Random permutations with logarithmic cycle Random permutations - - PowerPoint PPT Presentation

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Random permutations with logarithmic cycle weights

Dirk Zeindler (joint with Nicolas Robles)

Lancaster University

30 January 2020

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Setting the scene

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

1 2 3 4 5 6 7 8 9 10 All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6 All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ. This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2
• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

3 7 5

• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

3 7 5 4

• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

3 7 5 4 8

• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

3 7 5 4 8

• Two cycles (s0 . . . sk−1) and (t0 . . . tm−1) are called disjoint

if the sets {s0, . . . , sk−1} and {t0, . . . , tm−1} are disjoint.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Permutations and their cycle structure

π = 1 2 3 4 5 6 7 8 9 10 9 1 7 4 3 2 5 8 10 6

• Consider {1, . . . , n} and denote by Sn the symmetric group.

π =

• 1 9 10 6 2

3 7 5 4 8

• All σ ∈ Sn decompose into disjoint cycles: σ = σ1 σ2 . . . σℓ.

This is a text you cannot see

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Cycle-type

If σ ∈ Sn is given, then it can be written as σ = σ1σ2 · · · σℓ where σ1, . . . , σℓ are disjoint cycles.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Cycle-type

If σ ∈ Sn is given, then it can be written as σ = σ1σ2 · · · σℓ where σ1, . . . , σℓ are disjoint cycles. We write λj for the length of the cycle σj.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Cycle-type

If σ ∈ Sn is given, then it can be written as σ = σ1σ2 · · · σℓ where σ1, . . . , σℓ are disjoint cycles. We write λj for the length of the cycle σj. W.l.o.g. we can assume λ1 ≥ λ2 ≥ · · · ≥ λℓ.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Cycle-type

If σ ∈ Sn is given, then it can be written as σ = σ1σ2 · · · σℓ where σ1, . . . , σℓ are disjoint cycles. We write λj for the length of the cycle σj. W.l.o.g. we can assume λ1 ≥ λ2 ≥ · · · ≥ λℓ. The partition λ = (λ1, . . . , λℓ) is called the cycle-type of σ.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Random permutations on Sn

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations:

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability. ◮ Ewens measure:

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability. ◮ Ewens measure: Let ϑ > 0.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability. ◮ Ewens measure: Let ϑ > 0. We write σ ∈ Sn as σ = σ1σ2 · · · σℓ with σ1, . . . , σℓ disjoint cycles.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability. ◮ Ewens measure: Let ϑ > 0. We write σ ∈ Sn as σ = σ1σ2 · · · σℓ with σ1, . . . , σℓ disjoint cycles. The Ewens measure is then defined as Pϑ[σ] := ϑℓ hnn!

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Classical measures on Sn

◮ Uniform permutations: P[σ] = 1 n! for all σ ∈ Sn. Studied since 1940, many applications in probability. ◮ Ewens measure: Let ϑ > 0. We write σ ∈ Sn as σ = σ1σ2 · · · σℓ with σ1, . . . , σℓ disjoint cycles. The Ewens measure is then defined as Pϑ[σ] := ϑℓ ϑ(ϑ + 1) · · · (ϑ + n − 1)

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance ◮ It has a connection with Kingman’s coalescent process (1982).

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance ◮ It has a connection with Kingman’s coalescent process (1982). ◮ It has been used to model the dynamics of tumour

• evolution. (Barbour and Tavar´

e (2010))

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance ◮ It has a connection with Kingman’s coalescent process (1982). ◮ It has been used to model the dynamics of tumour

• evolution. (Barbour and Tavar´

e (2010)) ◮ It appears in a Bayesian non parametric statistics

• stetting. (Antoniak (1974))

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance ◮ It has a connection with Kingman’s coalescent process (1982). ◮ It has been used to model the dynamics of tumour

• evolution. (Barbour and Tavar´

e (2010)) ◮ It appears in a Bayesian non parametric statistics

• stetting. (Antoniak (1974))

◮ It plays a crucial role for virtual permutations since it is central and stable under the restriction Sn → Sn−1

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Ewens measure

The Ewens measure was introduced by Ewens (1972) in population genetics. But it has various applications, for instance ◮ It has a connection with Kingman’s coalescent process (1982). ◮ It has been used to model the dynamics of tumour

• evolution. (Barbour and Tavar´

e (2010)) ◮ It appears in a Bayesian non parametric statistics

• stetting. (Antoniak (1974))

◮ It plays a crucial role for virtual permutations since it is central and stable under the restriction Sn → Sn−1 ◮ .......

