Distribution for Non Symmetric Position Operators on the Free Toy Fock Space and Its Approximation on the Full Fock Space

In this paper, we investigate the “Poisson-type” limit theorem with respect to the vacuum state of a family of partial sums of non-symmetric position operators under an appropriate scaling on the free toy Fock space. We give a formula for the vacuum moment in relation with the combinatorics of non-crossing partitions. We show that the asymptotic measure associated to the limit of the partial sums of these operators is the free Meixner law with an atomic and an absolutely continuous part, whereas the probability distribution of any single operator is the two-point probability. The approximation of such operators on the full Fock space is given.


Introduction
Free probability is a branch of noncommutative probability theory.It was started by Voiculescu in the 80's [18][19][20] in order to deal with some problems on operator algebras and calculating distributions of noncommutative random variables.A noncommutative probability space is a couple (A, ϕ) of a (unital) * -algebra A and a state ϕ on it.
The state ϕ plays the role of the expectation in classical probability.The self-adjoint elements a = a * ∈ A are called noncommutative random variables.
The notion of independence in free probability is called free independence (or freeness) which is analogous to the classical (or tensor) independence in classical probability.It was introduced by Voiculescu [18] and it allows to compute the mixed moments using the marginal distributions as for tensor independence in classical probability.Recall that a family of unital sub-algebras {A i , i ∈ N} with the same unit of A is called free independent if ϕ(a 1 • • • a n ) = 0 for a j ∈ A i( j) whenever ϕ(a k ) = 0 for all k = 1, . . ., n and i(1 Here, only the neighboring random variables do not belong to the same algebra, there is no assumption otherwise.Noncommutative random variables are called free independent if the subalgebras they generate are free independent.Many tools in classical probability have been established in the free setting.For instance, one gets free central limit theorem, where the Gaussian distribution is replaced by the Wigner distribution, or the Poisson theorem, where the Poisson law is replaced by Marchenko-Pastur distribution. The free (full) Fock space is a basic structure for free probability theory.It is described as a direct sum of n-folds tensor product of a given Hilbert space H, i.e., n≥0 H ⊗n , where H ⊗0 := C , and is a distinguished norm one vector, called the vacuum vector.
We recall that in classical probability, the Brownian motion can be realized on the symmetric Fock space with the tensor independence as the sum of creation and annihilation operators [6], where the Gaussian law appears to be the distribution of such operators.This realization can be made also in the free probability by considering the sum of free creation and free annihilation operators acting on the free Fock space with freeness and the distribution of such operators is the Wigner law.
The free toy Fock space was introduced by Attal and Nechita [2] as a discrete type of the full Fock space, based on a free product of Hilbert spaces.The free toy Fock space can be seen, from a probabilistic point of view, as the smallest noncommutative probability space supporting a free family of Bernoulli random variables.It is shown in [2] that the free toy Fock space can be embedded into the full Fock space.Moreover, with this embedding, some elementary operators on the free toy Fock space approach its continuous counterparts operators on the full Fock space.
Generally, the (noncommutative) Poisson limit theorem (known also as law of small numbers) can be formulated by considering limit of arrays of (noncommutative) independent random variables.It was established by Speicher [17] for free independence, Muraki [11] for monotone independence, Wysoczański and the present author [14] for bm-independence.Another way of formulating Poisson-type limit theorem related to the discrete Fock space and its basic operators, was firstly established by Muraki [10] for monotone independence, Crismale et al. [3] for weakly-monotone independence, and recently the present author and Wysoczański [13] for bm-independence.The construction considered is as follows: For i ∈ N, consider the sum i and a scalar multiple λ ≥ 0 (which called the intensity) of a preservation operator A • i on an appropriate discrete model Fock space.For N ∈ N, set Then, for every p ∈ N, the limits describe a moments sequence of a distribution ν λ , called the "Poisson-type measure".
Here, ϕ(•) := • , is the vacuum state, where is the so-called vacuum vector.For weakly-monotone case [3], the associated measure belongs to the free Meixner law.For monotone independence, the probability measure has been identified by Muraki [10], whereas for bm-independence [13], the measure is still unknown.However, its moments m p (λ) are related to all noncrossing partitions consisting of pair or singleton blocks with bm-order.
In this paper, we extend this framework to the free toy Fock space, considered as a discrete type of the full Fock space.It turns out that in this case, the Poisson-type measure ν λ of the operators S N (λ) belongs to the free Meixner class as for the single operator S 1 (λ) i in the case of weakly-monotone independence [3], whereas the measure ν 1,λ associated to the single operator S 1 (λ) in our case is the Bernoulli distribution as for monotone case [10].
The paper is structured as follows.In Sect.2, we recall some basic knowledge about the noncommutative probability and the free toy Fock space with related creation, annihilation and conservation operators on it.We also recall some elementary information about noncrossing partitions of a finite set.Section 3 contains our main results: we perform analogous of the free Poisson-type limit theorem on the free toy Fock space.We start with the single case S 1 (λ), and show that the measure is the Bernoulli distribution, whereas the general case S N (λ) has the free Meixner distribution.In addition, we describe the moments sequence of the operators S N (λ) in relation with the combinatorics of noncrossing partitions.Finally, using the embedding of the free toy Fock space into the full Fock space, we investigate an approximation of the operators S N (λ) on the free toy Fock space with a continuous-time operators in the full Fock space.

