Limit theorems for conditioned non-generic Galton-Watson trees

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Limit theorems for conditioned non-generic Galton-Watson trees

Igor Kortchemski (Université Paris-Sud, Orsay) PIMS 2012

Igor Kortchemski (Université Paris-Sud, Orsay) Condensation in Galton-Watson trees

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State of the art Non-generic trees Limit theorems Extensions

What is this about?

Goal: understand the structure of largeconditionedGalton-Watsontrees.

Typical framework:

- the offspring distribution µis critical (P

i>0iµ(i)=1).

Two approaches: - Scaling limits

- Local limits

What happens when µis not critical?

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What is this about?

Typical framework:

i>0iµ(i)=1).

- µhas finite variance.

- one studies GWµtreesconditionedon having a fixed (large) number of vertices (or edges).

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What is this about?

Typical framework:

i>0iµ(i)=1).

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What is this about?

Typical framework:

i>0iµ(i)=1).

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What is this about?

Typical framework:

i>0iµ(i)=1).

Two approaches:

- Scaling limits

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What is this about?

Typical framework:

i>0iµ(i)=1).

Two approaches:

- Scaling limits - Local limits

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Typical framework:

i>0iµ(i)=1).

Two approaches:

- Scaling limits - Local limits

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I. State of the art (critical case) II. Non-generic trees

III. Limit theorems for non-generic trees IV. One conjecture and one problem

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I. State of the art

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Recap on Galton-Watson trees

Trees will be planar and rooted.

such that:

1. k_∅ has distributionρ, wherek_∅ is the number of children ofthe root. 2. for everyj>1withρ(j)>0, underPρ(·|k_∅=j), the number of children

of thejchildren ofthe rootare independent with distributionρ. Here,k_∅=2.

Here,ζ(τ) =5.

Letζ(τ)denote the total number of vertices ofτ.

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Recap on Galton-Watson trees

Letρ be a probability measure onN={0, 1, 2, . . .}withP

iiρ(i)61and ρ(1)<1.

Here,k_∅=2. Here,ζ(τ) =5.

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Recap on Galton-Watson trees

iiρ(i)61and ρ(1)<1. AGalton-Watson tree with offspring distributionρis a random tree such that:

Here,k_∅=2. Here,ζ(τ) =5.

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Recap on Galton-Watson trees

1. k_∅ has distributionρ, wherek_∅ is the number of children ofthe root.

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Recap on Galton-Watson trees

2. for everyj>1withρ(j)>0, underPρ(·|k_∅=j), the number of children of thejchildren ofthe rootare independent with distributionρ.

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Recap on Galton-Watson trees

Here,k_∅=2.

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Recap on Galton-Watson trees

Here,k_∅=2.

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Here,k_∅=2.

Here,ζ(τ) =5.

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Scaling limits

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Order the vertices in thethe lexicographical order: k_∅=u(0)< u(1)<· · ·< u(ζ(τ) −1).

Letku be the number of children of the vertexu.

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Definition

The Lukasiewicz pathW(τ) = (Wn(τ), 06n6ζ(τ))of a treeτis defined by : W0(τ) =0, Wn+1(τ) =Wn(τ) +k_u(n)(τ) −1.

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Definition

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Definition

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Definition

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Definition

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Definition

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Definition

The Lukasiewicz pathW(τ) = (Wn(τ), 06n6ζ(τ))of a treeτis defined by : W0(τ) =0, Wn+1(τ) =Wn(τ) +ku(n)(τ) −1.

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Definition

The Lukasiewicz pathW(τ) = (Wn(τ), 06n6ζ(τ))of a treeτis defined by : W0(τ) =0, Wn+1(τ) =Wn(τ) +ku(n)(τ) −1.

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Proposition

The Lukasiewicz path of a GWµtree has the same distribution as a random walkwith jump distributionν(k) =µ(k+1),k>−1, started from0, stopped when it hits−1.

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Definition (of the contour function)

A capybara explores the tree at unit speed. For 06t62(ζ(τ) −1),C_t(τ)is the distance between the beast at timetand the root.

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Definition (of the contour function)

A capybara explores the tree at unit speed. For 06t62(ζ(τ) −1),Ct(τ)is the distance between the beast at timetand the root.

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Definition (of the contour function)

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Definition (of the contour function)

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Figure: The Lukasiewicz path and the contour function.

I The Lukasiewicz path behaves like arandom walk.

