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Self-similar growth-fragmentations as scaling limits of Markov branching processes Benjamin Dadoun Institut fr Mathematik, Universitt Zrich Les probabilits de demain 3 mai 2018 Outline Introduction 1 Motivating example 2

  1. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 28 ) = ( 4 , 4 , 3 , 2 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  2. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 29 ) = ( 4 , 4 , 3 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  3. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 30 ) = ( 4 , 3 , 3 , 2 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  4. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 31 ) = ( 4 , 3 , 3 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  5. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 32 ) = ( 4 , 3 , 2 , 2 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  6. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 34 ) = ( 4 , 3 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  7. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 35 ) = ( 4 , 2 , 2 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  8. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 37 ) = ( 4 , 1 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  9. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 38 ) = ( 4 , 2 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  10. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 39 ) = ( 4 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  11. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 40 ) = ( 5 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  12. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 41 ) = ( 4 , 2 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  13. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 42 ) = ( 4 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  14. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 43 ) = ( 5 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  15. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 44 ) = ( 4 , 2 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  16. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 45 ) = ( 4 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  17. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 46 ) = ( 3 , 2 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  18. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 47 ) = ( 3 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  19. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 48 ) = ( 2 , 2 , 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  20. Motivating example Peeling a random Boltzmann triangulation of the n -gon Hole perimeters: X ( n ) ( 50 ) = ( 0 , . . . ) B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  21. Motivating example Peeling a random Boltzmann triangulation of the n -gon Fact (Bertoin et al., 2016, 2017) . There is a scaling limit:   ( n ) ( ⌊ a n t ⌋ )  � X ( d )  “ : t ≥ 0 − − − → Y ” n n →∞ B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 1/10

  22. Self-similar growth-fragmentation Y Y is a positive, self-similar, Markov process B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  23. Self-similar growth-fragmentation Y � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  24. Self-similar growth-fragmentation Y ∅ � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  25. Self-similar growth-fragmentation x Y ∅ b 1 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  26. Self-similar growth-fragmentation x − y Y ∅ y b 1 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  27. Self-similar growth-fragmentation Y ∅ Y 1 y b 1 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  28. Self-similar growth-fragmentation Y ∅ Y 1 b 1 b 2 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  29. Self-similar growth-fragmentation Y ∅ Y 2 Y 1 b 1 b 2 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  30. Self-similar growth-fragmentation Y ∅ Y 2 Y 1 b 1 b 2 b 21 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  31. Self-similar growth-fragmentation Y ∅ Y 2 Y 21 Y 1 b 1 b 2 b 21 � � y − γ · self-similar: law of Y under P y = law of yY under P 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  32. Self-similar growth-fragmentation Y ∅ Y 2 Y 21 Y 1 b 1 b 2 b 21 � � � � � � Y ( t ) = Y u ( t − b u ): b u ≤ t = Y 1 ( t ) ≥ Y 2 ( t ) ≥ · · · B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 2/10

  33. Markov branching process X ( 5 ) : B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  34. Markov branching process X ( 5 ) : B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  35. Markov branching process X ( 5 ) : B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  36. Markov branching process X ( 5 ) : B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  37. Markov branching process X ( 5 ) : B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  38. Markov branching process p 4 , 3 X ( 5 ) : p 2 , 4 p 2 , 5 p 3 , 2 p 5 , 3 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  39. Markov branching process p 4 , 3 X ( 5 ) : p 2 , 4 p 2 , 5 p 3 , 2 p 5 , 3 The system is entirely described through the Markov transition kernel p n , k , n ≤ 2 k , of the locally largest particle X ( n ) . B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  40. Markov branching process p 4 , 3 X ( 5 ) : p 2 , 4 p 2 , 5 p 3 , 2 p 5 , 3 locally largest The system is entirely described through the Markov transition kernel p n , k , n ≤ 2 k , of the locally largest particle X ( n ) . B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  41. Markov branching process p 4 , 3 X ( 5 ) : p 2 , 4 p 2 , 5 p 3 , 2 p 5 , 3 locally largest 2 nd locally largest The system is entirely described through the Markov transition kernel p n , k , n ≤ 2 k , of the locally largest particle X ( n ) . B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 3/10

  42. Scaling limits Which asymptotic conditions on ( p n , k ) imply a scaling limit 1. for the Markov branching process X ( n ) ? B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 4/10

  43. Scaling limits Which asymptotic conditions on ( p n , k ) imply a scaling limit 1. for the Markov branching process X ( n ) ? 2. for the associated genealogical tree X ( n ) ? B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 4/10

  44. Scaling limits Which asymptotic conditions on ( p n , k ) imply a scaling limit 1. for the Markov branching process X ( n ) ? 2. for the associated genealogical tree X ( n ) ? Without growth: Haas and Miermont (2004, 2012) [self-similar fragmentation tree/process]. B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 4/10

  45. Scaling limits Which asymptotic conditions on ( p n , k ) imply a scaling limit 1. for the Markov branching process X ( n ) ? 2. for the associated genealogical tree X ( n ) ? Without growth: Haas and Miermont (2004, 2012) [self-similar fragmentation tree/process]. Starting point: scaling limit for the locally largest particle X ( n ) . B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 4/10

  46. Two difficulties induced by growth ◮ The total mass is no longer conserved. B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 5/10