Preliminaries
In this section we present some basic knowledge and definitions for our study.

Basic Notions of Noncommutative Probability
In what follows, (A, ϕ) stands for a noncommutative probability space, that is, a unital * -algebra A and a linear functional ϕ : A → C, called state, such that ϕ(a * a) ≥ 0 for all a ∈ A and ϕ(1 A ) = 1 for the unit element 1 A ∈ A. The state ϕ plays a similar role as the expectation in classical probability.A noncommutative random variable is a self-adjoint element a = a * ∈ A, and its distribution (with respect to ϕ) is given by the probability measure ν a such that where (ϕ(a n )) n≥0 denotes the moment sequence, when the moment problem is answered.
For more details in the framework of noncommutative probability, we refer the readers to [5,7,15] and references therein.

Partitions of a Finite Set
For a positive integer p, we denote by [ p] := {1, . . ., p}.The collection π := {B 1 , . . ., B k } of disjoint non-empty subsets For a partition π = {B 1 , . . ., B k } ∈ P( p), there is a natural partial order π defined on the blocks of π as follows: where min (resp.max) is the minimum (resp.maximum) element of the block B.
A block On the other hand, it is called pair partition if |B j | = 2, for all j = 1, . . ., k, and we will use the notation N C 2 ( p) for the set of all such a partitions.
The following two subsets of N C 2 1 ( p) contribute in our study in the subsequent section.
(1) The subset in which the partitions have no inner pair blocks and no outer singletons.
(2) The subset N C 2  1,i ( p) ⊂ N C 2 1 ( p) of all partitions consisting of inner singletons (pair block can be either inner or outer).
For a partition π ∈ N C 2 1 ( p), we denote by π the reduced partition after removing the singletons of π , that is, In particular, for π ∈ N C 2 ( p), we have π = π .
Graphically, for p ∈ N, we present a partition π = (B 1 , . . ., B k ) ∈ P( p) by putting all numbers from the set [ p] (in increasing order) on a horizontal line from the right to the left, and joint all numbers which belong to the same block by lines above.In particular, singletons are presented by vertical lines.For instance, consider the and B 4 = {6, 7, 8}.Then the partition π can be presented graphically as follows: Definition 2.2 For two blocks B i π B j , we say that a block B j is a direct successor of a block B i (equivalently, B i is a direct predecessor of B j ) if there is no any other block B l such that Let (i p , . . ., i 1 ) ∈ N p be a sequence of elements, and denote by k the cardinality of the set {i p , . . ., i 1 }.For j ∈ {i p , . . ., i 1 }, we define B( j and for m = n, we have either In this case, we say that the partition π is adapted to the sequence (i p , . . ., i 1 ) and we will use the notation (i p , . . ., i 1 ) ∼ π or π ∼ (i p , . . ., i 1 ).Moreover, for j ∈ {i p , . . ., i 1 }, L(B( j)) will be called the label of the block B( j) and (L(B k ), . . ., L(B 1 )) will be called the label sequence of the partition π .

