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Limit Laws for the Number of Groups formed by Social Animals under - - PowerPoint PPT Presentation

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Limit Laws for the Number of Groups formed by Social Animals under the Extra Clustering Model (joint with Michael Drmota and Yi-Wen Lee) Michael Fuchs Institute of Applied Mathematics National Chiao Tung University June 19th, 2014 Michael

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SLIDE 1

Limit Laws for the Number of Groups formed by Social Animals under the Extra Clustering Model

(joint with Michael Drmota and Yi-Wen Lee) Michael Fuchs

Institute of Applied Mathematics National Chiao Tung University

June 19th, 2014

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 1 / 30

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SLIDE 2

Probabilistic Analysis of a Genealogical Model of Animal Group Patterns

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 2 / 30

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SLIDE 3

Phylogenetic Tree

Ordered, binary, rooted tree with leafs representing the animals. Describes the genetic relatedness of animals.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 3 / 30

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SLIDE 4

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 5

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 6

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 7

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 8

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 9

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 10

Yule-Harding Model (Bottom-Up)

Fundamental random model in phylogenetics. Uniformly choose a pair of yellow nodes and let them coalesce.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 4 / 30

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SLIDE 11

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30

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SLIDE 12

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related. − → neutral model.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30

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SLIDE 13

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related. − → neutral model. Clade of a leaf: All leafs of the tree rooted at the parent.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30

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SLIDE 14

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related. − → neutral model. Clade of a leaf: All leafs of the tree rooted at the parent.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 5 / 30

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SLIDE 15

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related. − → neutral model. # of groups

  • # of maximal

clades

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 6 / 30

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SLIDE 16

Animal Groups under the Yule-Harding Model

Durand, Blum and Fran¸ cois (2007): Groups are formed more likely by animals which are genetically related. − → neutral model. # of groups

  • # of maximal

clades

  • 2

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 6 / 30

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SLIDE 17

Yule-Harding Model (Top-Down)

Alternative description of Yule-Harding model:

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30

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SLIDE 18

Yule-Harding Model (Top-Down)

Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30

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SLIDE 19

Yule-Harding Model (Top-Down)

Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30

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SLIDE 20

Yule-Harding Model (Top-Down)

Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30

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SLIDE 21

Yule-Harding Model (Top-Down)

Alternative description of Yule-Harding model: Uniformly choose a yellow node and replace it by a cheery.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 7 / 30

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SLIDE 22

# of Groups

Xn = # of groups under the Yule Harding model

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30

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SLIDE 23

# of Groups

Xn = # of groups under the Yule Harding model We have, Xn

d

=

  • 1,

if In = 1 or In = n − 1, XIn + X∗

n−In,

  • therwise,

where In = Uniform{1, . . . , n − 1} is the # of animals in the left subtree and X∗

n is an independent copy of Xn.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30

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SLIDE 24

# of Groups

Xn = # of groups under the Yule Harding model We have, Xn

d

=

  • 1,

if In = 1 or In = n − 1, XIn + X∗

n−In,

  • therwise,

where In = Uniform{1, . . . , n − 1} is the # of animals in the left subtree and X∗

n is an independent copy of Xn.

Theorem (Durand and Fran¸ cois; 2010) We have, E(Xn) ∼ an

  • a := 1 − e−2

4

  • .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 8 / 30

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SLIDE 25

Comparison with Real-life Data

Durand, Blum and Fran¸ cois (2007) presented the following data:

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 9 / 30

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SLIDE 26

Extra Clustering Model

Durand, Blum and Fran¸ cois (2007): Let p ≥ 0. We have, Xn

d

=

  • 1,

with probability p neutral model,

  • therwise.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30

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SLIDE 27

Extra Clustering Model

Durand, Blum and Fran¸ cois (2007): Let p ≥ 0. We have, Xn

d

=

  • 1,

with probability p neutral model,

  • therwise.

Remark: p = 0 is neutral model.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30

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SLIDE 28

Extra Clustering Model

Durand, Blum and Fran¸ cois (2007): Let p ≥ 0. We have, Xn

d

=

  • 1,

with probability p neutral model,

  • therwise.

