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Pinning models Weak disorder regime The marginally relevant regime Polynomial Chaos and Scaling Limits of Disordered Systems 4. Free energy estimates. Introduction to marginal relevance. Francesco Caravenna Universit` a degli Studi di

  1. Pinning models Weak disorder regime The marginally relevant regime The phase transition How are the typical paths τ of the pinning model P ω N ? � = � N n =1 1 { n ∈ τ } = � N � � C N := � τ ∩ (0 , N ] Contact number n =1 1 { S n =0 } Theorem (phase transition) ∃ continuous, non decreasing, deterministic critical curve h c ( β ) : ◮ Localized regime: for h > h c ( β ) one has C N ≈ N �� � C N � P ω � � ∃ µ = µ β, h > 0 : N − µ � > ε − − − − → 0 ω –a.s. � � N N →∞ � ◮ Deocalized regime: for h < h c ( β ) one has C N = O (log N ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 8 / 26

  2. Pinning models Weak disorder regime The marginally relevant regime The phase transition How are the typical paths τ of the pinning model P ω N ? � = � N n =1 1 { n ∈ τ } = � N � � C N := � τ ∩ (0 , N ] Contact number n =1 1 { S n =0 } Theorem (phase transition) ∃ continuous, non decreasing, deterministic critical curve h c ( β ) : ◮ Localized regime: for h > h c ( β ) one has C N ≈ N �� � C N � P ω � � ∃ µ = µ β, h > 0 : N − µ � > ε − − − − → 0 ω –a.s. � � N N →∞ � ◮ Deocalized regime: for h < h c ( β ) one has C N = O (log N ) � C N � P ω ∃ A = A β, h > 0 : − − − − → log N > A 0 ω –a.s. N N →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 8 / 26

  3. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  4. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  5. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  6. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  7. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ ( α ≥ 1 2 ) disorder is relevant: h c ( β ) > 0 for all β > 0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  8. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ ( α ≥ 1 2 ) disorder is relevant: h c ( β ) > 0 for all β > 0 h c ( β ) ∼ C β 2 with explicit C = ◮ ( α > 1) 1 α 1+ α 2E( τ 1 ) [Berger, C., Poisat, Sun, Zygouras] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  9. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ ( α ≥ 1 2 ) disorder is relevant: h c ( β ) > 0 for all β > 0 h c ( β ) ∼ C β 2 with explicit C = ◮ ( α > 1) 1 α 1+ α 2E( τ 1 ) [Berger, C., Poisat, Sun, Zygouras] 2 α 2 α ◮ ( 1 2 α − 1 ≤ h c ( β ) ≤ C 2 β 2 < α < 1) C 1 β 2 α − 1 [Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  10. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ ( α ≥ 1 2 ) disorder is relevant: h c ( β ) > 0 for all β > 0 h c ( β ) ∼ C β 2 with explicit C = ◮ ( α > 1) 1 α 1+ α 2E( τ 1 ) [Berger, C., Poisat, Sun, Zygouras] 2 α 2 α ◮ ( 1 2 α − 1 ≤ h c ( β ) ≤ C 2 β 2 < α < 1) C 1 β 2 α − 1 [Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras] − c + o (1) ◮ ( α = 1 2 ) h c ( β ) = e β 2 [Giacomin, Lacoin, Toninelli] [Berger, Lacoin] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  11. Pinning models Weak disorder regime The marginally relevant regime Estimates on the critical curve For β = 0 (homogeneous pinning, no disorder) one has h c (0) = 0 What is the behavior of h c ( β ) for β > 0 small ? c K Theorem ( P( τ 1 = n ) ∼ n 1+ α ) ◮ ( α < 1 2 ) disorder is irrelevant: h c ( β ) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ ( α ≥ 1 2 ) disorder is relevant: h c ( β ) > 0 for all β > 0 h c ( β ) ∼ C β 2 with explicit C = ◮ ( α > 1) 1 α 1+ α 2E( τ 1 ) [Berger, C., Poisat, Sun, Zygouras] 2 α ◮ ( 1 h c ( β ) ∼ ˆ 2 < α < 1) C β using continuum model! 2 α − 1 [C., Torri, Toninelli] − c + o (1) ◮ ( α = 1 2 ) h c ( β ) = e β 2 [Giacomin, Lacoin, Toninelli] [Berger, Lacoin] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 9 / 26