Remark 2.3 For
1 ( p) and (i p , . . ., i 1 ) ∈ [1, N ] p such that (i p , . . ., i 1 ) π .We say that the sequence (i p , . . ., i 1 ) establishes free-labelling on the partition π (notation (i p , . . ., i 1 ) π ) if for all 1 ≤ l = m ≤ k the following conditions hold: In this case, we say that the label sequence (L(B k ), . . ., L(B 1 )) establishes freelabelling on π and we will use the notation (L(B k ), . . ., L(B 1 )) π .Definition 2. 5 Let N be a positive integer.For π ∈ N C 2  1 ( p, k), we define the following sets: Using the information above, one obtains the following result about the cardinality of the set FL(π, N ) for a non crossing pair partition π .

Proposition 2.6
For a noncrossing pair partition π , one has where the notation π = π 1 ∪ π 2 ∪ • • • ∪ π k is understood as a disjoint union of k subpartitions with exactly one outer block.Moreover,

Free Toy Fock Space
The free toy Fock space was constructed by Attal and Nechita in [2].In this section, we briefly recall this construction and some related aspects.Let H be a separable complex Hilbert space with a fixed orthonormal basis (e i ) i≥1 , and define the full Fock space as where H ⊗0 := C is a one -dimensional Hilbert space and is a distinguished vector with norm one, which is called the vacuum vector.
The free creation and annihilation operators with the "test function" f ∈ H, denoted by l + ( f ) and l − ( f ), respectively, are defined as For T ∈ B(H), we define the so-called gauge operator (or second quantization operator) (T ), as follows Since they can be extended by linearity and continuity to the whole space, where l + ( f ) and l − ( f ) are mutually adjoint.
In what follows, we will consider the case where H = L 2 (R + , C) is the complex Hilbert space of square integrable complex valued functions.Hence, one can redefine the creation, annihilation and gauge operators in the free (full) Fock space . For an arbitrary h ∈ L 2 (R + ), we denote the creation (resp.annihilation) operator by A + (h) (resp.A − (h)), and are defined as follows: Moreover, for an essentially bounded function g ∈ L ∞ (R + ), one can define the gauge operator A • (g) associated to the operator of multiplication by g as follows: In particular, for t ∈ R + and 1 t := 1 [0,t) denotes the indicator function of the interval [0, t), we set 7 For a positive integer n, we say that a given sequence (i n , . . ., i 1 ) ∈ N n is admissible, if i 1 = i 2 . . .= i n .We assume by convention that the empty set ∅ is admissible, and we denote by I n the set of all admissible sequences of size n, and by I the set of all admissible sequences of finite size.
For i ∈ N, let H i = C 2 be a two-dimensional complex Hilbert spaces, and denote by the free product.The free toy Fock space in the sequel denoted by T F(L 2 (R + , C)) is a countable free product of two-dimensional complex Hilbert spaces H i = C 2 , and is given by where C 2 (i) is the i-th copy of C 2 = H i which is endowed with the canonical basis { i = (1, 0) T , X i = (0, 1) T } and is the common identification of the vacuum vectors i with = 1.The orthonormal basis of T F(L 2 (R), C) is given by {X i } i∈I where X i is the tensor

The Embedding of the Free Toy Fock Space into the Full Fock Space
It is shown in [1,2] that the free toy Fock space can be embedded into the full Fock space F(L 2 (R + ; C)).Let us briefly recall this embedding: Consider a partition The associated full Fock space with this decomposition is given as follows and in each Fock space F(L 2 [t i , t i+1 )), we consider the vacuum vector i such that | i | = 1, and the normalized function where 1 [t i ,t i+1 ) denotes the indicator function of the interval [t i , t i+1 ).
The free toy Fock space T F(S) associated to the partition S is a closed subspace of the full Fock space F(L 2 (R + ; C)), and is given by: For t ∈ R + and ∈ {−, +, •}, the relation between the operators A t on the full Fock space F(L 2 (R + ; C)) and their discrete counterparts A i on the free toy Fock space T F(S) is given as follows: For a partition •}, the basic operators A i (S) associated to the partition S on the free toy Fock space T F(S) are given by where P S ∈ B(F(L 2 (R + ; C))) denotes the orthogonal projection on the free toy Fock space T F(S).

