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) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 1 / 15
Number of triangles in a random graph cumulant method A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each possible edge ( i , j ) belongs to G independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 2 / 15
Number of triangles in a random graph cumulant method A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each possible edge ( i , j ) belongs to G independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 Question Fix p ∈ ] 0 ; 1 [ . Describe asymptotically the fluctuations of the number T n of triangles. V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 2 / 15
Number of triangles in a random graph cumulant method A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each possible edge ( i , j ) belongs to G independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 Question Fix p ∈ ] 0 ; 1 [ . Describe asymptotically the fluctuations of the number T n of triangles. Answer (Rucińsky, 1988) The fluctuations are asymptotically Gaussian. V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 2 / 15
Number of triangles in a random graph cumulant method A good tool for that: mixed cumulants the r -th mixed cumulant k r of r random variables is a r -linear symmetric functional. It is a polynomial in joint moments. Examples: κ 2 ( X , Y ) = Cov ( X , Y ) = E ( XY ) − E ( X ) E ( Y ) κ 1 ( X ) = E ( X ) , κ 3 ( X , Y , Z ) = E ( XYZ ) − E ( XY ) E ( Z ) − E ( XZ ) E ( Y ) − E ( YZ ) E ( X ) + 2 E ( X ) E ( Y ) E ( Z ) . V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 3 / 15
Number of triangles in a random graph cumulant method A good tool for that: mixed cumulants the r -th mixed cumulant k r of r random variables is a r -linear symmetric functional. It is a polynomial in joint moments. Examples: κ 2 ( X , Y ) = Cov ( X , Y ) = E ( XY ) − E ( X ) E ( Y ) κ 1 ( X ) = E ( X ) , κ 3 ( X , Y , Z ) = E ( XYZ ) − E ( XY ) E ( Z ) − E ( XZ ) E ( Y ) − E ( YZ ) E ( X ) + 2 E ( 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) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 3 / 15
Number of triangles in a random graph cumulant method A good tool for that: mixed cumulants the r -th mixed cumulant k r of r random variables is a r -linear symmetric functional. It is a polynomial in joint moments. Examples: κ 2 ( X , Y ) = Cov ( X , Y ) = E ( XY ) − E ( X ) E ( Y ) κ 1 ( X ) = E ( X ) , κ 3 ( X , Y , Z ) = E ( XYZ ) − E ( XY ) E ( Z ) − E ( XZ ) E ( Y ) − E ( YZ ) E ( X ) + 2 E ( 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 κ r ( X n ) := κ r ( X n , . . . , X n ) converges towards 0 and if E ( X n ) and Var ( X n ) have a limit, then X n converges in distribution towards a Gaussian law. V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 3 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ But most of the terms vanish (because the variables are independent). V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ But most of the terms vanish (because the variables are independent). 11 7 3 is indepen- ∆ 7 { ∆ 1 , ∆ 2 , ∆ 5 , ∆ 7 } 1 ∆ 3 5 dent from { ∆ 3 , ∆ 4 , ∆ 6 } . 23 ∆ 5 ∆ 4 ∆ 1 19 ∆ 6 Reminder: different edges be- ∆ 2 13 8 ing in the graph are independent 9 Example: 2 events. κ ℓ ( B ∆ 1 , . . . , B ∆ 7 ) = 0 . V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ But most of the terms vanish (because the variables are independent). 11 7 ∆ 7 3 is indepen- { ∆ 1 , ∆ 2 , ∆ 5 , ∆ 7 } 5 ∆ 3 1 dent from { ∆ 3 , ∆ 4 , ∆ 6 } . ∆ 5 ∆ 1 ∆ 2 23 Triangles need to share an edge 13 ∆ 4 ∆ 6 8 to be dependent! 2 Example: 9 κ ℓ ( B ∆ 1 , . . . , B ∆ 7 ) = 0 . V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ But most of the terms vanish (because the variables are independent). 11 7 ∆ 7 3 This configuration contributes to 5 1 ∆ 3 the sum. Note that it has only ∆ 5 ∆ 1 ∆ 8 23 ∆ 2 13 ℓ + 2 vertices. ∆ 4 ∆ 6 8 2 Example: 9 κ ℓ ( B ∆ 1 , . . . , B ∆ 8 ) � = 0 . V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method Application to the number of triangles � 1 if G contains the triangle ∆ ; � B ∆ , where B ∆ ( G ) = T n = 0 otherwise. ∆= { i , j , k }⊂ [ n ] � By multilinearity, κ ℓ ( T n ) = κ ℓ ( B ∆ 1 , . . . , B ∆ ℓ ) . ∆ 1 ,..., ∆ ℓ But most of the terms vanish (because the variables are independent). 11 7 ∆ 7 3 This configuration contributes to 5 1 ∆ 3 the sum. Note that it has only ∆ 5 ∆ 1 ∆ 8 23 ∆ 2 13 ℓ + 2 vertices. ∆ 4 ∆ 6 8 2 Example: 9 | κ ℓ ( T n ) | = O ℓ ( n ℓ + 2 ) κ ℓ ( B ∆ 1 , . . . , B ∆ 8 ) � = 0 . V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 4 / 15
Number of triangles in a random graph cumulant method The central limit theorem for triangles Proposition (Leonov, Shirryaev, 1955) If X 1 , . . . , X ℓ can be split into two sets of mutually independent variables, then κ ℓ ( X 1 , · · · , X ℓ ) = 0 Corollary (Janson, 1988 ?) For each ℓ , there exists a constant C ℓ such that | κ ℓ ( T n ) | = C ℓ n ℓ + 2 V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 5 / 15
Number of triangles in a random graph cumulant method The central limit theorem for triangles Proposition (Leonov, Shirryaev, 1955) If X 1 , . . . , X ℓ can be split into two sets of mutually independent variables, then κ ℓ ( X 1 , · · · , X ℓ ) = 0 Corollary (Janson, 1988 ?) For each ℓ , there exists a constant C ℓ such that | κ ℓ ( T n ) | = C ℓ n ℓ + 2 Corollary (Ruciński, 1988) T n − E ( T n ) → N ( 0 , 1 ) � Var ( T n ) Proof: Var ( T n ) ≈ n 4 and κ ℓ ( T n / n 2 ) = n 2 − ℓ = o ℓ ( 1 ) for ℓ > 2. V. Féray (with PLM, AN) Cumulants mixtes et arbres couvrants (I-Math, UZH) ALÉA, mars 2014 5 / 15
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