k-divisible non-crossing partitions ∗

We derive a formula for the moments and the free cumulants of the multiplication of  $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $\mu^{\boxtimes k}$, given by Kargin, which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida.


Introduction and statement of results
Until recently, k-divisible non-crossing partitions have been overlooked in free probability and have barely appeared in the literature.However, their structure is very rich and there are, for instance, quite natural bijections between k-divisible non-crossing partitions and (k + 1)-equal partitions which preserve a lot of structure, (see e.g.[2]).Furthermore, as noticed in [1], the moments of µ k can be computed using k-divisible non-crossing partitions.
In this paper we exploit the fact that k-divisible and k-equal partitions are linked, by the Kreweras complement, to partitions which are involved in the calculation of moments and free cumulants of the product of k free random variables.For details on free cumulants and their relevance in free probability, see [10].
Given a, b ∈ A free random variables, with free cumulants κ n (a) and κ n (b), respectively, one can calculate the free cumulants of ab by κ n (ab) = π∈N C(n) κ π (a)κ Kr(π) (b), (1.1) where Kr(π) is the Kreweras complement of the non-crossing partition π.In particular, we are able to compute the free cumulants of the free multiplicative convolution of two compactly supported probability measures µ, ν, such that Supp(µ) ⊆ [0, ∞) by In principle, this formula could be inductively used to provide the free cumulants and moments of the convolutions of k (not necessarily equal) positive probability measures.This approach, however, prevents us from noticing a deeper combinatorial structure behind such products of free random variables.
Our fundamental observation is that, when π and Kr(π) are drawn together, the partition π∪Kr(π) ∈ N C(2n) is exactly the Kreweras complement of a 2-equal partition (i.e. a non-crossing pairing).Furthermore, one can show using the previous correspondence that Equation (1.1) may be rewritten as Since 2-equal partitions explain the free convolution of two variables, it is natural to try to describe the product of k free variables in terms of k-equal partitions.
The main result of this work is the following.Theorem 1.1.Let a 1 , . . ., a k ∈ (A, τ ) be free random variables.Then the free cumulants and the moments of a := a 1 . . .a k are given by where N C k (n) and N C k (n) denote, respectively, the k-equal and k-divisible partitions of [kn].
The main application of our formulas is a new proof of the fact, first proved by Kargin [5], that for positive measures centered at 1, the support of the free multiplicative convolution µ k grows at most linearly.Moreover, our approach enables us to generalize to the case µ 1 • • • µ k , as follows.

Theorem 1.2.
There exists a universal constant C > 0 such that for all k and any µ 1 , . . ., µ k probability measures supported on [0, L], satisfying E(µ i ) = 1 and V ar(µ i ) ≥ σ 2 , for i = 1, . . ., k, the supremum L k of the support of the measure µ In other words, for (not necessarily identically distributed) positive free random Let us point out that for the case µ 1 = • • • = µ k , the previous theorem can be proved as using the methods of [7].However, the norm estimates given there are meant to address more general situations (where certain linear combinations of products are allowed) and hence, the constants obtained using these methods for our specific problem are far from optimal.
The paper is organized as follows: Section 2 includes preliminaries on non-crossing partitions and the basic definitions from free probability that will be required.Some enumerative aspects of k-divisible partitions and their Kreweras complement are included.In Section 3 we prove our main formulae for the products of free random variables.We also express our results in terms of free multiplicative convolutions of compactly supported positive probability measures.Our formulas are used in Section 4 to derive bounds for the support of free multiplicative convolutions.Section 5 gives further applications and examples of our formulas.We calculate the free cumulants of products of free Poissons and shifted semicirculars.We show that the n-th cumulant of µ k eventually becomes positive as k grows.We also provide a new proof of the limit theorem of Sakuma and Yoshida [11].