Proposition 2.8 [2] The operators A + i (S), A − i (S) and A • i (S) act on the free toy Fock space T (S) in the same way as their discrete counterparts A
In what follows, we consider a partition S n := {0 = t ) → 0 for n going to infinite, and we denote by A i (n) := A i (S n ) for ∈ {−, +, •}.Finally, we get T F(n) := T F(S n ).The following result from [2] plays significant role in our study.Theorem 2.9 [2] For all t ∈ R + , the following operators converge strongly (as n → +∞) to A ± t and A • t , respectively.

Main Results
For n ≥ 1, let i := (i n , . . ., i 1 ) ∈ I be an arbitrary non-empty admissible sequence and let 1 denotes the indicator function.Consider the free toy Fock space T F(L 2 (R + , C)) with the orthonormal basis {X i } i∈I .We define the creation A + X i , annihilation A − X i and conservation A • X i operators in T F(L 2 (R + , C)) with the "test function" X i , as the image of A + , A − and A • , respectively, acting on C 2 i , as follows: (1) The creation operator

These operators are bounded, and A +
i and A − i are mutually adjoint (i.e., (A + i ) * = A − i ).Moreover, the conservation operator A • i is self-adjoint and satisfies Furthermore, the following relations hold Using ( 5) one can prove the following lemma.

Lemma 3.1 For any positive integers i, j and m, one has
Definition 3.2 For any integer p ≥ 2, we denote by E p the set of all sequences = ( p , . . ., 1 ) ∈ {−1, 0, +1} p which satisfy the following conditions: (1 It is known that there is a bijection between the set E p defined above and the set N C 2 1 ( p) of all noncrossing partitions consisting of pair or singleton blocks.Let us recall this bijection from [13] (see also [8]).
Given a sequence = ( p , . . ., 1 ) ∈ E p , for any 1 ≤ k < p for which k = +1, we define M(k) := min j : j > k and It follows that M(k) = −1, and for all k < l < M(k) such that l = +1, we have Therefore, for any sequence = ( p , . . ., 1 ) we can uniquely associate a noncrossing partition consisting of pair or singleton blocks as follows: In such a case, we will use the identification ≡ π , for ∈ E p and π ∈ N C 2 1 ( p).The following two lemmas, can be proven by a simple induction on p, and using the same arguments as in [13].
In the following result, we describe the relation between the sequence (i p , . . ., i 1 ) ∈ [1, N ] p and the mixed-moments ϕ(A ).

Lemma 3.5 [free-labelling and vanishing of mixed-moments]
Proof We proceed the proof by induction on p ≥ 2. Obviously, for p = 2 and p = 3, the sequences (i 2 , i 1 ) and (i 3 , i 2 , i 1 ) establish free-labelling.
(2) The case M(1) = p.We claim that the partition has only one outer block {1, M(1)} as follows: . Indeed, we define j := min{l, 2 ≤ l ≤ p, l = −1}, and we are looking at the first annihilation operator A j i j .For j = 2, we have one pair block {1, 2} and p = M(1) = 2. On the other hand, for j = p, the partition has only one outer block {1, p} and p − 2 singleton inner blocks.In any case, the sequence (i p , . . ., i 1 ) establishes free-labelling on π .
For 2 < j < p, we have k ∈ {+1, 0} for 2 ≤ k ≤ j − 1. Assume that there exists an index 2 ≤ k ≤ j − 1 which satisfies k = 0, and define Therefore, i = +1 for all 1 ≤ i ≤ k − 1.By induction assumption and by the commutation relation (5), one obtains . Taking into account that the singleton block {k } is a direct successor of the pair block {k − 1, M(k − 1)}, and the pair blocks {n, M(n)} are direct successors of the pair blocks {n −1, M(n −1)} for n = 2, . . ., k − 1, respectively.Then, Now, assume that k = +1 for all 1 ≤ k ≤ j − 1.Then, j = M( j − 1), and by induction assumption According to Lemma 3.1, we have Hence, the lemma follows, since and the pair block