Remark: p = 0 is neutral model. Introduced to test whether or not genetic relatedness is the sole driving force behind the group formation process.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 10 / 30

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SLIDE 29

Average Number of Groups

Theorem (Durand and Fran¸ cois; 2010) We have, E(Xn) ∼                  c(p) Γ(2(1 − p))n1−2p, if p < 1/2; log n 2 , if p = 1/2; p 2p − 1, if p > 1/2, where c(p) := 1 e2(1−p) 1 (1 − t)−2pe2(1−p)t 1 − (1 − p)t2 dt.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 11 / 30

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Testing for the Neutral Model

Durand, Blum and Fran¸ cois (2007):

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 12 / 30

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SLIDE 31

Yi-Wen’s Thesis (2012)

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 13 / 30

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SLIDE 32

Variance and SLLN

Theorem (Lee; 2012) We have, Var(Xn) ∼ (1 − e−2)2 4 n log n = 4a2n log n.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 14 / 30

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SLIDE 33

Variance and SLLN

Theorem (Lee; 2012) We have, Var(Xn) ∼ (1 − e−2)2 4 n log n = 4a2n log n. Theorem (Lee; 2012) We have, P

  • lim

n→∞

  • Xn

E(Xn) − 1

  • = 0
  • = 1.

For SLLN, Xn is constructed on the same probability space via the tree evolution process underlying the Yule-Harding model.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 14 / 30

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SLIDE 34

Higher Moments

Theorem (Lee; 2012) For all k ≥ 3, E(Xn − E(Xn))k ∼ (−1)k 2k k − 2aknk−1.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30

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Higher Moments

Theorem (Lee; 2012) For all k ≥ 3, E(Xn − E(Xn))k ∼ (−1)k 2k k − 2aknk−1. This implies that all moments larger than two of Xn − E(Xn)

  • Var(Xn)

tend to infinity!

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30

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SLIDE 36

Higher Moments

Theorem (Lee; 2012) For all k ≥ 3, E(Xn − E(Xn))k ∼ (−1)k 2k k − 2aknk−1. This implies that all moments larger than two of Xn − E(Xn)

  • Var(Xn)

tend to infinity! Question: Is there a limit distribution?

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 15 / 30

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SLIDE 37

Random Recursive Trees

Unordered, rooted trees.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 38

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 39

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 40

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 41

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 42

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 43

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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SLIDE 44

Random Recursive Trees

Unordered, rooted trees. Uniformly choose one of the nodes and attach a child.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 16 / 30

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Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 47

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 48

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 49

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 50

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 51

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒ ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 52

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒ ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 53

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒ ⇒ ⇒

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 54

Cutting Down Random Recursive Trees

Meir and Moon (1974): Randomly pick an edge and remove it; retain the tree containing the root. ⇒ ⇒ ⇒ ⇒ Yn = number of steps until tree is destroyed = number of edges cut = 4.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 17 / 30

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SLIDE 55

Mean, Variance and Higher Moments

Theorem (Panholzer; 2004) We have, E(Yn) ∼ n log n and for k ≥ 2 E(Yn − E(Yn))k ∼ (−1)k k(k − 1) · nk logk+1 n .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 18 / 30

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SLIDE 56

Mean, Variance and Higher Moments

Theorem (Panholzer; 2004) We have, E(Yn) ∼ n log n and for k ≥ 2 E(Yn − E(Yn))k ∼ (−1)k k(k − 1) · nk logk+1 n . Thus, again the limit law of Yn − E(Yn)

  • Var(Yn)

cannot obtained from the method of moments!

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 18 / 30

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SLIDE 57

Limit Law

Theorem (Drmota, Iksanov, Moehle, Roessler; 2009) We have, log2 n n Yn − log n − log log n

d

− → Y with E(eiλY ) = eiλ log |λ|−π|λ|/2. The law of Y is spectrally negative stable with index of stability 1.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 19 / 30

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SLIDE 58

Limit Law

Theorem (Drmota, Iksanov, Moehle, Roessler; 2009) We have, log2 n n Yn − log n − log log n

d

− → Y with E(eiλY ) = eiλ log |λ|−π|λ|/2. The law of Y is spectrally negative stable with index of stability 1. Different proofs of this result exist.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 19 / 30

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SLIDE 59

Limit Law of Xn

Recall that Xn

d

=

  • 1,

if In = 1 or In = n − 1, XIn + X∗

n−In,

  • therwise,

where In = Uniform{1, . . . , n − 1} and X∗

n is an independent copy of Xn.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 20 / 30

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SLIDE 60

Limit Law of Xn

Recall that Xn

d

=

  • 1,

if In = 1 or In = n − 1, XIn + X∗

n−In,

  • therwise,

where In = Uniform{1, . . . , n − 1} and X∗

n is an independent copy of Xn.

