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Cumulants mixtes et arbres couvrants Valentin Fray travail en commun avec Pierre-Loc Mliot (Orsay) and Ashkan Nighekbali (Zrich) Institut fr Mathematik, Universitt Zrich Journes ALA 2014 CIRM, Marseille, 17 mars 2014 V.

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SLIDE 1

Cumulants mixtes et arbres couvrants

Valentin Féray

travail en commun avec Pierre-Loïc Méliot (Orsay) and Ashkan Nighekbali (Zürich)

Institut für Mathematik, Universität Zürich

Journées ALÉA 2014 CIRM, Marseille, 17 mars 2014

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 1 / 15

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SLIDE 2

Number of triangles in a random graph cumulant method

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each possible edge (i, j) belongs to G independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 2 / 15

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SLIDE 3

Number of triangles in a random graph cumulant method

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each possible edge (i, j) belongs to G independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2 Question Fix p ∈]0; 1[. Describe asymptotically the fluctuations of the number Tn of triangles.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 2 / 15

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SLIDE 4

Number of triangles in a random graph cumulant method

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each possible edge (i, j) belongs to G independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2 Question Fix p ∈]0; 1[. Describe asymptotically the fluctuations of the number Tn of triangles. Answer (Rucińsky, 1988) The fluctuations are asymptotically Gaussian.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 2 / 15

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SLIDE 5

Number of triangles in a random graph cumulant method

A good tool for that: mixed cumulants

the r-th mixed cumulant kr of r random variables is a r-linear symmetric functional. It is a polynomial in joint moments. Examples: κ1(X) = E(X), κ2(X, Y ) = Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) = E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 3 / 15

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SLIDE 6

Number of triangles in a random graph cumulant method

A good tool for that: mixed cumulants

the r-th mixed cumulant kr of r random variables is a r-linear symmetric functional. It is a polynomial in joint moments. Examples: κ1(X) = E(X), κ2(X, Y ) = Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) = E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z). if the variables can be split in two mutually independent sets, then the cumulant vanishes.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 3 / 15

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SLIDE 7

Number of triangles in a random graph cumulant method

A good tool for that: mixed cumulants

the r-th mixed cumulant kr of r random variables is a r-linear symmetric functional. It is a polynomial in joint moments. Examples: κ1(X) = E(X), κ2(X, Y ) = Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) = E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z). if the variables can be split in two mutually independent sets, then the cumulant vanishes. if, for each r = 2, the sequence κr(Xn) := κr(Xn, . . . , Xn) converges towards 0 and if E(Xn) and Var(Xn) have a limit, then Xn converges in distribution towards a Gaussian law.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 3 / 15

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SLIDE 8

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 9

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 10

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). But most of the terms vanish (because the variables are independent).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 11

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). But most of the terms vanish (because the variables are independent). Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 13 8 3 11 5 2 7 23 9 1 19

κℓ(B∆1, . . . , B∆7) = 0. {∆1, ∆2, ∆5, ∆7} is indepen- dent from {∆3, ∆4, ∆6}. Reminder: different edges be- ing in the graph are independent events.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 12

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). But most of the terms vanish (because the variables are independent). Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 13 8 3 11 5 2 7 23 9 1

κℓ(B∆1, . . . , B∆7) = 0. {∆1, ∆2, ∆5, ∆7} is indepen- dent from {∆3, ∆4, ∆6}. Triangles need to share an edge to be dependent!

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 13

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). But most of the terms vanish (because the variables are independent). Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 ∆8 13 8 3 11 5 2 7 23 9 1

κℓ(B∆1, . . . , B∆8) = 0. This configuration contributes to the sum. Note that it has only ℓ + 2 vertices.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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SLIDE 14

Number of triangles in a random graph cumulant method

Application to the number of triangles

Tn =

  • ∆={i,j,k}⊂[n]

B∆, where B∆(G) =

  • 1

if G contains the triangle ∆;

  • therwise.