  12. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  13. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  14. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  15. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω N ≥ E Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  16. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω N ≥ E = P( τ ∩ (0 , N ] = ∅ ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  17. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  18. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α ◮ Critical curve h c ( β ) = sup { h ∈ R : F ( β, h ) = 0 } non analiticity! Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  19. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α ◮ Critical curve h c ( β ) = sup { h ∈ R : F ( β, h ) = 0 } non analiticity! � C N � ∂ F ( β, h ) N →∞ E ω (convexity) = lim N ∂ h N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  20. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α ◮ Critical curve h c ( β ) = sup { h ∈ R : F ( β, h ) = 0 } non analiticity! � � > 0 if h > h c ( β ) � C N ∂ F ( β, h ) N →∞ E ω (convexity) = lim N ∂ h N = 0 if h < h c ( β ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  21. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α ◮ Critical curve h c ( β ) = sup { h ∈ R : F ( β, h ) = 0 } non analiticity! � � > 0 if h > h c ( β ) � C N ∂ F ( β, h ) N →∞ E ω (convexity) = lim N ∂ h N = 0 if h < h c ( β ) F ( β, h ) and h c ( β ) depend on the law of τ and ω Universality as β, h → 0 ? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  22. Pinning models Weak disorder regime The marginally relevant regime Discrete free energy and critical curve � e H N ( τ ) � � � � N Z ω n =1 ( h + βω n − Λ( β )) 1 { n ∈ τ } Partition function N := E = E e Consider first the regime of N → ∞ with fixed β, h N log Z ω ◮ Free energy 1 N ≥ 0 F ( β, h ) := lim P (d ω )-a.s. N →∞ � � e H N ( τ ) 1 { τ ∩ (0 , N ]= ∅} Z ω = P( τ ∩ (0 , N ] = ∅ ) ∼ ( const . ) N ≥ E N α ◮ Critical curve h c ( β ) = sup { h ∈ R : F ( β, h ) = 0 } non analiticity! � � > 0 if h > h c ( β ) � C N ∂ F ( β, h ) N →∞ E ω (convexity) = lim N ∂ h N = 0 if h < h c ( β ) F ( β, h ) and h c ( β ) depend on the law of τ and ω Universality as β, h → 0 ? YES, connected to continuum model Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 10 / 26

  23. Pinning models Weak disorder regime The marginally relevant regime A word on critical exponents The free energy F ( β, h ) is non analytic at the critical point h = h c ( β ) F ( β, h ) = 0 ( h < h c ( β )) F ( β, h ) > 0 ( h > h c ( β )) What is the behavior of F ( β, h ) as h ↓ h c ( β ) ? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 11 / 26

  24. Pinning models Weak disorder regime The marginally relevant regime A word on critical exponents The free energy F ( β, h ) is non analytic at the critical point h = h c ( β ) F ( β, h ) = 0 ( h < h c ( β )) F ( β, h ) > 0 ( h > h c ( β )) What is the behavior of F ( β, h ) as h ↓ h c ( β ) ? For β = 0 the model is exactly solvable: h c (0) = 0 and 1 F (0 , h ) − F (0 , h c (0)) ∼ C ( h − h c (0)) ( α ∈ (0 , 1)) α Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 11 / 26

  25. Pinning models Weak disorder regime The marginally relevant regime A word on critical exponents The free energy F ( β, h ) is non analytic at the critical point h = h c ( β ) F ( β, h ) = 0 ( h < h c ( β )) F ( β, h ) > 0 ( h > h c ( β )) What is the behavior of F ( β, h ) as h ↓ h c ( β ) ? For β = 0 the model is exactly solvable: h c (0) = 0 and 1 F (0 , h ) − F (0 , h c (0)) ∼ C ( h − h c (0)) ( α ∈ (0 , 1)) α Smoothing inequality [Giacomin, Toninelli] F ( β, h ) − F ( β, h c ( β )) ≤ C β 2 ( h − h c (0)) 2 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 11 / 26