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. ◮ Ewens measure: Pϑ[σ] = ϑℓ hnn!

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. ◮ Ewens measure: Pϑ[σ] = ϑℓ hnn!

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. ◮ Ewens measure: Pϑ[σ] = ϑℓ hnn! = 1 hnn!

ℓ

• j=1

ϑ, ϑ > 0.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. ◮ Ewens measure: Pϑ[σ] = ϑℓ hnn! = 1 hnn!

ℓ

• j=1

ϑ, ϑ > 0. ◮ Let Θ := (θm)m≥1 non-negative weights.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. ◮ Ewens measure: Pϑ[σ] = ϑℓ hnn! = 1 hnn!

ℓ

• j=1

ϑ, ϑ > 0. ◮ Let Θ := (θm)m≥1 non-negative weights. The weighted measure is then defined as PΘ[σ] := 1 hnn!

ℓ

• i=1

θλi

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are

9 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are ◮ θm ≡ 1: Uniform measure,

9 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are ◮ θm ≡ 1: Uniform measure, ◮ θm ≡ ϑ: Ewens measure,

9 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are ◮ θm ≡ 1: Uniform measure, ◮ θm ≡ ϑ: Ewens measure, ◮ ’θm → ϑ’: Generalised Ewens measure.

9 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are ◮ θm ≡ 1: Uniform measure, ◮ θm ≡ ϑ: Ewens measure, ◮ ’θm → ϑ’: Generalised Ewens measure. ◮ θm = mγ with γ > 0.

9 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The weighted measure

Typical weights (θm)m≥1 studied in the literature are ◮ θm ≡ 1: Uniform measure, ◮ θm ≡ ϑ: Ewens measure, ◮ ’θm → ϑ’: Generalised Ewens measure. ◮ θm = mγ with γ > 0. We study in this talk weights of the form θm = logk(m), with k ≥ 1 fix.

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Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

.

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞.

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study?

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study? ◮ Cm: the cycle counts,

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study? ◮ Cm: the cycle counts, ◮ ℓ1: the length of the cycle containing 1,

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study? ◮ Cm: the cycle counts, ◮ ℓ1: the length of the cycle containing 1, ◮ Kn: the total number of cycles (Kn = n

m=1 Cm),

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study? ◮ Cm: the cycle counts, ◮ ℓ1: the length of the cycle containing 1, ◮ Kn: the total number of cycles (Kn = n

m=1 Cm),

◮ λ1: the length of the longest cycle.

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Interesting random variables

PΘ[σ] := 1 hnn!

n

• m=1

θCm(σ)

m

. We now use this measure on Sn and let n → ∞. What are interesting random variables to study? ◮ Cm: the cycle counts, ◮ ℓ1: the length of the cycle containing 1, ◮ Kn: the total number of cycles (Kn = n

m=1 Cm),

◮ λ1: the length of the longest cycle. ◮ (λ1, λ2, . . .): the length of the longest cycles.

10 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi.

11 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm

Let σ = σ1σ2 · · · σℓ ∈ Sn be given with σi disjoint cycles and cycle lengths λi. We then define Cm := #{i; λi = m}.

11 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

We have for the uniform and Ewens measure is

12 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

We have for the uniform and Ewens measure is

Theorem (Shepp,Loyd ϑ = 1, Watterson general ϑ)

We have for each b > 0 (C1, . . . , Cb)

d

− → (Y1, . . . , Yb) with Ym independent Poisson distributed with E [Ym] = θm m = ϑ m.

12 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

We have for the uniform and Ewens measure is

Theorem (Shepp,Loyd ϑ = 1, Watterson general ϑ)

We have for each b > 0 (C1, . . . , Cb)

d

− → (Y1, . . . , Yb) with Ym independent Poisson distributed with E [Ym] = θm m = ϑ m. This result can be improved further.

12 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

Let Ω be a countable set and P and P be two probability measures on Ω.

13 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

Let Ω be a countable set and P and P be two probability measures on Ω. The total variation distance between P and P is defined as dTV(P, P) = 1 2

• ω∈Ω

|P(ω) − P(ω)|.

13 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = ϑ

For instance

Theorem (Arratia and Tavar´ e)

Let Sn be endowed with the uniform measure. Then dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

if and only if b = o(n), where dTV denotes the total variation distance.

14 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = mγ

We have for the cycle weights θm = mγ with γ > 0

15 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = mγ

We have for the cycle weights θm = mγ with γ > 0

Theorem (Ercolani and Ueltschi)

We have for each b > 0 (C1, . . . , Cb)

d

− → (Y1, . . . , Yb) with Ym independent Poisson distributed with E [Ym] = θm m = mγ−1.