By multilinearity, κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). But most of the terms vanish (because the variables are independent). Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 ∆8 13 8 3 11 5 2 7 23 9 1

κℓ(B∆1, . . . , B∆8) = 0. This configuration contributes to the sum. Note that it has only ℓ + 2 vertices. |κℓ(Tn)| = Oℓ(nℓ+2)

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 4 / 15

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Number of triangles in a random graph cumulant method

The central limit theorem for triangles

Proposition (Leonov, Shirryaev, 1955) If X1, . . . , Xℓ can be split into two sets of mutually independent variables, then κℓ(X1, · · · , Xℓ) = 0 Corollary (Janson, 1988 ?) For each ℓ, there exists a constant Cℓ such that |κℓ(Tn)| = Cℓnℓ+2

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 5 / 15

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Number of triangles in a random graph cumulant method

The central limit theorem for triangles

Proposition (Leonov, Shirryaev, 1955) If X1, . . . , Xℓ can be split into two sets of mutually independent variables, then κℓ(X1, · · · , Xℓ) = 0 Corollary (Janson, 1988 ?) For each ℓ, there exists a constant Cℓ such that |κℓ(Tn)| = Cℓnℓ+2 Corollary (Ruciński, 1988) Tn − E(Tn)

  • Var(Tn)

→ N(0, 1) Proof: Var(Tn) ≈ n4 and κℓ(Tn/n2) = n2−ℓ = oℓ(1) for ℓ > 2.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 5 / 15

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Number of triangles in a random graph cumulant method

Our work

Theorem (F., Méliot, Nighekbali, 2014) Let X1, . . . , Xℓ be random variables with finite moments of order ℓ, |κℓ(X1, · · · , Xℓ)| ≤ 2ℓ−1X1ℓ · · · Xℓℓ · ST

  • Gdep(X1, · · · , Xℓ)
  • ,

where ST

  • Gdep(X1, · · · , Xℓ)
  • is the number of spanning trees of the

dependency graph of X1, · · · , Xℓ. Dependency graphs for a list (B∆1, · · · , B∆ℓ): B∆i ∼ B∆j ⇔ ∆i and ∆j share an edge Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 ∆8 B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 6 / 15

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Number of triangles in a random graph cumulant method

Our work

Theorem (F., Méliot, Nighekbali, 2014) Let X1, . . . , Xℓ be random variables with finite moments of order ℓ, |κℓ(X1, · · · , Xℓ)| ≤ 2ℓ−1X1ℓ · · · Xℓℓ · ST

  • Gdep(X1, · · · , Xℓ)
  • ,

where ST

  • Gdep(X1, · · · , Xℓ)
  • is the number of spanning trees of the

dependency graph of X1, · · · , Xℓ. Dependency graphs for a list (B∆1, · · · , B∆ℓ): B∆i ∼ B∆j ⇔ ∆i and ∆j share an edge Naive bound: ℓℓ(ℓ−1)!X1ℓ · · · Xℓℓ Example:

∆1 ∆7 ∆5 ∆2 ∆6 ∆3 ∆4 ∆8 B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 6 / 15

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SLIDE 19

Number of triangles in a random graph cumulant method

Our work

Theorem (F., Méliot, Nighekbali, 2014) Let X1, . . . , Xℓ be random variables with finite moments of order ℓ, |κℓ(X1, · · · , Xℓ)| ≤ 2ℓ−1X1ℓ · · · Xℓℓ · ST

  • Gdep(X1, · · · , Xℓ)
  • .

Corollary (FMN, 2014) There exists an absolute constant C such that |κℓ(Tn)| = (Cℓ)ℓnℓ+2 Bound in Janson’s proof: (Cℓ)3ℓnℓ+2 (Döring, Eichelsbacher, 2012)

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 6 / 15

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SLIDE 20

Number of triangles in a random graph cumulant method

Our work

Theorem (F., Méliot, Nighekbali, 2014) Let X1, . . . , Xℓ be random variables with finite moments of order ℓ, |κℓ(X1, · · · , Xℓ)| ≤ 2ℓ−1X1ℓ · · · Xℓℓ · ST

  • Gdep(X1, · · · , Xℓ)
  • .

Corollary (FMN, 2014) There exists an absolute constant C such that |κℓ(Tn)| = (Cℓ)ℓnℓ+2 Corollary (FMN, 2014) Very precise extension of the central limit theorem: if 1 < < v < < n1/2, P

  • Tn ≥

n

3

  • p3 + v · n2

∼

1

√

π p5(1−p)v 2 exp

  • −

v 2 p5 (1−p) + (7−8p) v 3 n√ p(1−p)/2

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(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 6 / 15

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A new bound on cumulants via spanning trees

Moment-cumulant relation

Mixed cumulants can be expressed in terms of mixed moments: κ(X1, . . . , Xr) =

  • π

µ(π)Mπ, where π runs over set-partitions of [ℓ], µ(π) = µ(π, {[ℓ]}) is the Möbius function of the set-partition poset (it is explicit!), Mπ =

B∈π E

  • i∈B Xi
  • .