Theorem (Drmota, F., Lee; 2014) We have, Xn − E(Xn)

  • Var(Xn)/2

d

− → N(0, 1).

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 20 / 30

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SLIDE 61

Some Ideas of the Proof (i)

Set X(y, z) =

  • n≥2

E

  • eyXn

zn. Then, z ∂ ∂z X(y, z) = X(y, z) + X2(y, z) + eyz2 2eyz3 1 − z . This is a Riccati DE.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 21 / 30

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SLIDE 62

Some Ideas of the Proof (i)

Set X(y, z) =

  • n≥2

E

  • eyXn

zn. Then, z ∂ ∂z X(y, z) = X(y, z) + X2(y, z) + eyz2 2eyz3 1 − z . This is a Riccati DE. Set ˜ X(y, z) = X(y, z) z . Then, ∂ ∂z ˜ X(y, z) = ˜ X2(y, z) + ey 1 + z 1 − z .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 21 / 30

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SLIDE 63

Some Ideas of the Proof (ii)

Set ˜ X(y, z) = −V ′(y, z) V (y, z) . Then, V ′′(y, z) + ey 1 + z 1 − z V (y, z) = 0. This is Whittaker’s DE.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 22 / 30

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SLIDE 64

Some Ideas of the Proof (ii)

Set ˜ X(y, z) = −V ′(y, z) V (y, z) . Then, V ′′(y, z) + ey 1 + z 1 − z V (y, z) = 0. This is Whittaker’s DE. Solution is given by V (y, z) = M−ey/2,1/2

  • 2ey/2(z − 1)
  • + c(y)W−ey/2,1/2
  • 2ey/2(z − 1)
  • ,

where c(y) = −

  • ey/2 − 1
  • M−ey/2+1,1/2
  • −2ey/2

W−ey/2+1,1/2

  • −2ey/2

.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 22 / 30

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SLIDE 65

Some Ideas of the Proof (iii)

Lemma V (y, z) is analytic in ∆ = {z ∈ C : |z| < 1 + δ, arg(z) = π} for all |y| < η. Moreover, V (y, z) has only one (simple) zero with z0(y) = 1 − ay + 2a2y2 log y + O(y2).

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 23 / 30

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SLIDE 66

Some Ideas of the Proof (iii)

Lemma V (y, z) is analytic in ∆ = {z ∈ C : |z| < 1 + δ, arg(z) = π} for all |y| < η. Moreover, V (y, z) has only one (simple) zero with z0(y) = 1 − ay + 2a2y2 log y + O(y2).

δ z0(y) 1

∆

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 23 / 30

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SLIDE 67

Some Ideas of the Proof (iv)

Let y = it/(2a√n log n). Then, E

  • eyXn

= 1 2πi

  • C

X(y, z) zn+1 dz.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 24 / 30

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SLIDE 68

Some Ideas of the Proof (iv)

Let y = it/(2a√n log n). Then, E

  • eyXn

= 1 2πi

  • C

X(y, z) zn+1 dz. Lemma We have, E

  • eyXn

= z0(y)−n + O log3 n n

  • .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 24 / 30

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SLIDE 69

Some Ideas of the Proof (iv)

Let y = it/(2a√n log n). Then, E

  • eyXn

= 1 2πi

  • C

X(y, z) zn+1 dz. Lemma We have, E

  • eyXn

= z0(y)−n + O log3 n n

  • .

This together with the expansion of z0(y) yields E

  • eyXn

= exp it√n 2√log n − t2 4 1 + O log log n log n

  • .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 24 / 30

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SLIDE 70

Extra Clustering Model: 0 < p < 1/2 (i)

Theorem (Drmota, F., Lee; 2014) We have, Xn n1−2p

d

− → X, where the distribution of X is the sum of a discrete distribution with mass p/(1 − p) at 0 and a continuous distribution on [0, ∞) with density f(x) = 4(1 − 2p)3 1 − p

  • k≥0

(−δ(p))k k!Γ(2(k + 1)p − k)xk, where δ(p) = (1 − 2p)2Wp,(1−2p)/p (−2(1 − p)) 4p−1(1 − p)2pMp,(1−2p)/p (−2(1 − p)).