15 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = mγ

As for the Ewens measure, we can also compute the total variation distance.

16 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = mγ

As for the Ewens measure, we can also compute the total variation distance.

Theorem (Storm and Zeindler)

We then have as n → ∞ dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

iff b = o(n

1 1+γ ),

where Ym are independent Poisson distributed with E [Ym] = θm

m = mγ−1.

16 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = logk(m)

We have for the cycle weights θm = logk(m) with k ≥ 1

17 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = logk(m)

We have for the cycle weights θm = logk(m) with k ≥ 1

Theorem (Robles and Zeindler)

We have for each b > 0 (C1, . . . , Cb)

d

− → (Y1, . . . , Yb) with Ym independent Poisson distributed with E [Ym] = θm m = logk(m) m .

17 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = logk(m)

Theorem (Robles and Zeindler)

Suppose that b(n) = o

• nc

with 0 < c < (3k + 3)−

1 k+1 . We

then have dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

18 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle counts Cm under θm = logk(m)

Theorem (Robles and Zeindler)

Suppose that b(n) = o

• nc

with 0 < c < (3k + 3)−

1 k+1 . We

then have dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

We expect that that the dTV in this theorem goes to 0 if and only if b(n) = o

• n

logk(n)

• .

18 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

We define ℓ1 := the length of the cycle containing 1.

19 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

We define ℓ1 := the length of the cycle containing 1. A simple computation gives that we have under the uniform measure on Sn Pn [ℓ1 = k] = 1 n for all 1 ≤ k ≤ n.

19 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1 and the uniform measure

We thus immediately get

Theorem

We have under the uniform measure ℓ1 n

d

− → U(0, 1), where U(0, 1) is the uniform measure on the interval [0, 1].

20 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1 and the Ewens measure

Furthermore

Theorem

We have under the Ewens measure with parameter ϑ > 0 ℓ1 n

d

− → Beta(1, ϑ). where Beta(1, ϑ) is the probability measure on the interval [0, 1] with density (1 − x)ϑ−1.

21 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1 under θm = mγ

Theorem (Ercolani, Ueltschi)

We have in the case θm = mγ ℓ1 n

1 1+γ

d

− → Γ

• 1 + γ, Γ(1 + γ)

1 1+γ

• where Γ (α, β) is the probability measure on [0, ∞[ with

density

βα Γ(α)xαe−βx and Γ(s) the Gamma function.

22 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1 under θm = logk(m)

Theorem (Robles, Zeindler)

We have in the case θm = θm = logk(m) ℓ1

n logk(n) d

− → Exp(1), where Exp(λ) is the probability measure on [0, ∞[ with density λe−λx.

23 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

θm = ϑ = mγ = logk(m) dTV → 0 ℓ1

1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

θm = ϑ = mγ = logk(m) dTV → 0 b = o(n) ℓ1 ≍ n

1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

θm = ϑ = mγ = logk(m) dTV → 0 b = o(n) b = o

• n

1 1+γ

• ℓ1

≍ n ≍ n

1 1+γ 1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

θm = ϑ = mγ = logk(m) dTV → 0 b = o(n) b = o

• n

1 1+γ

• b = o
• n

logk(n)

• 1

ℓ1 ≍ n ≍ n

1 1+γ 1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The cycle containing 1

dTV

• (C1, . . . , Cb), (Y1, . . . , Yb)
• → 0

θm = ϑ = mγ = logk(m) dTV → 0 b = o(n) b = o

• n

1 1+γ

• b = o
• n

logk(n)

• 1

ℓ1 ≍ n ≍ n

1 1+γ

≍ n/ logk(n)

1conjectural 24 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles

The Total number of cycles is defined as Kn := C1 + · · · + Cn.

25 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles

The Total number of cycles is defined as Kn := C1 + · · · + Cn. We have under the uniform and the Ewens measure.

25 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles

The Total number of cycles is defined as Kn := C1 + · · · + Cn. We have under the uniform and the Ewens measure.

Theorem (Goncharov ϑ = 1, Watterson general ϑ)

Kn − ϑ log(n)

• ϑ log(n)

d

− → N(0, 1)

25 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles under ’θm → ϑ’

Theorem (Nikeghbali and Zeindler)

Consider the general Ewens measure. We then have EΘ

• exp
• sKn
• = nθ(es−1)

Γ(θ) Γ(θes) + O 1 n

• with O(.) uniform for bounded s ∈ C.

