Matricial R-circular Systems and Random Matrices

Transcription

1 Instytut Matematyki i Informatyki Politechnika Wrocławska UC Berkeley, March 2014

2 Motivations Motivations: describe asymptotic distributions of random blocks construct new random matrix models unify concepts of noncommutative independence

3 Fundamental results 1 If Y pu, nq is a standard complex Hermitian Gaussian random matrix, it converges under the trace to a semicircular operator under τpnq E Trpnq lim Y pu, nq Ñ ωpuq nñ8 2 If Y pu, nq is a standard complex non-hermitian Gaussian random matrix, it converges to a circular operator under τpnq E Trpnq. lim Y pu, nq Ñ ηpuq nñ8

4 Different approaches 1 free probability and freeness 2 operator-valued free probability and freeness with amalgamation 3 matricially free probability and matricial freeness

5 Voiculescu s asymptotic freeness and generalizations 1 Complex independent Hermitian Gaussian random matrices converge to a free semicircular family ty pu, nq : u P Uu Ñ tωpuq : u P Uu 2 Complex independent Non-Hermitian Gaussian random matrices converge to a free circular family ty pu, nq : u P Uu Ñ tηpuq : u P Uu 3 Generalizations (Dykema, Schlakhtyenko, Hiai-Petz, Benaych-Georges and others)

6 Philosophy By asymptotic freeness, large random matrix is a free random variable, so it is natural to 1 decompose it into large blocks 2 look for a concept of matricial independence for blocks 3 find operatorial realizations for large blocks 4 reduce computations to properties of these operators

7 Matricial decomposition 1 decompose the set rns : t1,..., nu into disjoint intervals rns N 1 Y... Y N r and set n j N j, 2 use normalized partial traces τ j pnq E Tr j pnq where Tr j pnqpaq n n j TrpnqpD j AD j q and D j is the n j n j unit matrix embedded in M n pcq at the right place.

8 Matricial decomposition 1 decompose random matrices Y pu, nq into independent blocks S i,j pu, nq D i Y pu, nqd j 2 decompose symmetric blocks T i,j pu, nq, built from blocks of the same color: Y pu, nq S 1,1 pu, nq S 1,2 pu, nq... S 1,r pu, nq S 2,1 pu, nq S 2,2 pu, nq... S 2,r pu, nq S r,1 pu, nq S r,2 pu, nq... S r,r pu, nq

9 Matricial freeness In order to define a matricial concept of independence, 1 replace families of variables and subalgebras by arrays ta i, i P I u Ñ pa i,j q pi,jqpj ta i, i P I u Ñ pa i,j q pi,jqpj 2 replace one distinguished state in a unital algebra by an array of states ϕ Ñ pϕ i,j q pi,jqpj where we set ϕ i,j ϕ j (state under condition j)

10 Matricial freeness The definition of matricial freeness is based on two conditions 1 freeness condition ϕ i,j pa 1 a 2... a n q 0 where a k P A ik,j k X Kerϕ ik,j k and neighbors come from different algebras 2 matriciality condition : subalgebras are not unital, but they have internal units 1 i,j, such that the unit condition 1 i,j w w holds only if w is a reduced word matricially adapted to pi, jq and otherwise it is zero.

11 Some results The concept of matricial freeness allows to 1 unify the main notions of independence 2 give a unified approach to sums and products of a large class of independent random matrices 3 find a unified combinatorial description of limit distributions (non-crossing colored partitions) 4 derive explicit formulas for arbitrary mutliplicative convolutions of Marchenko-Pastur laws 5 find random matrix models for boolean independence, monotone independence and s-freeness (noncommutative independence defined by subordination) 6 construct a natural random matrix model for free Meixner laws

12 Matricially free Fock space of tracial type Definition By the matricially free Fock space of tracial type we understand M rà j1 where each summand is of the form M j CΩ j ` 8à à m1 j 1,...,jm u 1,...,um M j, H j1,j 2 pu 1 q b... b H jm,jpu m q,

13 Creation operators Definition Define matricially free creation operators on M as partial isometries with the action onto basis vactors i,j puqω j e i,j puq i,j puqpe j,k psqq e i,j puq b e j,k psq i,j puqpe j,k psq b wq e i,j puq b e j,k psq b w for any i, j, k P rrs and u, s P U, where e j,k psq b w is a basis vector. Their actions onto the remaining basis vectors give zero.

