Random permutations with logarithmic cycle Random permutations - PowerPoint PPT Presentation

Cycle-type Random permutations with logarithmic cycle weights Setting the scene If σ ∈ S n is given, then it can be written as Random permutations Classical measures σ = σ 1 σ 2 · · · σ ℓ The weighted measure The cycle counts C m where σ 1 , . . . , σ ℓ are disjoint cycles. The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 4 / 46

Cycle-type Random permutations with logarithmic cycle weights Setting the scene If σ ∈ S n is given, then it can be written as Random permutations Classical measures σ = σ 1 σ 2 · · · σ ℓ The weighted measure The cycle counts C m where σ 1 , . . . , σ ℓ are disjoint cycles. The cycle containing 1 Total Number of We write λ j for the length of the cycle σ j . Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 4 / 46

Cycle-type Random permutations with logarithmic cycle weights Setting the scene If σ ∈ S n is given, then it can be written as Random permutations Classical measures σ = σ 1 σ 2 · · · σ ℓ The weighted measure The cycle counts C m where σ 1 , . . . , σ ℓ are disjoint cycles. The cycle containing 1 Total Number of We write λ j for the length of the cycle σ j . Cycles Long Cycles W.l.o.g. we can assume λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ . Case θ m = ϑ Case θ m = m γ Limit Shape 4 / 46

Cycle-type Random permutations with logarithmic cycle weights Setting the scene If σ ∈ S n is given, then it can be written as Random permutations Classical measures σ = σ 1 σ 2 · · · σ ℓ The weighted measure The cycle counts C m where σ 1 , . . . , σ ℓ are disjoint cycles. The cycle containing 1 Total Number of We write λ j for the length of the cycle σ j . Cycles Long Cycles W.l.o.g. we can assume λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ . Case θ m = ϑ Case θ m = m γ Limit Shape The partition λ = ( λ 1 , . . . , λ ℓ ) is called the cycle-type of σ . 4 / 46

Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures The weighted measure The cycle counts Random permutations on S n C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 5 / 46

Classical measures on S n Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle ◮ Ewens measure: containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle ◮ Ewens measure: containing 1 Let ϑ > 0. Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle ◮ Ewens measure: containing 1 Let ϑ > 0. We write σ ∈ S n as σ = σ 1 σ 2 · · · σ ℓ with Total Number of Cycles σ 1 , . . . , σ ℓ disjoint cycles. Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle ◮ Ewens measure: containing 1 Let ϑ > 0. We write σ ∈ S n as σ = σ 1 σ 2 · · · σ ℓ with Total Number of Cycles σ 1 , . . . , σ ℓ disjoint cycles. The Ewens measure is then Long Cycles defined as Case θ m = ϑ Case θ m = m γ ϑ ℓ Limit Shape P ϑ [ σ ] := h n n ! 6 / 46

Classical measures on S n Random permutations with logarithmic cycle weights ◮ Uniform permutations: Setting the scene Random permutations P [ σ ] = 1 n ! for all σ ∈ S n . Classical measures The weighted measure The cycle counts C m Studied since 1940, many applications in probability. The cycle ◮ Ewens measure: containing 1 Let ϑ > 0. We write σ ∈ S n as σ = σ 1 σ 2 · · · σ ℓ with Total Number of Cycles σ 1 , . . . , σ ℓ disjoint cycles. The Ewens measure is then Long Cycles defined as Case θ m = ϑ Case θ m = m γ ϑ ℓ Limit Shape P ϑ [ σ ] := ϑ ( ϑ + 1) · · · ( ϑ + n − 1) 6 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure ◮ It has a connection with Kingman’s coalescent process The cycle counts C m (1982). The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure ◮ It has a connection with Kingman’s coalescent process The cycle counts C m (1982). The cycle ◮ It has been used to model the dynamics of tumour containing 1 Total Number of evolution. (Barbour and Tavar´ e (2010)) Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure ◮ It has a connection with Kingman’s coalescent process The cycle counts C m (1982). The cycle ◮ It has been used to model the dynamics of tumour containing 1 Total Number of evolution. (Barbour and Tavar´ e (2010)) Cycles ◮ It appears in a Bayesian non parametric statistics Long Cycles Case θ m = ϑ stetting. (Antoniak (1974)) Case θ m = m γ Limit Shape 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure ◮ It has a connection with Kingman’s coalescent process The cycle counts C m (1982). The cycle ◮ It has been used to model the dynamics of tumour containing 1 Total Number of evolution. (Barbour and Tavar´ e (2010)) Cycles ◮ It appears in a Bayesian non parametric statistics Long Cycles Case θ m = ϑ stetting. (Antoniak (1974)) Case θ m = m γ ◮ It plays a crucial role for virtual permutations since it is Limit Shape central and stable under the restriction S n → S n − 1 7 / 46

