are assumed to have covariancesindependent of u , but otherwise arbitrary nonnegative numbers, i.e. b p,q p u q “ b p,q forany u . When we express the RHS in terms of operators of the form ℘ p g p,q , u q and theiradjoints, where g p,q “ b p,q χ I p ˆ I q for any p, q , we can write the above sum as η p g r , u q “ ℘ p g r , u q ` ℘ ˚ p g tr , u q , where g r p x, y q “ ř rp,q “ g p,q and g tr stands for the transpose of g r . Now, if p g r q convergesto g in L p Γ q , then, by Definition 3.2, the mixed *-moments of ℘ p g r , u q in the state ϕ converge to the corresponding mixed *-moments of ℘ p g, u q , which entails convergence ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 19 of the mixed *-moments of η p g r , u q in the state ϕ to the mixed *-moments of η p g, u q .This completes the proof. (cid:4) Triangular operator and labeled ordered trees
Let us apply Theorem 5.1 to independent strictly upper triangular Gaussian randommatrices, whose limits are free triangular operators [4]. We express the limit *-momentsof such matrices in terms of operators η p u q “ ζ p χ ∆ , u q where χ ∆ is the characteristicfunction of the triangle ∆. Our result gives a new Hilbert space realization of the limit*-moments, equivalent to the von Neumann algebra approach of Dykema and Haagerupin [4], where the triangular operator T was introduced. Note that our approach to thecombinatorics of its *-moments is also different than the algorithm in [4, Lemma 2.4].By a family of independent strictly upper triangular Gaussian random matrices wewill understand a family of complex n ˆ n matrices Y p u, n q , where u P U , whose entriesabove the main diagonal form a family of complex Gaussian random variables whosereal and imaginary parts form a family of n p n ´ q i.i.d. Gaussian random variablesfor each u (also independent for different u P U ), each having mean zero and variance1 { n . Theorem 6.1.
Let t Y p u, n q : u P U u be a family of independent strictly upper triangularGaussian random matrices for any n P N . Then lim n Ñ8 τ p n qp Y ǫ p u , n q . . . Y ǫ m p u m , n qq “ ϕ p η ǫ p u q . . . η ǫ m p u m qq for any ǫ , . . . , ǫ m P t , ˚u and u , . . . , u m P U u , where η p u j q “ η p χ ∆ , u j q , j P r m s , with ∆ “ tp x, y q P Γ : x ă y u and ϕ “ ş ‘ I ϕ p γ q dγ .Proof. We know from [4] that the limits of the mixed *-moments of independentstrictly upper triangular Gaussian random matrices under τ p n q as n Ñ 8 exist andare, by definition, equal to the mixed *-moments of free triangular operators T p u q ,namely lim n Ñ8 τ p n qp Y ǫ p u , n q ¨ ¨ ¨ Y ǫ m p u m , n qq “ ϕ p T ǫ p u q ¨ ¨ ¨ T ǫ m p u m qq for any ǫ , . . . , ǫ m P t , ˚u and any u , . . . , u m P U u . At the same time, it can be justifiedthat the above limits are equal to the limit moments of Theorem 5.1, where matrices t Y p u, n, r q : u P U u are n ˆ n independent block strictly upper triangular matrices with r blocks for all natural n and r , the non-vanishing blocks being S p,q p u, n, r q for p ă q .For instance, an estimate in terms of Schatten p -norms } A } p “ p a tr p n qp| A | p q for p ě p n q is the normalized trace, can be used. It holds that | tr p n qp A q| ď k A k ď k A k p ď k A k for any p ě
1. Therefore, let Y j “ Y ǫ j p u j , n q and Y j “ Y ǫ j p u j , n, r q for j P r m s and any n, r . Then, applying the above inequalities to the tracetr p n qp Y . . . Y m ´ Y . . . Y m q “ m ÿ j “ tr p n qp Y . . . Y j ´ p Y j ´ Y j q Y j ` . . . Y m q , and using repeatedly the H¨older inequality k AB k s ď k A k p k B k q , where s ´ “ p ´ ` q ´ ,we obtain an upper bound for the absolute value of this trace of the form p m ` q ¨ max ď j ď m } Y j ´ Y j } m ` ¨ p max ď k ď m t} Y k } m , } Y k } m uq m ´ . Therefore, for large n there exists R such that if r ą R , then the difference betweenthe mixed *-moments of primed and unprimed matrices can be made arbitrarily smallsince the norms } Y j ´ Y j } m ` can be made arbitrarily small for large n and large r ą R .Therefore, we can use Theorem 5.1 to express the limit mixed *-moments of the strictlyupper triangular matrices in terms of the operators η p ξ ∆ , u q , respectively, where ξ ∆ isthe characteristic function of the triangle ∆ “ tp x, y q P Γ : x ă y u . In other words,we can identify the triangular operators T p u q with η p ξ ∆ , u q , u P U . This completes theproof. (cid:4) In order to give these *-moments in a more explicit form, let us assign continuouscolors to blocks of π P N C pp ǫ , u q , . . . , p ǫ m , u m qq , where m “ s , and analyze relationbetween these colors. By N C pp ǫ , u q , . . . , p ǫ m , u m qq we denote the set of noncrossingpair partitions of r m s , such that u i “ u j and ǫ i ‰ ǫ j whenever t i, j u is a block. Weshould remember that stars refer here to operators of the form η p f, u q “ ℘ p f, u q ` ℘ ˚ p f t , u q , where f “ χ ∆ . For simplicity, let us write η p f, u q “ η p u q , ℘ p f, u q “ ℘ p u q and ℘ p f t , u q “ ℘ p u q . Each pairing that gives a nonzero contribution must be of theform p ℘ ˚ p u q , ℘ p u qq or p ℘ ˚ p u q , ℘ p u qq . The first one is obtained when η ˚ p u q is associ-ated with the left leg of a block and η p u q is associated with the right leg, whereas inthe second one the stars are interchanged. In any case, only one leg of a block can bemarked with a star. We do not star the legs of the imaginary block. Definition 6.1.