14 Toeplitz-Cuntz-Krieger algebras Relations One square matrix of creation operators p i,j q gives an array of partial isometries satisfying relations ŗ j1 i,j i,j k,i k,i i for any k ŗ j1 k,j k,j 1 for any k where i is the projection onto CΩ j. The corresponding C -algebras are Toeplitz-Cuntz-Krieger algebras.

15 Matricially free Gaussians Arrays of matricially free Gaussians operators b ω i,j puq d j p i,j puq i,jpuqq play the role of matricial semicircular operators rωpuqs ω 1,1 puq ω 1,2 puq... ω 1,r puq ω 2,1 puq ω 2,2 puq... ω 2,r puq ω r,1 puq ω r,2 puq... ω r,r puq and generalize semicircular operators. From now on we incorporate scalaras like a d j or more general a b i,j puq in the operators.

16 Free probability Hermitian version Theorem [Voiculescu] If Y pu, nq are Hermitian standard Gaussian independent random matrices with complex entries, then lim Y pu, nq ωpuq nñ8 in the sense of mixed moments under the complete trace τpnq.

17 Asymptotic distributions of Hermitian symmetric blocks Theorem If Y pu, nq are Hermitian Gaussian independent random matrices with i.b.i.d. complex entries, then lim nñ8 T i,jpu, nq pω i,j puq in the sense of mixed moments under partial traces τ j pnq, where " ωj,j puq if i j pω i,j puq ω i,j puq ω i,j puq if i j are symmetrized Gaussian operators.

18 Types of blocks The symmetric blocks are called 1 balanced if d i 0 and d j 0 2 unbalanced if d i 0 ^ d j 0 or d i 0 ^ d j 0 3 evanescent if d i 0 and d j 0 where the numbers N j lim nñ8 n d j 0 are called asymptotic dimensions.

19 Special cases Special cases In the general formula for mixed moments, we get 1 T i,j pu, nq Ñ pω i,j puq if block is balanced 2 T i,j pu, nq Ñ ω i,j puq if block is unbalanced, j 0 ^ i 0 3 T i,j pu, nq Ñ ω j,i puq if block is unbalanced, j 0 ^ i 0 4 T i,j pu, nq Ñ 0 if block is evanescent

20 Free probability non-hermitian version Theorem [Voiculescu] If Y pu, nq are standard complex non-hermitian Gaussian independent random matrices, then lim Y pu, nq ηpuq nñ8 in the sense of moments under τpnq, where each ηpuq is circular.

21 Non-Hermitian case Theorem If Y pu, nq are non-hermitian Gaussian independent random matrices with i.b.i.d. complex entries, then lim nñ8 T i,jpu, nq η i,j puq where η i,j puq p i,j p2u 1q p i,jp2uq are called matricial circular operators and p i,j psq are symmetrizations of i,j psq (two partial isometries for each η i,j puq).

22 Matricial realization of partial isometries Lemma We can identify partial isometries i,j puq with where i,j puq lpi, j, uq b epi, jq P M r paq 1 the family tlpi, j, uq : i, j P rrs, u P Uu is a system of *-free creation operators w.r.t. state ϕ on A 2 pe i,j q is a system of matrix units in M r pcq 3 Ψ j ϕ b ψ j, where ψ j is the state associated with epjq P C r.

23 Matricial realization - Hermitian case Lemma Consequently, ω i,j puq lpi, j, uq b epi, jq lpi, j, uq b epj, iq " gpi, j, uq b epi, jq gpi, j, uq pω i,j puq b epj, iq if i j f pi, uq b epi, iq if i j where tgpi, j, uq : i, j P rrs, u P Uu, tf pi, uq : i P rrs, u P Uu are *-free generalized circular and semicircular systems, respectively.