  26. Pinning models Weak disorder regime The marginally relevant regime A word on critical exponents The free energy F ( β, h ) is non analytic at the critical point h = h c ( β ) F ( β, h ) = 0 ( h < h c ( β )) F ( β, h ) > 0 ( h > h c ( β )) What is the behavior of F ( β, h ) as h ↓ h c ( β ) ? For β = 0 the model is exactly solvable: h c (0) = 0 and 1 F (0 , h ) − F (0 , h c (0)) ∼ C ( h − h c (0)) ( α ∈ (0 , 1)) α Smoothing inequality [Giacomin, Toninelli] F ( β, h ) − F ( β, h c ( β )) ≤ C β 2 ( h − h c (0)) 2 ◮ For α > 1 2 disorder makes phase transition smoother! Also h c ( β ) � = h c (0) for every β > 0 � disorder relevance Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 11 / 26

  27. Pinning models Weak disorder regime The marginally relevant regime A word on critical exponents The free energy F ( β, h ) is non analytic at the critical point h = h c ( β ) F ( β, h ) = 0 ( h < h c ( β )) F ( β, h ) > 0 ( h > h c ( β )) What is the behavior of F ( β, h ) as h ↓ h c ( β ) ? For β = 0 the model is exactly solvable: h c (0) = 0 and 1 F (0 , h ) − F (0 , h c (0)) ∼ C ( h − h c (0)) ( α ∈ (0 , 1)) α Smoothing inequality [Giacomin, Toninelli] F ( β, h ) − F ( β, h c ( β )) ≤ C β 2 ( h − h c (0)) 2 ◮ For α > 1 2 disorder makes phase transition smoother! Also h c ( β ) � = h c (0) for every β > 0 � disorder relevance ◮ For α < 1 2 and for β > 0 small F ( β, h ) ≈ F (0 , h ) � irrelevance Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 11 / 26

  28. Pinning models Weak disorder regime The marginally relevant regime Outline 1. Pinning models 2. Weak disorder regime 3. The marginally relevant regime Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 12 / 26

  29. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Build continuum partition functions for Pinning Model with α ∈ ( 1 2 , 1) (disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 13 / 26

  30. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Build continuum partition functions for Pinning Model with α ∈ ( 1 2 , 1) (disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+] We need to rescale ˆ ˆ β h β = β N = h = h N = N α − 1 / 2 N α 2 α 2 α − 1 (Note that h N ≈ β ≈ h c ( β N )) N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 13 / 26

  31. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Build continuum partition functions for Pinning Model with α ∈ ( 1 2 , 1) (disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+] We need to rescale ˆ ˆ β h β = β N = h = h N = N α − 1 / 2 N α 2 α 2 α − 1 (Note that h N ≈ β ≈ h c ( β N )) N d One has Z ω Z W − − − − → with N N →∞ β, ˆ ˆ β, ˆ ˆ β, ˆ ˆ h h h � d W � d W d W Z W := 1 + C t t t ′ + C 2 t 1 − α ( t ′ − t ) 1 − α + . . . t 1 − α 0 < t < t ′ < 1 0 < t < 1 β, ˆ ˆ h := ˆ β W t + ˆ h t and C = α sin( απ ) where W t π c K Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 13 / 26

  32. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Exercise Recalling that c P ref ( n ∈ τ ) = P ref ( S n = 0) ∼ n 1 − α check that β N and h N are the correct scaling (polynomial chaos) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 14 / 26

  33. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Exercise Recalling that c P ref ( n ∈ τ ) = P ref ( S n = 0) ∼ n 1 − α check that β N and h N are the correct scaling (polynomial chaos) Like for DPRE we build constrained partition functions: 0 ≤ s < t < ∞ e H ω Z W � = scaling limit of E ref � � � � s , t � Ns ∈ τ [ Ns , Nt ] 1 { Nt ∈ τ } Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 14 / 26

  34. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Exercise Recalling that c P ref ( n ∈ τ ) = P ref ( S n = 0) ∼ n 1 − α check that β N and h N are the correct scaling (polynomial chaos) Like for DPRE we build constrained partition functions: 0 ≤ s < t < ∞ e H ω Z W � = scaling limit of E ref � � � � s , t � Ns ∈ τ [ Ns , Nt ] 1 { Nt ∈ τ } We show that they satisfy continuity, strict positivity, semigroup Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 14 / 26

  35. Pinning models Weak disorder regime The marginally relevant regime Continuum partition functions Exercise Recalling that c P ref ( n ∈ τ ) = P ref ( S n = 0) ∼ n 1 − α check that β N and h N are the correct scaling (polynomial chaos) Like for DPRE we build constrained partition functions: 0 ≤ s < t < ∞ e H ω Z W � = scaling limit of E ref � � � � s , t � Ns ∈ τ [ Ns , Nt ] 1 { Nt ∈ τ } We show that they satisfy continuity, strict positivity, semigroup Theorem [C., Sun, Zygouras 2015+b] We can build a continuum disordered Pinning model P W as a random probability law on the space of closed subsets of [0 , 1] Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 14 / 26

