Free cumulants and Schroder trees

Matthieu Josuat-VergesJoint work with F. Menous, J.-C. Novelli, J.-Y. Thibon

Universite de Marne-la-Vallee

Saarbrucken, January 4th 2017

1 / 25

Free cumulants

If X is a real random variable, its moments are Mn = E(X n),and the moment generating function is

M(z) =∑n≥0

Mnzn+1.

Often, the moments (Mn)n≥0 characterize the distribution of X .

Alternatively, one can use the free cumulants (Cn)n≥1 and theirgenerating function

C(z) = 1 +∑n≥1

Cnzn,

which is Voiculescu’s R-transform of M(z), i.e. C(z) = zM(z)〈−1〉 .

2 / 25

Free cumulants

Speicher’s combinatorial definition of free cumulants relies on

noncrossing partitions such as1 2 3 4 5 6

.

Mn =∑

π noncrossingpartition of {1,...,n}

Cπ,

Cn =∑

π noncrossingpartition of {1,...,n}

µ(π, 1n)Mπ,

where Mπ =∏

B∈πM#B, Cπ =

∏B∈π

C#B, µ is the Mobius function of

the noncrossing partitions lattice and 1n its maximal element.

3 / 25

Free cumulants for several random variables

Let A be an algebra (noncommutative probability space) andφ : A→ C a linear form (expectation).

The mixed moment of a1, . . . , an ∈ A is φ(a1 · · · an).

The mixed cumulants κ(a1, . . . , an) are also defined withnoncrossing partitions:

φ(a1, . . . , an) =∑

π noncrossingpartition of {1,...,n}

κπ(a1, . . . , an),

where κπ(a1, . . . , an) =∏

B∈π, B={i1<i2<... }κ(ai1 , ai2 , . . . ).

4 / 25

Free cumulants for several random variables

More generally, one can consider the operator-valued framework:we have a subalgebra B ⊂ A and φ : A→ B a linear map(conditional expectation).

In place of the mixed moment of (a1, . . . , an), we need themultilinear maps:

(x1, . . . , xn−1) 7→ φ(a1x1a2x2 · · · xn−1an) (xi ∈ B).

The free cumulants are linear maps κn : A⊗B · · · ⊗B A→ B, againdefined using noncrossing partitions.

5 / 25

The goal is to present a generalization of the R-transform coveringthe case of mixed cumulants, in the operator-valued framework.

The construction comes from calculations in an operad of Schrodertrees, and gives combinatorial formula in terms of trees rather thannoncrossing partitions.

Other operadic interpretation of the link between moments andfree cumulants have been studied by Ebrahimi-Fard and Patras,Drummond-Cole.

6 / 25

Partial compositionsLet f (x1, . . . , xm) and g(x1, . . . , xn) be functions of arity m and n.The partial composition f ◦i g is the function

f (x1, . . . , xi−1, g(xi , . . . , xi+n−1), xi+n, . . . , xm+n−1).

To avoid indexing problems of x -variables, a function of arity n isdenoted by a corolla with n leaves, and internal vertex decoratedby f :

f

Partial composition is denoted by plugging the output of afunction to one of the inputs of another one, for example (withm = 5, n = 3, i = 2):

f ◦2 g =

fg

7 / 25

Partial compositions

Partial compositions satisfy some relations.Let f , g , h, of arity m, n, p. If i > j , we have

(f ◦i g) ◦j h = (f ◦j h) ◦i+p−1 g =

f

gh .

If i ≤ j ≤ i + n − 1,

(f ◦i g) ◦j h = f ◦i (g ◦j−i+1 h) =

fg

h .

8 / 25

Operad

An operad is a graded set O =⊎

n≥1On (On is to be thought of as a

set of functions of arity n), together with maps

◦i : Om ×On −→ Om+n−1 (1 ≤ i ≤ m)

satisfying the same relations as partial compositions.(Additionally, we require that O1 contains a unit element.)

We can also consider linear operads: On is a vector space (to bethought of a a space of n-linear maps), and ◦i is bilinear.

A morphism of operads is a graded map commuting with partialcompositions ◦i .

9 / 25

The Schroder operadSchroder trees are plane trees where an internal vertex has at leasttwo descendants. Those with 4 leaves are:

, , , , , ,

, , , , .

The operad of Schroder trees is S =⊎

n≥1Sn where Sn is the set of

(or, vector space based on) Schroder trees with n leaves, andt1 ◦i t2 is obtained by plugging t2 on the ith leave of t1.

It is the free operad generated by corollas:

, , , , . . .

10 / 25

Multilinear mixed moments

The multilinear mixed moments are the maps:

φn : (a1, . . . , an, x1, . . . , xn+1) 7→ φ(x1a1x2a2 · · · anxn+1).

We also let φ0 = x1 (identity map of B).

Consider a sequence (κn)n≥0 of multilinear maps where κn dependson (x1, a1, x2, a2, . . . , an, xn+1). Interpreting ◦i as substitution of xi(leaving ai untouched), they are elements of an operad and there isa morphism defined by t 7→ κt for t ∈ Sn.This defines tree-indexed multilinear maps κt , t ∈ Sn.

11 / 25

Tree-indexed multilinear maps

Concretely, κt(x1, a1, x2, a2, . . . , an, xn+1) is defined by:I if t is the corolla with n + 1 leaves, then κt = κn,I if t = t1 ◦i t2, we let κt be κt1(. . . ) where xi is specialized toκt2(. . . ).

Example

t = x1 x2 x3 x4 x5 x6a1 a2 a3 a4 a5

,

κt = κ2(κ2(x1, a1, x2, a2, x3), a3, κ1(x4, a4, x5), a5, x6

).