24 Matricial realization - non-hermitian case Lemma We can also identify matricial circular operators with η i,j puq " gpi, j, uq b epi, jq gpj, i, uq b epj, iq if i j gpi, i, uq b epi, iq if i j

25 Asymptotic realization of blocks Theorem If Y pu, nq are complex Gaussian independent random matrices with i.b.i.d. entries, then lim nñ8 S i,jpu, nq ζ i,j puq where ζ i,j puq gpi, j, uq b epi, jq are called matricial R-circular operators.

26 Example Consider one off-diagonal ζ i,j of the form ζ i,j pl 1 l 2q b epi, jq where i j and l 1, l 2 are free creation operators with covariances γ 1 and γ 2, respectively. We have Ψ j pζ i,jζ i,j ζ i,jζ i,j q ϕpl 1l 1 l 1l 1 q ϕpl 1l 2l 2 l 1 q γ 2 1 γ 1 γ 2 whereas all remaining *-moments of ζ i,j in the state Ψ j vanish.

27 Matricial freeness Theorem The array of *-subalgebras pm i,j q of M r paq, each generated by tlpi, j, uq b epi, jq : u P Uu, with the unit 1 i,j tpi, jq b epi, iq 1 b epj, jq where tpi, jq " upu lpi, j, uqlpi, j, uq if i j 0 if i j is matricially free with respect to pψ i,j q, where Ψ j,j ϕ b ψ j.

28 Symmetric matricial freeness Theorem The array pm i,j q of *-subalgebras of M r paq, each generated by tlpi, j, uq b epi, jq, lpj, i, uq b epj, iq : u P Uu and the symmetrized unit 1 i,j 1 b epi, iq 1 b epj, jq, is symmetrically matricially free with respect to pψ i,j q.

29 Symmetric matricial freeness Corollary The array pm i,j q of *-subalgebras of M r paq such that one of the following cases holds: 1 each M i,j is generated by tpω i,j puq : u P Uu, 2 each M i,j is generated by tη i,j puq : u P Uu, 3 each M i,j is generated by tζ i,j puq : u P Uu, is symmetrically matricially free with respect to pψ i,j q.

30 Adapted partitions If a k c k b epi k, j k q P M r paq for k 1,..., m, where c k P A, we will denote by N C m pa 1,..., a m q the set of non-crossing partitions of rms which are adapted to pepi 1, j 1 q,..., epi m, j m qq which means that this tuple is cyclic j 1 i 2, j 2 i 3,..., j n i 1 together with all tuples associated with the blocks of π. Its subset consisting of pair partitions will be denoted N C 2 mpa 1,..., a m q.

31 Example For pepi, jq, epj, iq, epi, jq, epj, iqq, where i j, there are three non-crossing partitions adapted to it: tt1, 2, 3, 4uu, tt1, 4u, t2, 3uu, tt1, 2u, t3, 4uu. In turn, the partitions tt1, 2, 5u, t3, 4uu, tt1, 2, 3, 4, 5uu tt1, 2, 3u, t4, 5uu are the only non-crossing partitions adapted to the tuple pepi, jq, epj, kq, epk, iq, epi, mq, epm, iqq, where i j k i m.

32 Combinatorics of mixed *-moments Lemma With the above notations, let a k ζ ɛ k i k,j k pu k q, where i k, j k P rrs, u k P U, and ɛ k P t1, u for j P rms and m P N. Then Ψ j pa 1... a m q πpn C 2 mpa 1,...,a mq b j pπ, f q where j is equal to the second index of last matrix unit (associated with a m ) and f is the unique coloring of π adapted to pa 1,..., a m q, b j pπ, f q ¹ k b j pπ k, f q with b j pπ k q b cpkq,cpopkqq puq, where cpkq, cpopkqq, j are colors assigned to π k, its nearest outer block π opkq and to the imaginary block. In the remaining cases, the moment vanishes.