  36. Pinning models Weak disorder regime The marginally relevant regime Continuum free energy In analogy with the discrete model, define 1 F (ˆ β, ˆ t log Z W Continuum free energy h ) := lim h (0 , t ) a.s. β, ˆ ˆ t →∞ (existence and self-averaging need some work) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 15 / 26

  37. Pinning models Weak disorder regime The marginally relevant regime Continuum free energy In analogy with the discrete model, define 1 F (ˆ β, ˆ t log Z W Continuum free energy h ) := lim h (0 , t ) a.s. β, ˆ ˆ t →∞ (existence and self-averaging need some work) Again F (ˆ β, ˆ h ) ≥ 0 and define H c (ˆ � ˆ h ∈ R : F (ˆ β, ˆ � Continuum critical curve β ) := sup h ) = 0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 15 / 26

  38. Pinning models Weak disorder regime The marginally relevant regime Continuum free energy In analogy with the discrete model, define 1 F (ˆ β, ˆ t log Z W Continuum free energy h ) := lim h (0 , t ) a.s. β, ˆ ˆ t →∞ (existence and self-averaging need some work) Again F (ˆ β, ˆ h ) ≥ 0 and define H c (ˆ � ˆ h ∈ R : F (ˆ β, ˆ � Continuum critical curve β ) := sup h ) = 0 Scaling relations d Z W = Z W ∀ c > 0 : h ( c t ) h ( t ) (Wiener chaos exp.) β, ˆ ˆ c α − 1 2 ˆ β, c α ˆ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 15 / 26

  39. Pinning models Weak disorder regime The marginally relevant regime Continuum free energy In analogy with the discrete model, define 1 F (ˆ β, ˆ t log Z W Continuum free energy h ) := lim h (0 , t ) a.s. β, ˆ ˆ t →∞ (existence and self-averaging need some work) Again F (ˆ β, ˆ h ) ≥ 0 and define H c (ˆ � ˆ h ∈ R : F (ˆ β, ˆ � Continuum critical curve β ) := sup h ) = 0 Scaling relations d Z W = Z W ∀ c > 0 : h ( c t ) h ( t ) (Wiener chaos exp.) β, ˆ ˆ c α − 1 2 ˆ β, c α ˆ c α − 1 2 ˆ β, c α � = c F (ˆ β, ˆ � F h ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 15 / 26

  40. Pinning models Weak disorder regime The marginally relevant regime Continuum free energy In analogy with the discrete model, define 1 F (ˆ β, ˆ t log Z W Continuum free energy h ) := lim h (0 , t ) a.s. β, ˆ ˆ t →∞ (existence and self-averaging need some work) Again F (ˆ β, ˆ h ) ≥ 0 and define H c (ˆ � ˆ h ∈ R : F (ˆ β, ˆ � Continuum critical curve β ) := sup h ) = 0 Scaling relations d Z W = Z W ∀ c > 0 : h ( c t ) h ( t ) (Wiener chaos exp.) β, ˆ ˆ c α − 1 2 ˆ β, c α ˆ c α − 1 2 α 2 ˆ β, c α � = c F (ˆ β, ˆ H c (ˆ β ) = H c (1) ˆ � F h ) β 2 α − 1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 15 / 26