Ewens measure Random permutations with logarithmic cycle weights The Ewens measure was introduced by Ewens (1972) in Setting the scene population genetics. Random permutations But it has various applications, for instance Classical measures The weighted measure ◮ It has a connection with Kingman’s coalescent process The cycle counts C m (1982). The cycle ◮ It has been used to model the dynamics of tumour containing 1 Total Number of evolution. (Barbour and Tavar´ e (2010)) Cycles ◮ It appears in a Bayesian non parametric statistics Long Cycles Case θ m = ϑ stetting. (Antoniak (1974)) Case θ m = m γ ◮ It plays a crucial role for virtual permutations since it is Limit Shape central and stable under the restriction S n → S n − 1 ◮ ....... 7 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations ◮ Ewens measure: Classical measures The weighted measure ϑ ℓ The cycle counts C m P ϑ [ σ ] = h n n ! The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations ◮ Ewens measure: Classical measures The weighted measure ϑ ℓ The cycle counts C m P ϑ [ σ ] = h n n ! The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations ◮ Ewens measure: Classical measures The weighted measure ℓ � ϑ ℓ The cycle counts 1 C m P ϑ [ σ ] = h n n ! = ϑ, ϑ > 0 . h n n ! The cycle j =1 containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations ◮ Ewens measure: Classical measures The weighted measure ℓ � ϑ ℓ The cycle counts 1 C m P ϑ [ σ ] = h n n ! = ϑ, ϑ > 0 . h n n ! The cycle j =1 containing 1 Total Number of Cycles ◮ Let Θ := ( θ m ) m ≥ 1 non-negative weights. Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and Setting the scene cycle lengths λ i . Random permutations ◮ Ewens measure: Classical measures The weighted measure ℓ � ϑ ℓ The cycle counts 1 C m P ϑ [ σ ] = h n n ! = ϑ, ϑ > 0 . h n n ! The cycle j =1 containing 1 Total Number of Cycles ◮ Let Θ := ( θ m ) m ≥ 1 non-negative weights. The weighted Long Cycles measure is then defined as Case θ m = ϑ Case θ m = m γ Limit Shape ℓ � 1 P Θ [ σ ] := θ λ i h n n ! i =1 8 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 9 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations ◮ θ m ≡ 1: Uniform measure, Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 9 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations ◮ θ m ≡ 1: Uniform measure, Classical measures The weighted measure ◮ θ m ≡ ϑ : Ewens measure, The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 9 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations ◮ θ m ≡ 1: Uniform measure, Classical measures The weighted measure ◮ θ m ≡ ϑ : Ewens measure, The cycle counts C m ◮ ’ θ m → ϑ ’: Generalised Ewens measure. The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 9 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations ◮ θ m ≡ 1: Uniform measure, Classical measures The weighted measure ◮ θ m ≡ ϑ : Ewens measure, The cycle counts C m ◮ ’ θ m → ϑ ’: Generalised Ewens measure. The cycle containing 1 ◮ θ m = m γ with γ > 0. Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 9 / 46