Example: κ3(X, Y , Z) = E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 7 / 15

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A new bound on cumulants via spanning trees

Using independence to simplify Mπ

Example: π =

  • {1, 2, 3, 4}, {5, 6}
  • and

H := Gdep(X1, . . . , X6) =

1 2 3 4 5 6

Then Mπ := E(X1X2X3X4)E(X5X6) = E(X1X2)E(X3X4)E(X5)E(X6).

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A new bound on cumulants via spanning trees

Using independence to simplify Mπ

Example: π =

  • {1, 2, 3, 4}, {5, 6}
  • and

H := Gdep(X1, . . . , X6) =

1 2 3 4 5 6

Then Mπ := E(X1X2X3X4)E(X5X6) = E(X1X2)E(X3X4)E(X5)E(X6). In general, Mπ = MφH(π), with the following definition of φH(π). replace each part πi of π by the connected components of the induced graph H[πi].

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 8 / 15

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A new bound on cumulants via spanning trees

Rewriting the summation

κ(X1, . . . , Xr) =

  • π

µ(π)Mπ =

  • π

µ(π)MφH(π) =

  • π′

Mπ′   

  • π s.t.

φH (π)=π′

µ(π)   

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SLIDE 25

A new bound on cumulants via spanning trees

Rewriting the summation

κ(X1, . . . , Xr) =

  • π

µ(π)Mπ =

  • π

µ(π)MφH(π) =

  • π′

Mπ′   

  • π s.t.

φH (π)=π′

µ(π)    φH(π) = π′ ⇒ for all part π′

i of π′, the induced graph H[π′ i] is

connected.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 9 / 15

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SLIDE 26

A new bound on cumulants via spanning trees

Rewriting the summation

κ(X1, . . . , Xr) =

  • π

µ(π)Mπ =

  • π

µ(π)MφH(π) =

  • π′

Mπ′   

  • π s.t.

φH (π)=π′

µ(π)    φH(π) = π′ ⇒ for all part π′

i of π′, the induced graph H[π′ i] is

connected. if so, we have to compute απ′

H :=

  • π s.t.

φH (π)=π′

µ(π).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 9 / 15

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A new bound on cumulants via spanning trees

απ′

H and Tutte polynomial

Consider the contracted graph H/π. Example:

1 2 3 4 5 6

π =

  • {1, 2}, {3, 4}, {5, 6}
  • →

1, 2 3, 4 5, 6

It is a multigraph.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 10 / 15

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A new bound on cumulants via spanning trees

απ′

H and Tutte polynomial

Consider the contracted graph H/π. Example:

1 2 3 4 5 6

π =

  • {1, 2}, {3, 4}, {5, 6}
  • →

1, 2 3, 4 5, 6

Lemma απ′

H =

  • E⊂E(H/π′)

(−1)|E|, where the sum runs over spanning connected subgraphs of H/π′. If H/π′ is connected,

  • απ′

H

  • is Tutte polynomial evaluated at (1, 0).
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 10 / 15

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A new bound on cumulants via spanning trees

απ′

H and Tutte polynomial

Consider the contracted graph H/π. Example:

1 2 3 4 5 6

π =

  • {1, 2}, {3, 4}, {5, 6}
  • →

1, 2 3, 4 5, 6

Lemma απ′

H =

  • E⊂E(H/π′)

(−1)|E|, where the sum runs over spanning connected subgraphs of H/π′. If H/π′ is connected,

  • απ′

H

  • is Tutte polynomial evaluated at (1, 0).

Corollary:

  • απ′

H

  • ≤ ST(H/π′).
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 10 / 15

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A new bound on cumulants via spanning trees

Bounding everything

Reminder: κ(X1, . . . , Xℓ) =

  • π′

Mπ′απ′

H

where the sum runs over set-partition π′ such that the induced graphs H[π′

i] are connected.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 11 / 15

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SLIDE 31

A new bound on cumulants via spanning trees

Bounding everything

Reminder: κ(X1, . . . , Xℓ) =

  • π′

Mπ′απ′

H

  • i

1H[π′

i ] connected

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 11 / 15

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SLIDE 32

A new bound on cumulants via spanning trees

Bounding everything

Reminder: κ(X1, . . . , Xℓ) =

  • π′

Mπ′απ′

H

  • i

1H[π′

i ] connected

We have the following inequalities |Mπ| ≤ X1ℓ · · · Xℓℓ (Hölder inequality);