26 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles under ’θm → ϑ’

Recall, the Kolmogorov distance between two integer valued random variables X and Y is defined as dK(X, Y ) := sup

j∈Z

• Pn [X ≤ j] − Pn [Y ≤ j]
• 27 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles under ’θm → ϑ’

Recall, the Kolmogorov distance between two integer valued random variables X and Y is defined as dK(X, Y ) := sup

j∈Z

• Pn [X ≤ j] − Pn [Y ≤ j]
• Corollary

Let Pθ log(n) be a Poisson distributed random variable with parameter θ log(n). Then dK(Kn, Pθ log(n)) = O

• 1
• log(n)
• .

27 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles under θm = mγ

Theorem (Maples, Nikeghbali, Zeindler)

We have in the case θm = mγ Kn − cγn

γ 1+γ

• dγn

γ 1+γ

d

− → N(0, 1)

28 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Total Number of Cycles under θm = logk(m)

Theorem (Robles, Zeindler)

We have for θm = logk(m) Kn − logk+1(n)

k+1

• logk+1(n)

k+1 d

− → N(0, 1)

29 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles

Let σ = σ1σ2 · · · σℓ. We write λj for the length of the cycle σj.

30 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles

Let σ = σ1σ2 · · · σℓ. We write λj for the length of the cycle σj. W.l.o.g. we can assume λ1 ≥ λ2 ≥ . . . .

30 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles

Let σ = σ1σ2 · · · σℓ. We write λj for the length of the cycle σj. W.l.o.g. we can assume λ1 ≥ λ2 ≥ . . . .

Theorem (Vershik and Shmidt resp. Kingman )

λ1 n , λ2 n , . . .

• d

− → PD(ϑ), (n → ∞) with PD(ϑ) the Poisson–Dirichlet distribution with parameter ϑ.

30 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution?

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering.

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ).

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ). Consider a stick of length 1

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ). Consider a stick of length 1

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ). Consider a stick of length 1

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ). Consider a stick of length 1

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Poisson–Dirichlet distribution

What is the Poisson–Dirichlet distribution? Keyword: stick breaking process with size ordering. Let (Bk)k∈N be iid Beta distributed with parameters (1, ϑ). Consider a stick of length 1 Ordering the sticks obtained by this process by size then has a Poisson–Dirichlet distribution.

31 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Let us now consider the case θm = mγ.

32 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Let us now consider the case θm = mγ. Do have in this case also cycles of order n?

32 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Let us now consider the case θm = mγ. Do have in this case also cycles of order n? No!

32 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Let us now consider the case θm = mγ. Do have in this case also cycles of order n? No!

32 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Define n∗ := n

1 1+γ

and ℓ := γ log(n∗) + (γ − 1) log (γ log(n∗)) .

33 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

Define n∗ := n

1 1+γ

and ℓ := γ log(n∗) + (γ − 1) log (γ log(n∗)) .

Theorem (Zeindler)

Let K ∈ N be given. We have convergence in distribution of 1 n∗ ·

• L1 − n∗ℓ, . . . ,

LK − n∗ℓ

• d

− →  − log(E1), . . . , − log  

K

• j=1

Ej     . as n → ∞, where (Ej)K

j=1 is a sequence of iid Exp(1)

distributed random variables.

33 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

There is an important difference between the cases θm = ϑ and θm = mγ.

34 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

There is an important difference between the cases θm = ϑ and θm = mγ. Let us consider 1 n

• m≥d

mCm

34 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

There is an important difference between the cases θm = ϑ and θm = mγ. Let us consider 1 n

• m≥d

mCm This can be interpreted as the fraction of indices in cycles of length at least d.

34 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

We have in the case θm = ϑ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn

mCm   = 1

35 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

We have in the case θm = ϑ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn

mCm   = 1 and in the case θm = mγ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn∗ℓn

mCm   = 0.