The weighted measure Random permutations with logarithmic cycle weights Setting the scene Typical weights ( θ m ) m ≥ 1 studied in the literature are Random permutations ◮ θ m ≡ 1: Uniform measure, Classical measures The weighted measure ◮ θ m ≡ ϑ : Ewens measure, The cycle counts C m ◮ ’ θ m → ϑ ’: Generalised Ewens measure. The cycle containing 1 ◮ θ m = m γ with γ > 0. Total Number of Cycles We study in this talk weights of the form Long Cycles Case θ m = ϑ θ m = log k ( m ) , with k ≥ 1 fix. Case θ m = m γ Limit Shape 9 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of ◮ C m : the cycle counts, Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of ◮ C m : the cycle counts, Cycles ◮ ℓ 1 : the length of the cycle containing 1, Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of ◮ C m : the cycle counts, Cycles ◮ ℓ 1 : the length of the cycle containing 1, Long Cycles ◮ K n : the total number of cycles ( K n = � n Case θ m = ϑ Case θ m = m γ m =1 C m ), Limit Shape 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of ◮ C m : the cycle counts, Cycles ◮ ℓ 1 : the length of the cycle containing 1, Long Cycles ◮ K n : the total number of cycles ( K n = � n Case θ m = ϑ Case θ m = m γ m =1 C m ), Limit Shape ◮ λ 1 : the length of the longest cycle. 10 / 46

Interesting random variables Random permutations with logarithmic cycle weights Setting the scene n � Random 1 θ C m ( σ ) permutations P Θ [ σ ] := . m Classical measures h n n ! The weighted measure m =1 The cycle counts C m We now use this measure on S n and let n → ∞ . The cycle What are interesting random variables to study? containing 1 Total Number of ◮ C m : the cycle counts, Cycles ◮ ℓ 1 : the length of the cycle containing 1, Long Cycles ◮ K n : the total number of cycles ( K n = � n Case θ m = ϑ Case θ m = m γ m =1 C m ), Limit Shape ◮ λ 1 : the length of the longest cycle. ◮ ( λ 1 , λ 2 , . . . ): the length of the longest cycles. 10 / 46

The cycle counts C m Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures The weighted measure Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and The cycle counts C m cycle lengths λ i . The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 11 / 46

The cycle counts C m Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures The weighted measure Let σ = σ 1 σ 2 · · · σ ℓ ∈ S n be given with σ i disjoint cycles and The cycle counts C m cycle lengths λ i . We then define The cycle containing 1 C m := # { i ; λ i = m } . Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 11 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene We have for the uniform and Ewens measure is Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 12 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene We have for the uniform and Ewens measure is Random permutations Theorem (Shepp,Loyd ϑ = 1, Watterson general ϑ ) Classical measures The weighted measure We have for each b > 0 The cycle counts C m d − → ( Y 1 , . . . , Y b ) ( C 1 , . . . , C b ) The cycle containing 1 Total Number of with Y m independent Poisson distributed with Cycles Long Cycles E [ Y m ] = θ m m = ϑ Case θ m = ϑ m . Case θ m = m γ Limit Shape 12 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene We have for the uniform and Ewens measure is Random permutations Theorem (Shepp,Loyd ϑ = 1, Watterson general ϑ ) Classical measures The weighted measure We have for each b > 0 The cycle counts C m d − → ( Y 1 , . . . , Y b ) ( C 1 , . . . , C b ) The cycle containing 1 Total Number of with Y m independent Poisson distributed with Cycles Long Cycles E [ Y m ] = θ m m = ϑ Case θ m = ϑ m . Case θ m = m γ Limit Shape This result can be improved further. 12 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures Let Ω be a countable set and P and � P be two probability The weighted measure measures on Ω. The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 13 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures Let Ω be a countable set and P and � P be two probability The weighted measure measures on Ω. The cycle counts C m The total variation distance between P and � P is defined as The cycle containing 1 � P ) = 1 d TV ( P , � | P ( ω ) − � Total Number of P ( ω ) | . Cycles 2 ω ∈ Ω Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 13 / 46

The cycle counts C m under θ m = ϑ Random permutations with logarithmic cycle weights Setting the scene Random permutations For instance Classical measures The weighted measure Theorem (Arratia and Tavar´ e) The cycle counts C m Let S n be endowed with the uniform measure. Then The cycle containing 1 � � ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 d TV Total Number of Cycles Long Cycles if and only if b = o ( n ) , where d TV denotes the total Case θ m = ϑ Case θ m = m γ variation distance. Limit Shape 14 / 46

The cycle counts C m under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene We have for the cycle weights θ m = m γ with γ > 0 Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 15 / 46