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 25 / 30

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SLIDE 71

Extra Clustering Model: 0 < p < 1/2 (ii)

We have E(Xk) = dk/Γ(k(1 − 2p) + 1) with d1 = 1 e2(1−p) 1 (1 − t)−2pe2(1−p)t 1 − (1 − p)t2 dt and for k ≥ 2 dk = 2(1 − p) (k − 1)(1 − 2p)

k−2

  • j=0

k − 1 j

  • dk−1−jdj+1.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 26 / 30

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SLIDE 72

Extra Clustering Model: 0 < p < 1/2 (ii)

We have E(Xk) = dk/Γ(k(1 − 2p) + 1) with d1 = 1 e2(1−p) 1 (1 − t)−2pe2(1−p)t 1 − (1 − p)t2 dt and for k ≥ 2 dk = 2(1 − p) (k − 1)(1 − 2p)

k−2

  • j=0

k − 1 j

  • dk−1−jdj+1.

Moreover, E

  • eyX

= 1 2πi

  • H

Φ(y, t)e−tdt, where H is the Hankel contour and Φ(y, t) = 4(1 − 2p)2 − ypm(p)4p(1 − p)2p−1t2p−1 4(1 − 2p)2t − ym(p)4p(1 − p)2pt2p and m(p) = Mp,(1−2p)/2(−2(1 − p))/Wp,(1−2p)/2(−2(1 − p)).

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 26 / 30

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SLIDE 73

Extra Clustering Model: p = 1/2

Theorem (Drmota, F., Lee; 2014) We have, E(Xk

n) ∼

k!J2k−1 (2k − 1)!22k−1 log2k−1 n, where J2k−1 are the tangent numbers (or Euler numbers of odd index).

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 27 / 30

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SLIDE 74

Extra Clustering Model: p = 1/2

Theorem (Drmota, F., Lee; 2014) We have, E(Xk

n) ∼

k!J2k−1 (2k − 1)!22k−1 log2k−1 n, where J2k−1 are the tangent numbers (or Euler numbers of odd index). Theorem (Drmota, F., Lee; 2014) We have, Xn

d

− → X, where X is the discrete distribution with E

  • uX

= 1 − √ 1 − u.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 27 / 30

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SLIDE 75

Extra Clustering Model: 1/2 < p < 1

Theorem (Drmota, F., Lee; 2014) We have, Xn

d

− → X, where X is the discrete distribution with E

  • uX

= 1 −

  • 1 − 4p(1 − p)u

2(1 − p) .

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 28 / 30

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SLIDE 76

Extra Clustering Model: 1/2 < p < 1

Theorem (Drmota, F., Lee; 2014) We have, Xn

d

− → X, where X is the discrete distribution with E

  • uX

= 1 −

  • 1 − 4p(1 − p)u

2(1 − p) . For the moments, we have E(Xk) = ek with e1 = p/(2p − 1) and for k ≥ 2 ek = 2(1 − p) 2p − 1

k−2

  • j=0

k − 1 j

  • ek−1−jej+1 +

p 2p − 1.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 28 / 30

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SLIDE 77

Summary

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 29 / 30

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SLIDE 78

Summary

Complete analysis of the extra clustering model.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 29 / 30

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SLIDE 79

Summary

Complete analysis of the extra clustering model. Surprising central limit law for p = 0.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 29 / 30

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SLIDE 80

Summary

Complete analysis of the extra clustering model. Surprising central limit law for p = 0. Only for 0 < p < 1/2 and 1/2 < p ≤ 1 the method of moments applies.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 29 / 30

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SLIDE 81

Summary

Complete analysis of the extra clustering model. Surprising central limit law for p = 0. Only for 0 < p < 1/2 and 1/2 < p ≤ 1 the method of moments applies. Analytic proof in all cases via singularity perturbation theory.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 29 / 30

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SLIDE 82

Open Problems

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 30 / 30

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SLIDE 83

Open Problems

Better explanation of the curious central limit law.

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 30 / 30

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SLIDE 84

Open Problems

Better explanation of the curious central limit law. Probabilistic proof?

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 30 / 30

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SLIDE 85

Open Problems

Better explanation of the curious central limit law. Probabilistic proof? Similar curious central limit law for the total length of external branches in Kingman’s coalescent: Janson and Kersting (2011). On the total external length of the Kingman coalescent, Electronic J. Probability, 16, 2203-2218. Any relationship?

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 30 / 30

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SLIDE 86

Open Problems

Better explanation of the curious central limit law. Probabilistic proof? Similar curious central limit law for the total length of external branches in Kingman’s coalescent: Janson and Kersting (2011). On the total external length of the Kingman coalescent, Electronic J. Probability, 16, 2203-2218. Any relationship? How about limit laws for Xn for other random tree models?

Michael Fuchs (NCTU) Animal Group Patterns June 19th, 2014 30 / 30