  41. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  42. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  43. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 F (ˆ β, ˆ log Z W � � h ) = lim t E h ( t ) β, ˆ ˆ t →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  44. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 1 F (ˆ β, ˆ log Z ω log Z W � � � � h ) = lim t E h ( t ) = lim N →∞ E lim β N , h N ( Nt ) β, ˆ ˆ t t →∞ t →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  45. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 1 F (ˆ β, ˆ log Z ω log Z W � � � � h ) = lim t E h ( t ) = lim N →∞ E lim β N , h N ( Nt ) β, ˆ ˆ t t →∞ t →∞ Assuming we can interchange the limits N → ∞ and t → ∞ 1 F (ˆ β, ˆ log Z ω � � h ) = lim lim β N , h N ( Nt ) t E t →∞ N →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  46. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 1 F (ˆ β, ˆ log Z ω log Z W � � � � h ) = lim t E h ( t ) = lim N →∞ E lim β N , h N ( Nt ) β, ˆ ˆ t t →∞ t →∞ Assuming we can interchange the limits N → ∞ and t → ∞ 1 F (ˆ β, ˆ log Z ω � � h ) = N →∞ N lim lim β N , h N ( Nt ) Nt E t →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  47. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 1 F (ˆ β, ˆ log Z ω log Z W � � � � h ) = lim t E h ( t ) = lim N →∞ E lim β N , h N ( Nt ) β, ˆ ˆ t t →∞ t →∞ Assuming we can interchange the limits N → ∞ and t → ∞ 1 F (ˆ β, ˆ log Z ω � � h ) = N →∞ N lim lim β N , h N ( Nt ) = N →∞ N F ( β N , h N ) lim Nt E t →∞ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  48. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Can we relate continuum free energy to the discrete one? By construction of continuum partition functions d Z W N →∞ Z ω h ( t ) = lim β N , h N ( Nt ) β, ˆ ˆ Assuming uniform integrability of log Z ω (OK) 1 1 F (ˆ β, ˆ log Z ω log Z W � � � � h ) = lim t E h ( t ) = lim N →∞ E lim β N , h N ( Nt ) β, ˆ ˆ t t →∞ t →∞ Assuming we can interchange the limits N → ∞ and t → ∞ 1 F (ˆ β, ˆ log Z ω � � h ) = N →∞ N lim lim β N , h N ( Nt ) = N →∞ N F ( β N , h N ) lim Nt E t →∞ Setting δ = 1 N for clarity, we arrive at. . . Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 16 / 26

  49. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Conjecture � ˆ βδ α − 1 2 , ˆ h δ α � F F (ˆ β, ˆ h ) = lim δ δ → 0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 17 / 26

  50. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Conjecture � ˆ βδ α − 1 2 , ˆ h δ α � F F (ˆ β, ˆ h ) = lim δ δ → 0 Theorem [C., Toninelli, Torri 2015] For all ˆ β > 0, ˆ h ∈ R and η > 0 there is δ 0 > 0 such that ∀ δ < δ 0 βδ α − 1 � ˆ 2 , ˆ h δ α � h − η ) ≤ F F (ˆ β, ˆ ≤ F (ˆ β, ˆ h + η ) δ Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 17 / 26

  51. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Conjecture � ˆ βδ α − 1 2 , ˆ h δ α � F F (ˆ β, ˆ h ) = lim δ δ → 0 Theorem [C., Toninelli, Torri 2015] For all ˆ β > 0, ˆ h ∈ R and η > 0 there is δ 0 > 0 such that ∀ δ < δ 0 βδ α − 1 � ˆ 2 , ˆ h δ α � h − η ) ≤ F F (ˆ β, ˆ ≤ F (ˆ β, ˆ h + η ) δ 2 α This implies Conj. and h c ( β ) ∼ H c ( β ) ∼ H c (1) β 2 α − 1 For any discrete Pinning model with α ∈ ( 1 2 , 1), the free energy F ( β, h ) and the critical curve h c ( β ) have a universal shape in the regime β, h → 0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 17 / 26

  52. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  53. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? ◮ Assume that there is a continuum Hamiltonian: Z ω = E Z W = E e H ω e H W � Nt � � t � Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  54. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? ◮ Assume that there is a continuum Hamiltonian: Z ω = E Z W = E e H ω e H W � Nt � � t � ◮ Couple H ω Nt and H W on the same probability space in such a way t that the difference ∆ N , t := H ω Nt − H W is “small” t Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  55. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? ◮ Assume that there is a continuum Hamiltonian: Z ω = E Z W = E e H ω e H W � Nt � � t � ◮ Couple H ω Nt and H W on the same probability space in such a way t that the difference ∆ N , t := H ω Nt − H W is “small” t ◮ Deduce that log Z ω � e ∆ N , t � � � log Z W � � ≤ E + log E E E and show that the last term is “negligible” Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  56. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? ◮ Assume that there is a continuum Hamiltonian: Z ω = E Z W = E e H ω e H W � Nt � � t � ◮ Couple H ω Nt and H W on the same probability space in such a way t that the difference ∆ N , t := H ω Nt − H W is “small” t ◮ Deduce that log Z ω � e ∆ N , t � � � log Z W � � ≤ E + log E E E and show that the last term is “negligible” Problem: there is no continuum Hamiltonian! Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  57. Pinning models Weak disorder regime The marginally relevant regime Interchanging the limits Very delicate result. How to prove it? ◮ Assume that there is a continuum Hamiltonian: Z ω = E Z W = E e H ω e H W � Nt � � t � ◮ Couple H ω Nt and H W on the same probability space in such a way t that the difference ∆ N , t := H ω Nt − H W is “small” t ◮ Deduce that log Z ω � e ∆ N , t � � � log Z W � � ≤ E + log E E E and show that the last term is “negligible” Problem: there is no continuum Hamiltonian! Solution: perform coarse-graining and define an “effective” Hamiltonian (drawing!) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 18 / 26