Let V j be a block of π P N C pp ǫ , u q , . . . , p ǫ m , u m qq and let V o p j q be itsnearest outer block. We distinguish four types of blocks:(1) type 1 : the right leg of V j and the left leg of V o p j q are starred,(2) type 2 : the left leg of V j and the right leg of V o p j q are starred,(3) type 3 : the left legs of both V j and V o p j q are starred,(4) type 4 : the right legs of both V j and V o p j q are starred.All types of pairs p V, o p V qq are shown in Fig. 2. Remark 6.1.
Our combinatorics is based on coloring the blocks of noncrossing pairpartitions with numbers from r , s and finding relations betwen them. This is a con-tinuous analog of coloring blocks with numbers from the discrete set r r s .(1) Let us color the blocks of π P N C pp ǫ , u q , . . . , p ǫ m , u m qq , where m “ s , and theimaginary block with s ` r , s : x , . . . , x s ` , assignedto V , . . . , V s ` , respectively. It is convenient to number those blocks and colorsstarting from the right, as shown in Fig. 4 (thus, for instance, V is the imaginaryblock and its color is x ).(2) Now, using these inequalities, we can associate a region V p π q Ă Γ to each π .Namely, let V p π q “ t x P Γ s : x j ă x o p j q if V j P B p π q ^ x j ą x o p j q if V j P B p π qu , where B p π q and B p π q stand for blocks of π whose left legs are starred andunstarred, respectively. ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 21 * *
Type 1 * *
Type 2 * *
Type 3 * *
Type 4
Figure 2.
Four types of pairs p V, o p V qq , where V is a block and o p V q isthe nearest outer block of V , which depend on which legs are starred. Corollary 6.1.
The non-vanishing mixed *-moments of the free triangular operatorsin the state ϕ take the form ϕ p T ǫ p u q . . . T ǫ m p u m qq “ ÿ π P N C pp ǫ ,u q ,..., p ǫ m ,u m qq V ol p π q , where m “ s and V ol p π q is the volume of the region V p π q .Proof. We know that we can replace T p u j q by η p u j q . Now, without loss of generality,we can assume that u j “ u for all j since the general case just gives the additionalcondition on π that u i “ u j whenever t i, j u is a block. We will omit u and write ℘ “ ℘ p u q and ℘ “ ℘ p u q and replace N C pp ǫ , u q , . . . , p ǫ m , u qq by N C p ǫ , . . . , ǫ m q .We have ϕ p T ǫ . . . T ǫ m q “ ÿ π P N C p ǫ ,...,ǫ m q ϕ p ℘ ǫ p π q j p π q . . . ℘ ǫ m p π q j m p π q q , where the partition π ‘chooses’ whether to take the pairing p ℘ ˚ , ℘ q or p ℘ ˚ , ℘ q , whichformally can be written as ℘ ǫ i p π q j i p π q “ $’’&’’% ℘ if ǫ i “ i P R p π q ℘ if ǫ i “ ˚ and i P R p π q ℘ ˚ if ǫ i “ i P L p π q ℘ ˚ if ǫ i “ ˚ and i P L p π q , where R p π q and L p π q stand for the right and left legs of π , respectively. If j ą
1, thenwe choose the color x j assigned to block V j to be the first coordinate of χ ∆ associatedwith the pairing of type p ℘ ˚ , ℘ q , or the first coordinate of χ t ∆ associated with the pair-ing of type p ℘ ˚ , ℘ q , depending on whether the left leg of V j is starred or unstarred,respectively. Since we have f “ χ ∆ in each operator η that appears in the moment ϕ p η ǫ . . . η ǫ m q , let us observe that if the left leg of V j is starred, then x j ă x o p j q is obtainedfrom Remark 3.3 on the form of M p k q , with k p x o p j q q “ ş x j ă x o p j q dx j . In turn, if the leftleg of V j is unstarred, then instead of χ ∆ , we take its transpose in the correspondingpairing of type p ℘ ˚ , ℘ q which amounts to taking M p k q with k p x o p j q q “ ş x j ą x o p j q dx j (since u j “ u o p j q , by the adaptedness assumption on π ). This completes the proof. (cid:4) Remark 6.2.