33 Cyclic cumulants Definition A family of multilinear functions κ π r. ; js, where j P rrs, of matricial variables a k c k b epi k, j k q P M r paq is cyclically multiplicative over the blocks π 1,..., π s of π P N C m pa 1,..., a m q if κ π ra 1,..., a m ; qs for any a 1,..., a m, where s¹ k1 κpπ k qra 1,..., a m ; jpkqs, κpπ k qra 1,..., a m ; jpkqs κ s pa q1,..., a qs ; j qs q for the block π k pq 1... q n q, where tκ s p.; jq : s 1, j P rrsu is a family of multilinear functions. If π R N C m pa 1,..., a m q, we set κ π ra 1,..., a m ; js 0 for any j.

34 Cyclic cumulants Definition By the cyclic cumulants we shall understand the family of multilinear cyclically multiplicative functionals over the blocks of non-crossing partitions π Ñ κ π r. ; js, defined by r moment-cumulant formulas Ψ j pa 1... a m q where j P rrs and Ψ j ϕ b ψ j. πpn C mpa 1,...,a mq κ π ra 1,..., a m ; js,

35 Example Take ζ i,j pl 1 l 2q b epi, jq and ζ i,j pl 1 l 2 q b epj, iq, where i j. The non-trivial second order cumulants are κ 2 pζ i,j, ζ i,j ; jq Ψ j pζ i,jζ i,j q ϕpl 1l 1 q γ 1 κ 2 pζ i,j, ζ i,j; iq Ψ i pζ i,j ζ i,jq ϕpl 2l 2 q γ 2 whereas all higher order cumulants vanish. For instance, κ 2 pζ i,j, ζ i,j, ζ i,j, ζ i,j ; jq Ψ j pζ i,jζ i,j ζ i,jζ i,j q κ 2 pζ i,j, ζ i,j ; jqκ 2 pζ i,j, ζ i,j ; jq κ 2 pζ i,j, ζ i,j ; jqκ 2 pζ i,j, ζ i,j; iq γ1 2 γ 1 γ 2 γ1 2 γ 1 γ 2 0,

36 Cyclic R-transforms Cyclic R-transforms Let ζ : pζ i,j q and ζ : pζ i,j q. Cyclic R-transforms of this pair of arrays are formal power series in z : pz i,j q and z : pz i,j q of the form R ζ,ζ pz, z ; jq 8 n1 i 1,...,in j 1,...,jn ɛ 1,...,ɛ n κ n pζ ɛ 1 i 1,j 1,..., ζ ɛn i n,j n ; jqz ɛ 1 i 1,j 1... z ɛn i n,j n, where i 1,..., i n, j, j 1,... j n P rrs and ɛ 1,..., ɛ n P t1, u.

37 Cyclic R-transforms of R-circular systems Theorem If ζ : pζ i,j q is the square array of matricial R-circular operators and ζ : pζ i,j q, then its cyclic R-transforms are of the form R ζ,ζ pz, z ; jq ŗ i1 pb i,j z i,jz i,j b i,j z j,i z j,iq where j P rrs, where b i,j κ 2 pξ i,j, ξ i,j; jq κ 2 pξ j,i, ξ j,i ; jq.

38 Cyclic R-transforms of single R-circular operators Theorem If ζ : ζ i,j, where i j, then non-trivial cyclic R-transforms of the pair tζ, ζ u are of the form and if ζ : ζ j,j, then R ζ,ζ pz, z ; jq b i,j z i,jz i,j R ζ,ζ pz, z ; iq b j,i z i,j z i,j R ζ,ζ pz, z ; jq b j,j pz j,jz j,j z j,j z j,jq

39 Cyclic R-transforms of single R-circular operators Corollary If b i,j d i for any i, j, then c : i,j ζ i,j is circular with respect to Ψ r j1 d jψ j and the corresponding R-transform takes the form R c,c pz 1, z 2 q ŗ j1 d j R ζ,ζ pz, z ; jq z 1 z 2 z 2 z 1, where R ζ,ζ pz, z ; jq are the cyclic R-transforms considered above in which all entries of arrays z and z are identified with z 1 and z 2, respectively.