The cycle counts C m under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene We have for the cycle weights θ m = m γ with γ > 0 Random permutations Classical measures Theorem (Ercolani and Ueltschi) The weighted measure We have for each b > 0 The cycle counts C m The cycle d ( C 1 , . . . , C b ) − → ( Y 1 , . . . , Y b ) containing 1 Total Number of Cycles with Y m independent Poisson distributed with Long Cycles Case θ m = ϑ E [ Y m ] = θ m Case θ m = m γ m = m γ − 1 . Limit Shape 15 / 46

The cycle counts C m under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene Random As for the Ewens measure, we can also compute the total permutations Classical measures variation distance. The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 16 / 46

The cycle counts C m under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene Random As for the Ewens measure, we can also compute the total permutations Classical measures variation distance. The weighted measure The cycle counts Theorem (Storm and Zeindler) C m We then have as n → ∞ The cycle containing 1 � � Total Number of 1 1+ γ ) , d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 iff b = o ( n Cycles Long Cycles Case θ m = ϑ where Y m are independent Poisson distributed with Case θ m = m γ E [ Y m ] = θ m m = m γ − 1 . Limit Shape 16 / 46

The cycle counts C m under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene We have for the cycle weights θ m = log k ( m ) with k ≥ 1 Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 17 / 46

The cycle counts C m under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene We have for the cycle weights θ m = log k ( m ) with k ≥ 1 Random permutations Theorem (Robles and Zeindler) Classical measures The weighted measure We have for each b > 0 The cycle counts C m d The cycle ( C 1 , . . . , C b ) − → ( Y 1 , . . . , Y b ) containing 1 Total Number of Cycles with Y m independent Poisson distributed with Long Cycles Case θ m = ϑ m = log k ( m ) E [ Y m ] = θ m Case θ m = m γ . Limit Shape m 17 / 46

The cycle counts C m under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene Random permutations Theorem (Robles and Zeindler) Classical measures � n c � The weighted measure 1 with 0 < c < (3 k + 3) − k +1 . We Suppose that b ( n ) = o The cycle counts C m then have The cycle � � containing 1 ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 d TV Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 18 / 46

The cycle counts C m under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene Random permutations Theorem (Robles and Zeindler) Classical measures � n c � The weighted measure 1 with 0 < c < (3 k + 3) − k +1 . We Suppose that b ( n ) = o The cycle counts C m then have The cycle � � containing 1 ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 d TV Total Number of Cycles Long Cycles We expect that that the d TV in this theorem goes to 0 if � � Case θ m = ϑ Case θ m = m γ n and only if b ( n ) = o . log k ( n ) Limit Shape 18 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations We define Classical measures The weighted measure ℓ 1 := the length of the cycle containing 1 . The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 19 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations We define Classical measures The weighted measure ℓ 1 := the length of the cycle containing 1 . The cycle counts C m The cycle A simple computation gives that we have under the uniform containing 1 measure on S n Total Number of Cycles P n [ ℓ 1 = k ] = 1 Long Cycles n for all 1 ≤ k ≤ n . Case θ m = ϑ Case θ m = m γ Limit Shape 19 / 46

The cycle containing 1 and the uniform measure Random permutations with logarithmic cycle weights Setting the scene Random permutations We thus immediately get Classical measures The weighted measure Theorem The cycle counts C m We have under the uniform measure The cycle containing 1 ℓ 1 d Total Number of − → U (0 , 1) , Cycles n Long Cycles Case θ m = ϑ where U (0 , 1) is the uniform measure on the interval [0 , 1] . Case θ m = m γ Limit Shape 20 / 46

The cycle containing 1 and the Ewens measure Random permutations with logarithmic cycle weights Setting the scene Random permutations Furthermore Classical measures The weighted measure Theorem The cycle counts C m We have under the Ewens measure with parameter ϑ > 0 The cycle containing 1 ℓ 1 d − → Beta (1 , ϑ ) . Total Number of n Cycles Long Cycles where Beta (1 , ϑ ) is the probability measure on the interval Case θ m = ϑ Case θ m = m γ [0 , 1] with density (1 − x ) ϑ − 1 . Limit Shape 21 / 46