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Scaling limits

Letµ be a critical offspring distribution with finite variance. Lett_n be a P^µ[·|ζ(τ) =n]tree. What doestn look like fornlarge ?

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Scaling limits

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Scaling limits

Theorem (Aldous ’93, Duquesne ’04)

Letσ² be the variance ofµ. Then :

1

√nW[nt](t_n), 1 2√

nC_2nt(t_n)

06t61

−→(d) n→∞

σ·e(t),1 σe(t)

06t61

,

where eis the normalized Brownian excursion.

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Scaling limits

Theorem (Aldous ’93, Duquesne ’04)

1

√nW[nt](t_n), 1 2√

nC_2nt(t_n)

06t61

−→(d) n→∞

σ·e(t),1 σe(t)

06t61

,

0.2 0.4 0.6 0.8 1

0.

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Scaling limits

Theorem (Aldous ’93, Duquesne ’04)

1

√nW[nt](t_n), 1 2√

nC_2nt(t_n)

06t61

−→(d) n→∞

σ·e(t),1 σe(t)

06t61

,

Remark:

I Duquesne ’04: extension to the case whereµ is in the domain of attraction of a stable law.

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Theorem (Aldous ’93, Duquesne ’04)

1

√nW[nt](t_n), 1 2√

nC_2nt(t_n)

06t61

−→(d) n→∞

σ·e(t),1 σe(t)

06t61

,

Consequences:

- limit theorem for the height oftn,

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Theorem (Aldous ’93, Duquesne ’04)

1

√nW[nt](t_n), 1 2√

nC_2nt(t_n)

06t61

−→(d) n→∞

σ·e(t),1 σe(t)

06t61

,

Consequences:

- limit theorem for the height oftn,

- convergence in the Gromov-Hausdorff sense oftn, suitably rescaled, towards the Brownian CRT.

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II. Non-generic trees

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II. 1) Exponential families

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Exponential families

Letµ be an offspring distribution with0<µ(0)<1.

Lemma (Kennedy ’75)

Letλ >0 be such that

Z_λ=X

i>0

µ(i)λⁱ<∞.

Then a GWµ treeconditionedon having nvertices has the same distribution as a GW_µ(λ) treeconditionedon havingnvertices.

Consequence:

I if there existsλ >0 such thatZ_λ<∞andµ^(λ)is critical, then we are back to the critical case.

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Exponential families

Lemma (Kennedy ’75)

Z_λ=X

i>0

µ(i)λⁱ<∞.

Consequence:

I if there existsλ >0 such thatZ_λ<∞andµ^(λ) is critical, then we are back to the critical case.

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Exponential families

Lemma (Kennedy ’75)

Z_λ=X

i>0

µ(i)λⁱ<∞. Set

µ^(λ)(i)= 1

Z_λµ(i)λⁱ, i>0.

Consequence:

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Exponential families

Lemma (Kennedy ’75)

Z_λ=X

i>0

µ(i)λⁱ<∞. Set

µ^(λ)(i)= 1

Then a GWµtreeconditionedon having nvertices has the same distribution as a GW_µ(λ) treeconditionedon havingnvertices.

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Lemma (Kennedy ’75)

Z_λ=X

i>0

µ(i)λⁱ<∞. Set

µ^(λ)(i)= 1

Then a GWµtreeconditionedon having nvertices has the same distribution as a GW_µ(λ) treeconditionedon havingnvertices.

Consequence:

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Exponential families

Definition

We say that µisnon-genericif there exist noλ >0such that Z_λ<∞and µ^(λ)is critical.

Example:

Example: µ(i)∼c/i^β withc >0andβ >2.

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Exponential families

Definition

Example:

- µis subcritical (P

iiµ(i)<1)

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Exponential families

Definition

Example:

iiµ(i)<1) - and the radius of convergence ofX

i>0

µ(i)zⁱ is1.

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Definition

Example:

iiµ(i)<1) - and the radius of convergence ofX

i>0

µ(i)zⁱ is1.

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II. 2) Large non-generic trees

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Large non-generic trees

Fixµnon-generic. What does aPµ[·|ζ(τ) =n]tree look like for nlarge (Jonsson & Stefánsson 11’)?

Condensationphenomenon

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Large non-generic trees

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Large non-generic trees

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Condensationphenomenon (which also appears in the zero-range process !).