  • απ′

H

  • ≤ ST(H/π′);

1H[π′

i ] connected ≤ ST(H[π′

i])

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 11 / 15

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SLIDE 33

A new bound on cumulants via spanning trees

Bounding everything

Reminder: κ(X1, . . . , Xℓ) =

  • π′

Mπ′απ′

H

  • i

1H[π′

i ] connected

We have the following inequalities |Mπ| ≤ X1ℓ · · · Xℓℓ (Hölder inequality);

  • απ′

H

  • ≤ ST(H/π′);

1H[π′

i ] connected ≤ ST(H[π′

i])

Thus |κ(X1, . . . , Xℓ)| ≤ X1ℓ · · · Xℓℓ

  • π′

ST(H/π′)

  • i

ST(H[π′

i])

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 11 / 15

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A new bound on cumulants via spanning trees

A combinatorial identity

Lemma 2ℓ−1 ST(H) =

  • π′

ST(H/π′)

  • i

ST(H[π′

i])

  • ,

where the sum runs over all set-partitions of [ℓ].

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 12 / 15

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A new bound on cumulants via spanning trees

A combinatorial identity

Lemma 2ℓ−1 ST(H) =

  • π′

ST(H/π′)

  • i

ST(H[π′

i])

  • ,

where the sum runs over all set-partitions of [ℓ]. T =

1 2 3 4 5 6

↔                              π =

  • {1, 2, 3}, {4, 5, 6}
  • ;

T1 =

1 2 3

, T2 =

4 5 6

; T =

π1 π2

.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 12 / 15

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A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤

  • ∆1,...,∆ℓ

2ℓ−1 · ST

  • Gdep(B∆1,...,∆ℓ)
  • .
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

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SLIDE 37

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-38
SLIDE 38

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ?

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-39
SLIDE 39

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ?

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

Choose any triangle for ∆1: n

3

  • choices ;
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-40
SLIDE 40

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ?

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

Choose any triangle for ∆1: n

3

  • choices ;

∆5 should have an edge in common with ∆1: 3 for an edge of ∆1 and n − 2 choices for the other vertex of ∆5 ⇒ 3n − 6 choices ;

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-41
SLIDE 41

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ?

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

Choose any triangle for ∆1: n

3

  • choices ;

∆5 should have an edge in common with ∆1: 3 for an edge of ∆1 and n − 2 choices for the other vertex of ∆5 ⇒ 3n − 6 choices ; ∆2 should have an edge in common with ∆5. Also 3n − 6 choices. . . .

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-42
SLIDE 42

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ? n

3

  • (3n−6)ℓ−1

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

Choose any triangle for ∆1: n

3

  • choices ;

∆5 should have an edge in common with ∆1: 3 for an edge of ∆1 and n − 2 choices for the other vertex of ∆5 ⇒ 3n − 6 choices ; ∆2 should have an edge in common with ∆5. Also 3n − 6 choices. . . .

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-43
SLIDE 43

A new bound on cumulants via spanning trees

A precise bound on cumulants of Tn

Recall that κℓ(Tn) =

  • ∆1,...,∆ℓ

κℓ(B∆1, . . . , B∆ℓ). Thus |κℓ(Tn)| ≤ 2ℓ−1

  • T Cayley tree
  • (∆1, . . . , ∆ℓ) s.t. T ⊂ Gdep(B∆1,...,∆ℓ)
  • .

Fix a Cayley tree. For how many lists of triangles is it contained in Gdep(B∆1,...,∆ℓ) ? n

3

  • (3n−6)ℓ−1

B∆1 B∆7 B∆5 B∆2 B∆6 B∆3 B∆4 B∆8

Choose any triangle for ∆1: n

3

  • choices ;

∆5 should have an edge in common with ∆1: 3 for an edge of ∆1 and n − 2 choices for the other vertex of ∆5 ⇒ 3n − 6 choices ; ∆2 should have an edge in common with ∆5. Also 3n − 6 choices. . . . |κℓ(Tn)| ≤ ℓℓ−2 n 3

  • (6n − 12)ℓ−1 ≤ (6ℓ)ℓnℓ+2
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 13 / 15

slide-44
SLIDE 44

A new bound on cumulants via spanning trees

Moderate deviations

Let Xn = (Tn − E(Tn))/n5/3, then log E(exp(zXn)) =

  • ℓ≥2

κ(ℓ)(Xn)zℓ/ℓ! = n2/3σ2 z2/2 + L z3/6 +

  • ℓ≥4

n5/3 κ(ℓ)(Tn) zℓ/ℓ!