Non-crossing partitions
A partition of a finite, totally ordered set S is a decomposition π = {V 1 , . . ., V r } of S into pairwise disjoint, non-empty subsets V i , (1 The number of blocks of π is denoted by |π|. A partition π = {V 1 , . . ., V r } is called non-crossing if for every 4-tuple a < b < c < d ∈ S such that a, c ∈ V i and b, d ∈ V j , we have that V j = V i .We denote by N C(S) the set of non-crossing partitions of S. For S = [n] := {1, 2, . . ., n} we simply write N C(n).
N C(S) can be equipped with the partial order of reverse refinement (π σ if and only if every block of π is completely contained in a block of σ).This turns N C(S) into a lattice, in particular, we can talk about the least upper bound π 1 ∨ π 2 ∈ N C(S) of two non-crossing partitions π 1 , π 2 ∈ N C(S).We denote the minimum and maximum partitions of N C(n) by 0 n , 1 n , respectively.We will write ρ n k := {(1, . . ., k), (k + 1, . . ., 2k), . . ., ((k − 1)n + 1, . . ., kn)} for the partition in N C(kn) consisting of n consecutive blocks of size k.Definition 2.1.We say that a non-crossing partition π is k-divisible if the sizes of all the blocks of are multiples of k.If, furthermore, all the blocks are exactly of size k, we say that π is k-equal.A partition π ∈ N C(nk) is called k-preserving if all its blocks contain numbers with the same congruence modulo k.
Later we will see that these concepts are closely related.We denote the set of kdivisible non-crossing partitions of [kn] It is helpful to picture partitions via their circular representation: We think of [n] as the clockwise labelling of the vertices of a regular n-gon.If we identify each block of π ∈ N C(n) with the convex hull of its corresponding vertices, then we see that π is non-crossing precisely when its blocks are pairwise disjoint (that is, they don't cross).
It is well known that the number of non-crossing partitions is given by the Catalan The following formula (proved by Kreweras [8]) counts the number of partitions of a given type.Proposition 2.2.Let n be a positive integer and let r 1 , r 2 , . . ., r n ∈ N ∪ {0} be such that Then the number of partitions of π in N C(n) with r 1 blocks with 1 element, r 2 blocks with 2 elements, . . ., r n blocks with n elements equals From the previous proposition one can easily count k-equal partitions.
Finally, the number of k-divisible partitions was essentially given by Edelman [4].
The Kreweras complement [8] satisfies many nice properties.The map Kr : Similar to Proposition 2.2, one can count the number of partitions π, such that π and Kr(π) have certain block structures.Let (r i ) 1≤i≤n , (b j ) 1≤j≤n be tuples satisfying Then the number of partitions such that π has r i blocks of size i and Kr(π) has b j blocks of size j is given by the formula As a consequence, we can show that for large k, the Kreweras complements of kequal partitions have "typically" small blocks.More precisely, for n, k In this case, the only possibility is that b (2.5) An easy application of Stirling's approximation formula shows that (2.6)

Free Probability and Free Cumulants
A C * -probability space is a pair (A, τ ), where A is a unital C * -algebra and τ : A → C is a positive unital linear functional.
The free cumulant functionals (κ n ) n≥1 , κ n : A n → C are defined inductively and indirectly by the moment-cumulant formula: where, for a family of functionals (f n ) n≥1 and a partition π ). ( If we put τ n (a 1 , . . ., a n ) := τ (a 1 • • • a n ), Formula 2.7 may be inverted as follows: where M ob : For a detailed exposition we refer to [10, Lecture 11].
The main property of the free cumulants is that they characterize free independence.Here we will actually employ this characterization as the definition for freeness (for the equivalence with the usual definition see e.g.[10]).
Consider random variables a 1 , . . ., a n ∈ A. Then the following equation holds: κ π (a 1 , . . ., a n ).Products of k free random variables is well known that for a self-adjoint element a ∈ A there exist a unique compactly supported measure µ a (its distribution) with the same moments as a, that is, In particular, if a ∈ A is positive (i.e. a = bb * for some b ∈ A), the measure µ a is supported in the positive half-line.Furthermore, given compactly supported probability measures µ 1 , . . ., µ k we can find free elements a 1 , . . ., a k in some C * -probability space such that µ ai = µ i .
For compactly supported measures on the positive half-line µ 1 , . . ., µ k , we can then consider positive elements a 1 , . . ., a k ∈ A such that µ i = µ ai .Positive elements have unique positive square roots, so let b 1 , . . ., b k be the (necessarily free) positive elements such that b 2 i = a i .
Then, on one hand, the free multiplicative convolution is defined as the distribution It is not hard to see that for tracial C * -probability spaces, the moments of b are exactly the moments of a := a 1 • • • a k .It will be convenient for us to work with a instead of b.
On the other hand, the free additive convolution is defined as the distribution µ 1 Finally, for a measure µ and c > 0 we denote by D c (µ) the measure such that