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Theorem (Jonsson & Stefánsson ’11)

Letµ be a subcritical offspring distribution such thatµ(i)∼c/i^β withc >0, β >2.

Lettn be aPµ[·|ζ(τ) =n]tree and mbe the mean ofµ.

1) We havet_n −→^(d)

n→∞ bTwhere bTis:

2) The maximal degree oft_n, divided byn, converges in probability towards 1−m.

Questions:

- Do the Lukasiewicz path and contour function oftn, properly rescaled, converge?

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Theorem (Jonsson & Stefánsson ’11)

Letµ be a subcritical offspring distribution such thatµ(i)∼c/i^β withc >0, β >2. Lettn be aPµ[·|ζ(τ) =n]tree andmbe the mean ofµ.

Questions:

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Theorem (Jonsson & Stefánsson ’11)

1) Then:

tn

−→(d) n→∞ Tb

We havetn

−→(d)

n→∞ Tbwhere bTis:

Questions:

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Theorem (Jonsson & Stefánsson ’11)

1) Then:

tn

−→(d) n→∞ Tb

where the convergence holds for the local convergence

We havetn

−→(d)

2) The maximal degree oftn, divided byn, converges in probability towards 1−m.

Questions:

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Theorem (Jonsson & Stefánsson ’11)

1) Then:

tn

−→(d) n→∞ Tb

where the convergence holds for the local convergence and the treebThas the following form:

We havetn

−→(d)

Questions:

- Do the Lukasiewicz path and contour function oft_n, properly rescaled, converge?

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Theorem (Jonsson & Stefánsson ’11)

1) Then:

tn

−→(d) n→∞ Tb

where the convergence holds for the local convergence and the treebThas the following form:

The spine has a finite random lengthS, where:

P[S=i] = (1−m)mⁱfori>0

We havetn

−→(d)

Questions:

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Theorem (Jonsson & Stefánsson ’11)

n→∞ bTwhereTb is:

Questions:

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Theorem (Jonsson & Stefánsson ’11)

Questions:

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Theorem (Jonsson & Stefánsson ’11)

Remarks:

I In the critical case the spine is infinite (Kesten ’86).

I Janson ’12: Assertion 1) holds for every non-genericµ.

I A GWµ tree has in expectation1+m+m²+· · ·=1/(1−m)vertices. Questions:

- What are the fluctuations of the maximal degree? - Where is located the vertex of maximal degree ? - What is the height oftn?

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Theorem (Jonsson & Stefánsson ’11)

Remarks:

I A GWµ tree has in expectation1+m+m²+· · ·=1/(1−m)vertices. Questions:

- What is the height oftn?

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Theorem (Jonsson & Stefánsson ’11)

Remarks:

I A GWµ tree has in expectation1+m+m²+· · ·=1/(1−m)vertices.

Hence a forest ofcntrees GWµhas in expectationcn/(1−m)vertices. Questions:

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Theorem (Jonsson & Stefánsson ’11)

Remarks:

I A GWµ tree has in expectation1+m+m²+· · ·=1/(1−m)vertices.

Hence a forest ofcntrees GWµhas in expectationcn/(1−m)vertices.

Questions:

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Theorem (Jonsson & Stefánsson ’11)

Questions:

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Theorem (Jonsson & Stefánsson ’11)

Questions:

- What are the fluctuations of the maximal degree?

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Theorem (Jonsson & Stefánsson ’11)

Questions:

- Where is located the vertex of maximal degree ?

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Questions:

- Where is located the vertex of maximal degree ? - What is the height oft_n?

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III. Limit theorems for non-generic trees

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Assumptions

We consider an offspring distribution µsuch that:

- µis subcritical (0<P

iiµ(i)<1)

Lettn be aPµ[·|ζ(τ) =n]tree.

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Assumptions

iiµ(i)<1)

- There exists a slowly varying functionLsuch that µ(n)= L(n)

n^1+θ, n>1 with fixedθ >1.

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Assumptions

iiµ(i)<1)

(Lis slowly varying ifL(tx)/L(x)→1whenx→∞, ∀t >0.)

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iiµ(i)<1)

(Lis slowly varying ifL(tx)/L(x)→1whenx→∞, ∀t >0.) Lettn be aP^µ[·|ζ(τ) =n]tree.

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III. 1) Convergence of the Lukasiewicz path

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Letk_u be the number of children of the vertexu.