  • call it R(z)

+O(n−1/3)

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 14 / 15

slide-45
SLIDE 45

A new bound on cumulants via spanning trees

Moderate deviations

Let Xn = (Tn − E(Tn))/n5/3, then log E(exp(zXn)) =

  • ℓ≥2

κ(ℓ)(Xn)zℓ/ℓ! = n2/3σ2 z2/2 + L z3/6 +

  • ℓ≥4

n5/3 κ(ℓ)(Tn) zℓ/ℓ!

  • call it R(z)

+O(n−1/3) But |R(z)| ≤

ℓ≥4 n2(3−ℓ)/3(Cℓ)ℓzℓ/ℓ! = O(n−2/3) locally uniformly for z

in C. Thus E(exp(zXn)) = exp

  • n2/3σ2 z2/2 + L z3/6
  • (1 + O(n−1/3)).
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 14 / 15

slide-46
SLIDE 46

A new bound on cumulants via spanning trees

Moderate deviations

Let Xn = (Tn − E(Tn))/n5/3, then log E(exp(zXn)) =

  • ℓ≥2

κ(ℓ)(Xn)zℓ/ℓ! = n2/3σ2 z2/2 + L z3/6 +

  • ℓ≥4

n5/3 κ(ℓ)(Tn) zℓ/ℓ!

  • call it R(z)

+O(n−1/3) But |R(z)| ≤

ℓ≥4 n2(3−ℓ)/3(Cℓ)ℓzℓ/ℓ! = O(n−2/3) locally uniformly for z

in C. Thus E(exp(zXn)) = exp

  • n2/3σ2 z2/2 + L z3/6
  • (1 + O(n−1/3)).

Then use P[X ≥ x] = lim

R→∞

  • 1

2π

R

−R exp(−x(h+iu)) h+iu

E

  • exp((h + iu)X)
  • du
  • .
  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 14 / 15

slide-47
SLIDE 47

A new bound on cumulants via spanning trees

Moderate deviations

Let Xn = (Tn − E(Tn))/n5/3, then log E(exp(zXn)) =

  • ℓ≥2

κ(ℓ)(Xn)zℓ/ℓ! = n2/3σ2 z2/2 + L z3/6 +

  • ℓ≥4

n5/3 κ(ℓ)(Tn) zℓ/ℓ!

  • call it R(z)

+O(n−1/3) But |R(z)| ≤

ℓ≥4 n2(3−ℓ)/3(Cℓ)ℓzℓ/ℓ! = O(n−2/3) locally uniformly for z

in C. Thus E(exp(zXn)) = exp

  • n2/3σ2 z2/2 + L z3/6
  • (1 + O(n−1/3)).

→ looks like, but is stronger than the hypotheses in Hwang’s quasi-power theorem (convergence on C!) ⇒ stronger results.

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 14 / 15

slide-48
SLIDE 48

A new bound on cumulants via spanning trees

Conclusion

very general bound on mixed cumulants, with a strong combinatorial flavor ; implies a good uniform bound on cumulants of sums of partially dependent random variables (number of copies of subgraphs, character

  • f a random irreducible representation, . . . ) ;

implies some precise deviation results (+ local limit laws, bound on Kolmogorov distance).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 15 / 15

slide-49
SLIDE 49

A new bound on cumulants via spanning trees

Conclusion

very general bound on mixed cumulants, with a strong combinatorial flavor ; implies a good uniform bound on cumulants of sums of partially dependent random variables (number of copies of subgraphs, character

  • f a random irreducible representation, . . . ) ;

implies some precise deviation results (+ local limit laws, bound on Kolmogorov distance). Questions: Large deviations P(Tn ≥ E(Tn) + v n3) ∼ ? ;

  • ther models: pn → 0, G(n, M) (fixed number of edges ⇒

almost-independence).

  • V. Féray (with PLM, AN)

(I-Math, UZH) Cumulants mixtes et arbres couvrants ALÉA, mars 2014 15 / 15