Main Formulas
The following characterization plays a central role in this work.The proof, elementary but cumbersome, can be found in the Appendix.
ii) π ∈ N C(kn) is k-completing if and only if π = Kr(σ) for some k-equal partition σ ∈ N C k (n).
We are ready to prove our main Theorem.
Proof of Theorem 1.1.By the formula for products as arguments, we have that Since the random variables are free, the sum runs actually over k-preserving partitions (otherwise there would be a mixed cumulant).But then by Proposition 3.1 ii), the partitions involved in the sum are exactly the Kreweras complements of k-equal partitions, and the formula follows.
For the proof of (1.5), we use the moment-cumulant formula κ π (a 1 , . . ., a n ). where Remark 3.5.From Equation (3.3) it is easy to see that for compactly supported measures with mean 1, the variance is additive with respect to free multiplicative convolution, that is V ar(µ i ).

Supports of free multiplicative convolutions
Our main result can be used to compute bounds for the supports of free multiplicative convolutions of positive measures.Our results in this section generalize those by recall that the cardinality of N C(n) is given by the Catalan number C n , which is easily bounded by 4 n .It is also known that the Möbius function M ob : [10,Prop 13.15]).Thus we are able to control the size of the free cumulants.Lemma 4.1.Let µ be a probability measure supported on [0, L] with variance σ 2 , such that Proof.It is easy to see that L ≥ 1 and therefore Then we have that κ and for n ≥ 4 we have We easily see that the growth of the support is no less than linear.
Proposition 4.2.Let µ 1 , . . ., µ k be compactly supported probability measures on R + , satisfying E(µ i ) = 1, V ar(µ i ) = σ 2 , and let L k be the supremum of the support of In general, C may be taken ≤ 26e.If the measures µ i , 1 ≤ i ≤ k have non-negative free cumulants, C may be taken ≤ e.
Proof.By Equation (3.3) we get Since a k-divisible partition π ∈ N C k (n) has at most n blocks, we know that Now, let L = 26L.By Lemma 4.1, we know that κ πi (µ i ) ≤ ( L) n−|πi| .Hence   If µ has non-negative free cumulants we may replace L by L.

More applications and examples
In this section we want to show some examples of how Theorem 1.1 may be used to calculate free cumulants.
We want to calculate the free cumulants of ω k + , k ≥ 2. So let a 1 , . . ., a k be free random variables with distribution ω + .By Theorem 1.1, the free cumulants of a := a κ Kr(π) (a 1 , . . ., a k ). (5.1) If Kr(π) contains a block of size greater than 2, then κ Kr(π) = 0. Hence the sum runs actually over N C(k, n) 2,1 .Therefore each summand has the common contribution of (σ 2 ) n−1 and by Equation (2.5) we know the number of summands.Then the free cumulants are another application of our main formula, we show that the free cumulants of µ k become positive for large k.This is of some relevance if we recall that our estimates of the support of µ k are better with the presence of non-negative free cumulants.Theorem 5.3.Let µ be a probability measure supported on [0, L] with mean α and variance σ 2 .Then for each n ≥ 1 there exist a constant N such that for all k ≥ N , the first n free cumulants of µ k are non-negative.
Proof.Clearly it is enough to show that, for each n ≥ 1, there exist N 0 such that the n-th free cumulant of µ k is positive for all k ≥ N 0 .
Let n > 1 and α := max{α n−1 , 1}.By the same arguments as in Lemma 4.1 one can show that |κ n (µ)| ≤ 16(L n ).Then by Theorem 1.1 we have The factor α (k−2)n+2 is positive and the rest of the expression becomes positive for all k larger than some N 0 , since, by Equation (2.6), N C(k, n) 2,1 /N C k (n) → 1 as k → ∞.
It would be interesting to investigate whether or not all free cumulants become positive.
In a recent paper [11], Sakuma and Yoshida found a probability measure h σ 2 , arising from a limit theorem, which is infinitely divisible with respect to both multiplicative and additive free convolutions.Using our methods, we give another proof of this limit theorem.We restrict to the case E(µ) = 1.The general case follows directly from this.

Example 5 . 1 .
(Product of free Poissons) Theorem 1.1 takes a very easy form in the particular case µ i = m, where m is the Marchenko-Pastur distribution of parameter 1.Indeed, since κ n (m) = 1, we get

. 2 )
Note that in this example m n (a) ≥ κ n (a) (since all the free cumulants are positive) and L = 1 + 2σ.By Proposition 4.3 and an application of Stirling's formula to Equation (5.2) the supremum L k of the support of ω k + satisfies