Definition

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Convergence of the Lukasiewicz path

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

LetU(tn)be the index of the first vertex with maximal degree oftn.

Theorem (K. 12’)

We have:

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Convergence of the Lukasiewicz path

LetU(t_n)be the index of the first vertex with maximal degree oft_n.

Theorem (K. 12’)

We have:

66 n

(iii)

W[nt]∨(U(t_n)+1)(t_n)

n , 06t61

(d)

n→−→∞ ((1−m)(1−t))₀₆_t₆₁.

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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Convergence of the Lukasiewicz path

Theorem (K. 12’)

We have:

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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Convergence of the Lukasiewicz path

Theorem (K. 12’)

We have:

(i) U(tn)/nconverges in probability towards0 asn→∞.

(iii) W[nt]∨(U(t_n)+1) n

n , 06t61 −→^(d)

n→∞ ((1−m)(1−t))₀₆_t₆₁.

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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Convergence of the Lukasiewicz path

Theorem (K. 12’)

We have:

(i) U(tn)/nconverges in probability towards0 asn→∞. (ii) sup

06i6U(t_n)

Wi(tn) n

(P) n→−→∞ 0.

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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Theorem (K. 12’)

We have:

06i6U(t_n)

Wi(tn) n

(P) n→−→∞ 0.

(iii)

W[nt]∨(U(t_n)+1)(t_n)

n , 06t61

(d)

n→−→∞ ((1−m)(1−t))₀₆_t₆₁.

0 0.2 0.4 0.6 0.8 1

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

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We have:

06i6U(t_n)

Wi(tn) n

(P) n→−→∞ 0.

(iii)

W[nt]∨(U(t_n)+1)(t_n)

n , 06t61

(d)

n→−→∞ ((1−m)(1−t))₀₆_t₆₁. Remarks:

I The limit is deterministic and depends only onm(the mean ofµ).

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Theorem (K. 12’)

We have:

06i6U(t_n)

Wi(tn) n

(P) n→−→∞ 0.

(iii)

W[nt]∨(U(t_n)+1)(t_n)

n , 06t61

(d)

n→−→∞ ((1−m)(1−t))₀₆_t₆₁.

Remarks:

I The limit is deterministic and depends only onm(the mean ofµ).

I With high probability, there is one vertex with degree roughly(1−m)nand the others have degree o(n).

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Idea of the proof

I We know thatW(tn)has the law of arandom walk(Wn)n>0 with jump distributionν(k) =µ(k+1),k>−1

all the other jumps are asymptotically independent (the distribution ofW1

is(0, 1]–subexponential).

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Idea of the proof

I We know thatW(tn)has the law of arandom walk(Wn)n>0 with jump distributionν(k) =µ(k+1),k>−1, conditioned on

W1>0,W2>0, . . . ,Wn−1>0 andWn = −1.

all the other jumps are asymptotically independent (the distribution ofW1

is(0, 1]–subexponential).

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Idea of the proof

W1>0,W2>0, . . . ,Wn−1>0 andWn = −1.

I ButE[W1] =m−1<0.

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Idea of the proof

W1>0,W2>0, . . . ,Wn−1>0 andWn = −1.

I ButE[W1] =m−1<0.

I By the “one big jump principle”,W(t_n)makes one macroscopic jump, and all the other jumps are asymptotically independent

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W1>0,W2>0, . . . ,Wn−1>0 andWn = −1.

I ButE[W1] =m−1<0.

I By the “one big jump principle”,W(t_n)makes one macroscopic jump, and all the other jumps are asymptotically independent (the distribution ofW₁ is(0, 1]–subexponential).

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Applications

Letu_?(t_n)be the vertex of maximal degree

Recall the local convergence oftn to

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Applications

Letu_?(t_n)be the vertex of maximal degree,∆(t_n)its degree

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Applications

Letu_?(t_n)be the vertex of maximal degree,∆(t_n)its degree and|u_?(t_n)|its height.

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Applications

Letu?(t_n)be the vertex of maximal degree,∆(t_n)its degree and|u?(t_n)|its height.

I The fluctuations of∆(t_n)around(1−m)nare of ordern²^∧^θ.

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Applications

I Fori>0,P[|u_?(t_n)|=i] −→

n→∞ (1−m)mⁱ.

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Applications

I Fori>0,P[|u_?(t_n)|=i] −→

n→∞ (1−m)mⁱ.