12 / 25

Multilinear free cumulants

Free cumulants are the multilinear maps

κn(x1, a1, x2, a2, . . . , an, xn+1)

implicitly defined by

φn =∑

t∈Sn+1,t left directed

κt

where left directed means every right edge arrives at a leaf.

Left directed Schroder trees are in bijections with noncrossingpartitions, so that in the case xi = 1 for all i , κn is the usual freecumulant κn.

13 / 25

Multilinear free cumulants

The bijection from left directed Schroder trees to noncrossingpartitions, on an example:

1 2 3 4 5 6 7

7→ 1|27|346|5

(The map is defined on all Schroder trees, and it is a bijectionwhen restricted to left directed ones.)

14 / 25

Multilinear free cumulants

Some computations:I φ1 = κ1 (the only prime tree with two leaves is ). Soκ1 : (x1, x2) 7→ φ(x1a1x2).

I For n = 2, the trees are and , so

φ2 = κ2 + κ1(κ1(x1, a1, x2), a2, x3)= κ2 + κ1(φ(x1a1x2), a2, x3)= κ2 + φ(φ(x1a1x2)a2x3)

andκ2 = φ(x1a1x2a2x3)− φ(φ(x1a1x2)a2x3).

15 / 25

The group of the Schroder operadTo define the analog of the R-transform, we need the group of theoperad (as defined by Chapoton).An element of the group is a formal series

F =∑

t∈Sn, n≥1ft · t

with f• = 1. The composition is defined by

F ◦ G =∑

t∈Sn, n≥1

∑t1∈Si1 , i1≥1

...tn∈Sin , in≥1

ftgt1 . . . gtn · t ◦ (t1, . . . , tn).

where t ◦ (t1, . . . , tn) = (((t ◦n tn) ◦n−1 tn−1) · · · ◦1 t1).

The inverse F 〈−1〉 exists (its coefficients can be computedrecursively).

16 / 25

The group of the Schroder operad

Another product on series∑

ft · t is the multiplication defined by

F × G =∑

t∈Sn, n≥1

∑u∈Sp , p≥1

ftgu · t ◦n u.

The inverse F−1 exists (its coefficients can be computedrecursively).

PropositionDefine Γ(t) = zn for t ∈ Sn and extend linearly to series of tree.Then Γ(F ◦ G) = Γ(F ) ◦ Γ(G) and Γ(F × G) = Γ(F )Γ(G)

z .

17 / 25

TheoremConsider the series of corollas:

F = • + + + + . . .

Then(F−1)〈−1〉 =

∑t∈Sn, n≥1

t left directed

t.

Apply the morphism t 7→ κt , we get that Φ =∑

n≥0φn and

K =∑

n≥0κn are related by (K−1)〈−1〉 = Φ.

18 / 25

Proof.First, we have

F−1 = (• − (• − F ))−1 =∑n≥0

(• − F )n =∑

t∈Sn, n≥1t right comb

(−1)int(t)t

where right comb means a tree where all internal vertices are inthe right branch, and int(t) is the number of internal vertices of t.Then consider:( ∑

t∈Sn, n≥1right comb

(−1)int(t)t)◦( ∑

u∈Sn, n≥1u left directed

u)

We define a pairing on trees t ◦ (u1, . . . , un) in the expansion toshow that only • remains.

19 / 25

If t ◦ (u1, . . . , un) with un 6= •, write un = cp ◦ (v1, . . . , vp−1, •)where cp is the corolla with p leaves, and vi is also left directed.

Then t ◦ (u1, . . . , un) is paired with

(t ◦n cp) ◦ (u1, . . . , un−1, v1, . . . , vp−1, •).

↔

20 / 25

TheoremLet

F = • + + + + . . .

Then(F 〈−1〉)−1 =

∑t∈Sn, n≥1

t prime

(−1)1+int(t)t

where prime means the right edge from the root arrives at a leaf.

Apply the morphism t 7→ φt , this gives

κn =∑

t∈Sn+1t prime

(−1)int(t)φt .

21 / 25

Proof.First, we show that F 〈−1〉 =

∑t∈Sn, n≥1

(−1)int(t) · t. Expand

F ◦( ∑

t∈Sn, n≥1(−1)int(t) · t

).

If u 6= •, it appears two times in the expansion: either as • ◦ u, oras u = cn ◦ (u1, . . . , un) where cn is the corolla with n leaves.They have different sign and cancel each other.Also, we have:( ∑

t∈Sn, n≥1t prime

(−1)int(t)t)−1

=∑k≥0

( ∑t∈Sn, n≥2

t prime

(−1)1+int(t)t)k

= F 〈−1〉.

22 / 25

Prime Schroder trees with 4 leaves are:

, , , , , .

So:

κ3 = φ(x1a1x2a2x3a3x4)− φ(φ(x1a1x2)a2x3a3x4)− φ(φ(x1a1x2a2x3)a3x4)− φ(x1a1φ(x2a2x3)a3x4)+ φ(φ(φ(x1a1x2)a2x3)a3x4) + φ(φx1a1φ(x2a2x3)a3x4).

23 / 25

To recover the known result using the Mobius function ofnoncrossing partitions, recall the map from Schroder trees tononcrossing partitions:

1 2 3 4 5 6 7

7→ 1|27|346|5

PropositionThe number of preimages of a noncrossing partition π is∏B∈πc

C#B−1 where: πc is the Kreweras complement of π, Cn is the

nth Catalan number.The proof is bijective and this number is also µ(π, 1n).

24 / 25

Thanks for your attention

25 / 25