Let us make some remarks on noncrossing pair partitions, colored non-crossing pair partitions, ordered rooted trees and labeled ordered rooted trees, whichwill be useful in establishing a bijective proof of the formula for the moments ϕ pp T ˚ T q n q for the triangular operator T . rrrt T ❅❅(cid:0)(cid:0) rr rt T ❅❅(cid:0)(cid:0) rr rt T ❅❅(cid:0)(cid:0) rr rt T ❅❅(cid:0)(cid:0) r r rt T Figure 3.
Ordered rooted trees on 4 vertices(1) There is an obvious bijection between
N C n and the set of associated extendedpair-partitions of r , n ` s , and for that reason we will use the same notationfor this set and we will understand from now on that each π P N C n is identifiedwith its extension π Y t , n ` u .(2) For any integer n , let O n be the set of ordered rooted trees on the set of n ` γ : O n Ñ N C n given by the following rule: a vertex v of O n is a child of vertex w if and only if γ p w q is the nearest outer block of γ p v q . Thus, the root of T corresponds to theimaginary block of π “ γ p T q .(3) Suppose now that to each vertex of an ordered rooted tree we assign a label fromsome set L . We will consider the case when trees on n ` r n ` s . The bijection between noncrossing pair partitionsand ordered rooted trees leads to a natural bijection γ : L n Ñ CN C n between labeled ordered rooted trees on n ` L n , where thevertices are labeled by different numbers from the set r n ` s , and colored non-crossing pair-partitions of the set r , n ` s , denoted CN C n , extended by the(colored) imaginary block (again, we identify π with π Y t , n ` u ), whereblocks (including the imaginary block) are colored by different numbers fromthe set r n ` s . Clearly, the labeling of each vertex is the same as the col-oring of the corresponding block. We prefer to speak of labeled ordered treesand colored noncrossing partitions since the first terminology is standard in thecombinatorics of trees and we used the second one in our previous works.(4) It is well known that the number of ordered labeled trees on n ` | L n | “ p n ` q ! C n “ p n q ! n ! , where C n “ n ` ` nn ˘ is the n th Catalan number. Example 6.1.
The set O of ordered rooted trees on 4 vertices consists of the treesgiven in Fig. 3, where the root is distinguished by a larger circle. In ordered trees,the children of any vertex are ordered and that is why T and T are inequivalent sincedifferent children of the roots have off-springs. In turn, N C consists of the non-crossingpair partitions shown in Fig. 4, where we also draw the imaginary blocks which can be ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 23 π R p π q S p π q V ol p π q * * * π : x x x x x ă x x ă x x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x * * * π : x x x x x ă x x ă x x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x * * * π : x x x x x ă x x ă x x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x * * * π : x x x x x ă x x ă x x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x * * * π : x x x x x ă x x ă x x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x x ă x ă x ă x Figure 4.
Noncrossing pair partitions of r s , each extended by an imag-inary block, corresponding to ϕ pp T ˚ T q q . Starred legs correspond to T ˚ .We assign continuous colors x j to blocks, where j P r s . The correspond-ing regions R p π q inside the cube r , s are given by three inequalities forcolors of the blocks of π and have volumes V ol p π q . Each region consistsof simplices S p π q defined by linearly ordered colors. Altogether we get27 “ simplices.identified with the roots of the corresponding trees. The natural bijection γ : O Ñ N C is given by γ p T k q “ π k . In fact, that is why there are 5 trees of this type since C “ γ . In turn, thereare 6! { “
120 different labeled ordered trees on 4 vertices if we label them by the4-element set in an arbitrary way. rrrt
423 1 rrrt
324 1 rrrt
314 2 rrrt
413 2 rrrt
214 3 ❅❅(cid:0)(cid:0) rr rt
32 4 1 ❅❅(cid:0)(cid:0) rr rt
23 4 1 ❅❅(cid:0)(cid:0) rr rt
13 4 2 ❅❅(cid:0)(cid:0) rr rt
31 4 2 ❅❅(cid:0)(cid:0) rr rt
21 4 3 ❅❅(cid:0)(cid:0) rr rt
12 4 3 ❅❅(cid:0)(cid:0) rr rt
32 41 ❅❅(cid:0)(cid:0) rr rt
42 31 ❅❅(cid:0)(cid:0) rr rt
43 21 ❅❅(cid:0)(cid:0) rr rt
41 32 ❅❅(cid:0)(cid:0) rr rt
31 42 ❅❅(cid:0)(cid:0) rr rt ❅❅(cid:0)(cid:0) r rrt ❅❅(cid:0)(cid:0) r rrt ❅❅(cid:0)(cid:0) r rrt ❅❅(cid:0)(cid:0) r rrt ❅❅(cid:0)(cid:0) r r rt
14 3 2 ❅❅(cid:0)(cid:0) r r rt
13 4 2 ❅❅(cid:0)(cid:0) r r rt
14 2 3 ❅❅(cid:0)(cid:0) r r rt
13 2 4 ❅❅(cid:0)(cid:0) r r rt
12 4 3 ❅❅(cid:0)(cid:0) r r rt
12 3 4
Figure 5.