  47. Two difficulties induced by growth ◮ The total mass is no longer conserved. ◮ The system may even explode, e.g. 1 1 2 1 ... 1 2 1 1 2 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 5/10

  48. Two difficulties induced by growth ◮ The total mass is no longer conserved. ◮ The system may even explode, e.g. 1 1 2 1 ... 1 2 1 1 2 1 Since we do not (want to) deal with the behaviour of p n , k for “small” n , we must prune the system: B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 5/10

  49. Two difficulties induced by growth ◮ The total mass is no longer conserved. ◮ The system may even explode, e.g. 1 1 2 1 ... 1 2 1 1 2 1 Since we do not (want to) deal with the behaviour of p n , k for “small” n , we must prune the system: Particles with size ≤ M (large but fixed) are frozen . = ⇒ Locally largest particle stopped below M ( p n , n := 1 , n ≤ M ) . B. Dadoun | Self-similar scaling limits of Markov branching processes › Introduction 5/10

  50. Outline Introduction 1 Motivating example 2 Self-similar growth-fragmentation 3 Markov branching process 4 Scaling limits 5 Two difficulties induced by growth Results 6 Assumptions 7 Scaling limit for the process 8 Scaling limit for the tree 9-10 Proof aspects B. Dadoun | Self-similar scaling limits of Markov branching processes › Results

  51. Assumptions n →∞ a ⌊ nx ⌋ / a n = x γ . Let γ > 0 and ( a n ) regularly varying: ∀ x > 0 , lim B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 6/10

  52. Assumptions n →∞ a ⌊ nx ⌋ / a n = x γ . Let γ > 0 and ( a n ) regularly varying: ∀ x > 0 , lim We suppose that there exist q ∗ > 0 and a Lévy process ξ such that B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 6/10

  53. Assumptions n →∞ a ⌊ nx ⌋ / a n = x γ . Let γ > 0 and ( a n ) regularly varying: ∀ x > 0 , lim We suppose that there exist q ∗ > 0 and a Lévy process ξ such that (H1) for all t ∈ R , �� m � ∞ � � i t log E [ e it ξ 1 ] =: Ψ( i t ); Ψ n ( i t ) := a n p n , m − 1 − − − → n n →∞ m = 1 B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 6/10

  54. Assumptions n →∞ a ⌊ nx ⌋ / a n = x γ . Let γ > 0 and ( a n ) regularly varying: ∀ x > 0 , lim We suppose that there exist q ∗ > 0 and a Lévy process ξ such that (H1) for all t ∈ R , �� m � ∞ � � i t log E [ e it ξ 1 ] =: Ψ( i t ); Ψ n ( i t ) := a n p n , m − 1 − − − → n n →∞ m = 1 ∞ � m � � q ∗ (H2) lim sup < ∞ ; a n p n , m n n →∞ m = 2 n B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 6/10

  55. Assumptions n →∞ a ⌊ nx ⌋ / a n = x γ . Let γ > 0 and ( a n ) regularly varying: ∀ x > 0 , lim We suppose that there exist q ∗ > 0 and a Lévy process ξ such that (H1) for all t ∈ R , �� m � ∞ � � i t log E [ e it ξ 1 ] =: Ψ( i t ); Ψ n ( i t ) := a n p n , m − 1 − − − → n n →∞ m = 1 ∞ � m � � q ∗ (H2) lim sup < ∞ ; a n p n , m n n →∞ m = 2 n (H3) κ ( q ∗ ) < 0, and for some ε > 0, � n − 1 � � � � 1 − e y � q ∗ − ε Λ( d y ) , q ∗ − ε 1 − m lim = n →∞ a n p n , m n R − m = 1 � R − ( 1 − e y ) q Λ( d y ) . with Λ Lévy measure of ξ , κ ( q ) := Ψ( q ) + B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 6/10

  56. Scaling limit for the process Theorem 1. Under (H1) – (H3) , we can fix M large so that � � X ( n ) ( ⌊ a n t ⌋ ) ( d ) : t ≥ 0 − − − → Y n n →∞ holds as càdlàg processes in ℓ q for q ≥ 1 ∨ q ∗ . B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 7/10

  57. Scaling limit for the process Theorem 1. Under (H1) – (H3) , we can fix M large so that � � X ( n ) ( ⌊ a n t ⌋ ) ( d ) : t ≥ 0 − − − → Y n n →∞ holds as càdlàg processes in ℓ q for q ≥ 1 ∨ q ∗ . The limit Y is the self-similar growth-fragmentation driven by Y , where �� t � Y ( s ) − γ d s log Y ( t ) = ξ , t ≥ 0 . 0 B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 7/10

  58. Scaling limit for the tree Theorem 2. Under (H1) – (H3) , q ∗ > γ , we can fix M so that X ( n ) ( d ) − n →∞ Y , − − → a n as compact real trees in the Gromov–Hausdorff topology. B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 8/10

  59. Scaling limit for the tree Theorem 2. Under (H1) – (H3) , q ∗ > γ , we can fix M so that X ( n ) ( d ) − n →∞ Y , − − → a n as compact real trees in the Gromov–Hausdorff topology. The limit Y is the continuum random tree associated with Y , as constructed by Rembardt and Winkel (2016). B. Dadoun | Self-similar scaling limits of Markov branching processes › Results 8/10

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