35 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

We have in the case θm = ϑ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn

mCm   = 1 and in the case θm = mγ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn∗ℓn

mCm   = 0. Recall, we have E [ℓ1] ≈ n

1 1+γ . 35 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Long Cycles under θm = mγ

We have in the case θm = ϑ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn

mCm   = 1 and in the case θm = mγ lim

ǫ→0 lim n→∞ E

 1 n

• m≥ǫn∗ℓn

mCm   = 0. Recall, we have E [ℓ1] ≈ n

1 1+γ . Thus it is natural to study

also the cycles the region n

1 1+γ . 35 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Young diagram

σ = (128)(3)(4579)(6) ∈ S9 ⇒ λ = (4, 3, 1, 1)

36 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Young diagram

σ = (128)(3)(4579)(6) ∈ S9 ⇒ λ = (4, 3, 1, 1)

36 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Young diagram

σ = (128)(3)(4579)(6) ∈ S9 ⇒ λ = (4, 3, 1, 1)

36 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Young diagram

σ = (128)(3)(4579)(6) ∈ S9 ⇒ λ = (4, 3, 1, 1) =: wn(x)

36 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Formula for wn(x)

We then obtain wn(x) =

n

• m≥x

Cm

37 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape

38 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape

What happens if we take a σ randomly?

38 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape

What happens if we take a σ randomly?

38 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape

What happens if we take a σ randomly?

38 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

The limit shape is a function ω(·) s.t. for all ǫ, δ > 0 lim

n→+∞ Pn

• sup

x≥δ

| ˜ wn(x) − ω(x)| ≤ ǫ

• = 1

˜ wn is an appropriate rescaling of wn.

39 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = mγ

Recall, we have ℓ1 ≍ n

1 1+γ . 40 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = mγ

Recall, we have ℓ1 ≍ n

1 1+γ .

Thus it is natural to use the scale wn as

• wn(x) := n−

γ 1+γ wn

• xn

1 1+γ

• 40 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = mγ

Recall, we have ℓ1 ≍ n

1 1+γ .

Thus it is natural to use the scale wn as

• wn(x) := n−

γ 1+γ wn

• xn

1 1+γ

• Proposition (Erlihson, Granovsky (2008))

The limit shape exist and is given by ω(x) := Γ(γ, x) Γ(γ + 1), where Γ( · , · ) is the incomplete Γ-function.

40 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = mγ

Recall, we have ℓ1 ≍ n

1 1+γ .

Thus it is natural to use the scale wn as

• wn(x) := n−

γ 1+γ wn

• xn

1 1+γ

• Proposition (Erlihson, Granovsky (2008))

The limit shape exist and is given by ω(x) := Γ(γ, x) Γ(γ + 1), where Γ( · , · ) is the incomplete Γ-function. For γ = 1, we have ω(x) = e−x

40 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = logk(m)

41 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = logk(m)

Recall, we have ℓ1 ≍

n logk(n).

41 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = logk(m)

Recall, we have ℓ1 ≍

n logk(n).

Thus it is natural to use the scale wn as

• wn(x) := log−k(n)wn
• x

n logk(n)

• 41 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = logk(m)

Recall, we have ℓ1 ≍

n logk(n).

Thus it is natural to use the scale wn as

• wn(x) := log−k(n)wn
• x

n logk(n)

• Theorem (Robles, Zeindler (2018))

Let k ≥ 3.

41 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shape under θm = logk(m)

Recall, we have ℓ1 ≍

n logk(n).

Thus it is natural to use the scale wn as

• wn(x) := log−k(n)wn
• x

n logk(n)

• Theorem (Robles, Zeindler (2018))

Let k ≥ 3. Then the limit shape exists and is given by w∞(x) = ∞

x

u−1e−u du. (1)

41 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Limit shapes

(a) θm = mγ (b) θm = logk(m) Figure: Plots of limit shapes

42 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

• 43 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Fluctuations under θm = mγ

We define

• wn(x) :=

wn

• x n

1 1+γ

• − n

γ 1+γ ω(x)

• n

γ 1+γ 44 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Fluctuations under θm = mγ

We define

• wn(x) :=

wn

• x n

1 1+γ

• − n

γ 1+γ ω(x)

• n

γ 1+γ

Theorem (Erlihson, Granovsky (2008))

We have as n → ∞

• wn(x) → N
• 0, ω(x) −

Γ(γ + 1, x)2 2Γ(γ + 1)Γ(γ + 2)

• .

44 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Fluctuations under θm = logk(m)

We define

• wn(x) :=

wn

• x

n logk(n)

• − logk(n) ω(x)
• logk(n)

45 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Fluctuations under θm = logk(m)

We define

• wn(x) :=

wn

• x

n logk(n)

• − logk(n) ω(x)
• logk(n)

Theorem (Robles, Zeindler (2018))

We have for k ≥ 3 and as n → ∞

• wn(x) → N
• 0, ω(x) + e−2x

.

45 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations

Classical measures The weighted measure

The cycle counts Cm The cycle containing 1 Total Number of Cycles Long Cycles

Case θm = ϑ Case θm = mγ

Limit Shape

Thank you!

46 / 46