Poisson-Type Limit Theorem
In this section, we will study the vacuum distribution for the operators on the free toy Fock space T F(L 2 (R + , C)), where For the case λ = 0, we obtain the free central limit theorem [9,12,21], where the associated (even) moments are the Catalan numbers C p := 1 p+1 2 p p , and the distribution limit is the Wigner measure [19] supported on [−2, 2].Namely, Let us first start with the single operators S 1 (λ), that is, It is clear that the family {A + i + A − i + λA • i } i∈N is free independent in the free toy Fock space T F(L 2 (R + , C)).
We denote by b p (λ) := ϕ((S 1 (λ)) p ) the p-th moment of the operators S 1 (λ).The following theorem can be proven in similar way as in [13] for bm-independence.Theorem 3.7 For any non-negative integer p, the moment sequence where in the sum appear the set of noncrossing partitions consisting of pair outer blocks or inner singleton blocks, and s(π ) denotes the number of singletons in π .Moreover, we have the following recursive formula Remark 3.8 By taking λ = 1 in ( 14), we obtain the Fibonacci sequence.
Theorem 3.9 The distribution law ν 1,λ of the operators S 1 (λ) under the vacuum state ϕ is the two-point distribution: where .
Proof By Theorem 3.7, the direct calculations of the moment generating function f λ (t) := p≥0 b p (λ)t p , yields (cf. [16]).Then, using the Cauchy transform [21] and the Stieltjes inversion formula, we obtain (15).Now let us turn to the general case by considering the limit distribution of the operators under the vacuum state ϕ.After denoting m p (λ) := lim N →+∞ ϕ((S N (λ)) p ), we have the following.
Theorem 3.10 For any non-negative integer p, the moment sequence (m p (λ)) p≥0 is given by the following V (π, λ) (17) where in the sum appear all noncrossing partitions consisting of pair or inner singleton blocks.
Theorem 3.12 For λ > 0, the vacuum distribution of S N (λ) converges for N → +∞ to the free Meixner law ν λ (dx) with parameters (λ, 1, 1, 0).Namely p) be a disjoint union of k sub-partitions with exactly one outer block, such that π j = π j , |π j | = p j ≥ 2 for j = 1, . . ., k, and k j=1 p j = p where π j is an arbitrarily partition consisting of pair or inner singleton blocks, which can be also empty.Then Here x denotes the greatest integer less than or equal to x.Hence, the associated moment generating function g λ (t) := p≥0 m p (λ)t p is given as follows where Note that in the last equality, we used the formula from [4] n≥0 n m t n = t m (1 − t) m+1 , and the moment generating function of the Wigner law where C n denote the Catalan numbers.Hence, the moment generating function g λ (t) is given by Therefore, the desired result follows through the Cauchy transform of the moment generating function g λ (t) (cf.[16,21]).

Remark 3.13
It is worthwhile to mention that for the weakly-monotone independence case [3], the single operator S 1 (λ) = A + i + A − i + λA • i on the weakly-monotone Fock space, has also the vacuum law belonging to the free Meixner class as for the general case S N (λ).

Examples
For illustration, in Table 1, we listed the first moments (m 0 (λ), . . ., m 6 (λ)) of free Poisson distribution in comparison with monotone and some of bm-analogues.
For λ ≥ 0 and n ≥ 1, we define the operators on the free toy Fock space T F(n).On the other hand, we define the operators Proof The proof of (23) follows from Proposition 2.8 and Theorem 3.9, by replacing λ with √ nλ.
For the proof of the second part, we have Hence, the proof follows as an application of Theorem 2.9.

8 7 6 5 4 3 2 1 Definition 2 . 1
Let I be an arbitrary set of indices and π = {B 1 , . . ., B k } be a partition of [ p].A label function with values in I is a map L
where |.| denotes the cardinality, and it is called pair block if |B i | = 2.The set of all partitions π of [ p] is denoted by P( p).
is called inner if there exists another block B j such that B j ≺ π B i .Otherwise, it is called outer.A noncrossing partition π = {B 1 , . . ., B k } ∈ N C( p, k) with |B j | ∈ {1, 2} for j = 1, . . ., k is called noncrossing partition consisting of singletons or pair blocks, and the set of such a partitions is denoted by be a noncrossing partition with blocks being pair or singletons, and let = ( p , . . ., 1 ) ∈ E p .Let us take i = (i p , . . ., i 1 ) ∈ [1, N ] p such that ≡ π and i

Table 1
The first moments m 0 (λ), . . ., m 6 (λ) of free Poisson distribution, monotone Poisson distribution and bm-Poisson distribution for the cones R 2 + and R 3