The cycle containing 1 under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene Random permutations Theorem (Ercolani, Ueltschi) Classical measures The weighted measure We have in the case θ m = m γ The cycle counts C m � � ℓ 1 The cycle 1 d − → Γ containing 1 1 + γ, Γ(1 + γ ) 1+ γ 1 n Total Number of 1+ γ Cycles where Γ ( α, β ) is the probability measure on [0 , ∞ [ with Long Cycles Case θ m = ϑ Γ( α ) x α e − β x and Γ( s ) the Gamma function. β α density Case θ m = m γ Limit Shape 22 / 46

The cycle containing 1 under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene Random permutations Theorem (Robles, Zeindler) Classical measures The weighted measure We have in the case θ m = θ m = log k ( m ) The cycle counts C m The cycle ℓ 1 d containing 1 − → Exp (1) , n Total Number of log k ( n ) Cycles Long Cycles where Exp ( λ ) is the probability measure on [0 , ∞ [ with Case θ m = ϑ Case θ m = m γ density λ e − λ x . Limit Shape 23 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 1 conjectural 24 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 = m γ = log k ( m ) θ m = ϑ Total Number of Cycles d TV → 0 Long Cycles Case θ m = ϑ Case θ m = m γ ℓ 1 Limit Shape 1 conjectural 24 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 = m γ = log k ( m ) θ m = ϑ Total Number of Cycles d TV → 0 b = o ( n ) Long Cycles Case θ m = ϑ Case θ m = m γ ℓ 1 ≍ n Limit Shape 1 conjectural 24 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 = m γ = log k ( m ) θ m = ϑ Total Number of � � Cycles 1 d TV → 0 b = o ( n ) b = o n 1+ γ Long Cycles Case θ m = ϑ Case θ m = m γ 1 ℓ 1 ≍ n ≍ n 1+ γ Limit Shape 1 conjectural 24 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 = m γ = log k ( m ) θ m = ϑ Total Number of � � � � Cycles 1 n 1 d TV → 0 b = o ( n ) b = o n b = o 1+ γ Long Cycles log k ( n ) Case θ m = ϑ Case θ m = m γ 1 ℓ 1 ≍ n ≍ n 1+ γ Limit Shape 1 conjectural 24 / 46

The cycle containing 1 Random permutations with logarithmic cycle weights Setting the scene Random permutations � � Classical measures d TV ( C 1 , . . . , C b ) , ( Y 1 , . . . , Y b ) → 0 The weighted measure The cycle counts C m The cycle containing 1 = m γ = log k ( m ) θ m = ϑ Total Number of � � � � Cycles 1 n 1 d TV → 0 b = o ( n ) b = o n b = o 1+ γ Long Cycles log k ( n ) Case θ m = ϑ Case θ m = m γ 1 ≍ n / log k ( n ) ℓ 1 ≍ n ≍ n 1+ γ Limit Shape 1 conjectural 24 / 46

Total Number of Cycles Random permutations with logarithmic cycle weights Setting the scene The Total number of cycles is defined as Random permutations Classical measures K n := C 1 + · · · + C n . The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 25 / 46

Total Number of Cycles Random permutations with logarithmic cycle weights Setting the scene The Total number of cycles is defined as Random permutations Classical measures K n := C 1 + · · · + C n . The weighted measure The cycle counts C m We have under the uniform and the Ewens measure. The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 25 / 46

Total Number of Cycles Random permutations with logarithmic cycle weights Setting the scene The Total number of cycles is defined as Random permutations Classical measures K n := C 1 + · · · + C n . The weighted measure The cycle counts C m We have under the uniform and the Ewens measure. The cycle containing 1 Theorem (Goncharov ϑ = 1, Watterson general ϑ ) Total Number of Cycles Long Cycles K n − ϑ log( n ) Case θ m = ϑ d Case θ m = m γ � − → N (0 , 1) ϑ log( n ) Limit Shape 25 / 46

Total Number of Cycles under ’ θ m → ϑ ’ Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures Theorem (Nikeghbali and Zeindler) The weighted measure The cycle counts Consider the general Ewens measure. We then have C m � Γ( θ ) � 1 �� The cycle � � �� containing 1 = n θ ( e s − 1) E Θ exp sK n Γ( θ e s ) + O Total Number of n Cycles Long Cycles with O ( . ) uniform for bounded s ∈ C . Case θ m = ϑ Case θ m = m γ Limit Shape 26 / 46