The set A of alternating ordered rooted trees of type I. Thisset is in bijection with the set of noncrossing colored pair partitions withtotally ordered colors given in Fig. 4, or with the corresponding simplices.The alternating ordered rooted trees are listed in the same order as thecorresponding simplices in Fig. 4. Definition 6.2.
Let p v , v , v , v , . . . q be a path in an ordered labeled rooted tree T on n ` v j is a son of v j ´ , and let p x , x , x , x , . . . q be thecorresponding sequence of labels. Then T is called alternating if this sequence satisfiesone of the inequalities, x ą x ă x ą x . . . , or x ă x ą x ă x . . . , i.e. the differences of labels corresponding to the neighboring vertices alternate in sign.These two types of conditions split the set of alternating ordered rooted trees on n ` of type I and oftype II , respectively. Denote by A n the set of alternating ordered rooted trees on n ` Remark 6.3.
There is a nice enumeration result of Chauve, Dulucq and Rechnitzer [2]which says that | A n | “ n n for any natural n . Let us recall that in our notation A n stands for the set of alternatingordered rooted trees of type I on n ` ANDOM MATRICES AND CONTINUOUS CIRCULAR SYSTEMS 25 by 2 to get the number of all alternating ordered rooted trees. Note also that a typicalformula refers to trees on n vertices. Example 6.2.
Among the ordered labeled rooted trees on 4 vertices, there are 2 ˆ “
54 alternating ones. In this example, each type contains 3 “
27 alternating orderedrooted trees. The complete set A of all alternating ordered rooted trees of type I isgiven in Figure 5. In view of the above bijection results, the cardinality of all non-crossing pair partitions of the 6-element set with alternating colorings is also 54 andthere are 27 partitions in which each block of odd depth has a smaller color than itsnearest outer block (equivalently, the color of the imaginary block, which is assumed tohave zero depth, is greater than the colors of all blocks for which the imaginary blockis the nearest outer block). Corollary 6.2.
The moments of T ˚ T , where T is the triangular operator, are M n “ ϕ pp T ˚ T q n q “ n n p n ` q ! for any n P N .Proof. Let ǫ j “ ˚ if j is odd and ǫ j “ j is even. In this special case, it is easyto see that N C p ǫ , . . . , ǫ n q – N C n . Observe that in the case of alternating starredand unstarred legs there can be no pairs p V, o p V qq of type 3 and 4 since otherwise therewould be unequal numbers of starred and unstarred legs between the right leg of V and the left leg of o p V q , which would mean that there must be a block between V and o p V q , which is a contradiction. Therefore, blocks with starred left and right legs mustalternate as we take a sequence of blocks p V i , . . . , V i p q , where each block is the nearestouter block of its successor. By Corollary 6.1, we need to compute V ol p π q for each π P N C n . Each of the corresponding regions R p π q is defined by a set of n inequalitiesfor colors x , . . . , x n ` . Irrespective of what symbols represent these colors, in order tosatisfy these inequalities, we have to find the number of total orderings of the form x j ă x j ă . . . ă x j n ` which satisfy the given inequalities. The number of these total orderings is equal to thenumber of n ` {p n ` q !, into which R p π q decomposes. Thekey observation is that in order to compute the number of these total orderings corre-sponding to π (under conditions given by n inequalities which express orders betweenthe colors of each V and o p V q ) it suffices to count in how many ways we can label blocksof π with natural numbers from r n ` s in such a way that orders between these num-bers alternate as we go down each sequence of type p V i , . . . , V i p q . More specifically, tocolors x j , x j , . . . , x j n ` in the total ordering (defining a simplex) given above we assignnumbers n ` , n, . . . ,
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Romuald Lenczewski,Wydzia l Matematyki, Politechnika Wroc lawska,Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc law, Poland
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