Total Number of Cycles under ’ θ m → ϑ ’ Random permutations with logarithmic cycle weights Recall, the Kolmogorov distance between two integer valued Setting the scene random variables X and Y is defined as Random permutations � � Classical measures � P n [ X ≤ j ] − P n [ Y ≤ j ] � d K ( X , Y ) := sup The weighted measure j ∈ Z The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 27 / 46

Total Number of Cycles under ’ θ m → ϑ ’ Random permutations with logarithmic cycle weights Recall, the Kolmogorov distance between two integer valued Setting the scene random variables X and Y is defined as Random permutations � � Classical measures � P n [ X ≤ j ] − P n [ Y ≤ j ] � d K ( X , Y ) := sup The weighted measure j ∈ Z The cycle counts C m The cycle containing 1 Corollary Total Number of Cycles Let P θ log( n ) be a Poisson distributed random variable with Long Cycles parameter θ log( n ) . Then Case θ m = ϑ Case θ m = m γ � � Limit Shape 1 � d K ( K n , P θ log( n ) ) = O . log( n ) 27 / 46

Total Number of Cycles under θ m = m γ Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures Theorem (Maples, Nikeghbali, Zeindler) The weighted measure The cycle counts We have in the case θ m = m γ C m The cycle containing 1 γ K n − c γ n 1+ γ d � − → N (0 , 1) Total Number of Cycles γ d γ n 1+ γ Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 28 / 46

Total Number of Cycles under θ m = log k ( m ) Random permutations with logarithmic cycle weights Setting the scene Random permutations Classical measures Theorem (Robles, Zeindler) The weighted measure The cycle counts We have for θ m = log k ( m ) C m The cycle K n − log k +1 ( n ) containing 1 d k +1 Total Number of � − → N (0 , 1) Cycles log k +1 ( n ) Long Cycles k +1 Case θ m = ϑ Case θ m = m γ Limit Shape 29 / 46

Long Cycles Random permutations with logarithmic cycle weights Setting the scene Let Random σ = σ 1 σ 2 · · · σ ℓ . permutations Classical measures We write λ j for the length of the cycle σ j . The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 30 / 46

Long Cycles Random permutations with logarithmic cycle weights Setting the scene Let Random σ = σ 1 σ 2 · · · σ ℓ . permutations Classical measures We write λ j for the length of the cycle σ j . The weighted measure The cycle counts W.l.o.g. we can assume λ 1 ≥ λ 2 ≥ . . . . C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 30 / 46

Long Cycles Random permutations with logarithmic cycle weights Setting the scene Let Random σ = σ 1 σ 2 · · · σ ℓ . permutations Classical measures We write λ j for the length of the cycle σ j . The weighted measure The cycle counts W.l.o.g. we can assume λ 1 ≥ λ 2 ≥ . . . . C m The cycle Theorem (Vershik and Shmidt resp. Kingman ) containing 1 Total Number of Cycles � λ 1 � n , λ 2 Long Cycles d n , . . . − → PD ( ϑ ) , ( n → ∞ ) Case θ m = ϑ Case θ m = m γ Limit Shape with PD ( ϑ ) the Poisson–Dirichlet distribution with parameter ϑ . 30 / 46

Poisson–Dirichlet distribution Random permutations with logarithmic cycle weights Setting the scene What is the Poisson–Dirichlet distribution? Random permutations Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 31 / 46

Poisson–Dirichlet distribution Random permutations with logarithmic cycle weights Setting the scene What is the Poisson–Dirichlet distribution? Random permutations Keyword: stick breaking process with size ordering. Classical measures The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 31 / 46

Poisson–Dirichlet distribution Random permutations with logarithmic cycle weights Setting the scene What is the Poisson–Dirichlet distribution? Random permutations Keyword: stick breaking process with size ordering. Classical measures Let ( B k ) k ∈ N be iid Beta distributed with parameters (1 , ϑ ). The weighted measure The cycle counts C m The cycle containing 1 Total Number of Cycles Long Cycles Case θ m = ϑ Case θ m = m γ Limit Shape 31 / 46