  58. Pinning models Weak disorder regime The marginally relevant regime The DPRE case What about the DPRE? We can still define discrete F ( β ) and continuum F (ˆ β ) free energy β ) ∼ F (1) β 4 we can hope that Since F (ˆ F ( β ) ∼ F (1) β 4 as β → 0 provided the “interchanging of limits” is justified N. Torri is currently working on this problem. A finer coarse-graining is needed, together with sharper estimates on continuum partition functions Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 19 / 26

  59. Pinning models Weak disorder regime The marginally relevant regime Outline 1. Pinning models 2. Weak disorder regime 3. The marginally relevant regime Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 20 / 26

  60. Pinning models Weak disorder regime The marginally relevant regime The marginal case We consider simultaneously different models that are marginally relevant: ◮ Pinning Models with α = 1 2 ◮ DPRE with d = 2 (RW attracted to BM) ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n ◮ Stochastic Heat Equation in d = 2 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 21 / 26

  61. Pinning models Weak disorder regime The marginally relevant regime The marginal case We consider simultaneously different models that are marginally relevant: ◮ Pinning Models with α = 1 2 ◮ DPRE with d = 2 (RW attracted to BM) ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n ◮ Stochastic Heat Equation in d = 2 All these different models share a crucial feature: logarithmic overlap  � P ref ( n ∈ τ ) 2    1 ≤ n ≤ N  R N = ∼ C log N � � P ref ( S n = x ) 2     1 ≤ n ≤ N x ∈ Z d Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 21 / 26

  62. Pinning models Weak disorder regime The marginally relevant regime The marginal case We consider simultaneously different models that are marginally relevant: ◮ Pinning Models with α = 1 2 ◮ DPRE with d = 2 (RW attracted to BM) ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n ◮ Stochastic Heat Equation in d = 2 All these different models share a crucial feature: logarithmic overlap  � P ref ( n ∈ τ ) 2    1 ≤ n ≤ N  R N = ∼ C log N � � P ref ( S n = x ) 2     1 ≤ n ≤ N x ∈ Z d More generally: R N diverges as a slowly varying function Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 21 / 26

  63. Pinning models Weak disorder regime The marginally relevant regime The marginal case Analogy between Pinning model with α = 1 2 and DPRE with d = 2 N � Z ω P ref ( n ∈ τ ) X n + . . . Pin = 1 + n =1 N � � � � Z ω P ref ( S n = x ) X n , x DPRE = 1 + + . . . n =1 x ∈ Z 2 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 22 / 26

  64. Pinning models Weak disorder regime The marginally relevant regime The marginal case Analogy between Pinning model with α = 1 2 and DPRE with d = 2 N � Z ω P ref ( n ∈ τ ) X n + . . . Pin = 1 + n =1 N � � � � Z ω P ref ( S n = x ) X n , x DPRE = 1 + + . . . n =1 x ∈ Z 2 Note that P ref ( S n = x ) ∼ 1 n g 1 ( x √ n ) (recall that d = 2) Then the random variable in parenthesis has variance � 2 ∼ � g 1 � 2 P ref ( S n = x ) 2 ∼ 1 1 � x � � 2 g 1 √ n n n n x ∈ Z 2 x ∈ Z 2 hence we can replace it by � g 1 � 2 √ n X n � Pinning! ( P ref ( n ∈ τ ) ∼ c √ n ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 22 / 26

  65. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 n 1 − α ) Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  66. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 n 1 − α ) N k Z ω � β k � � P ref ( n i − n i − 1 ∈ τ ) X n i N = k =0 0 < n 1 <...< n k ≤ N i =1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  67. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 n 1 − α ) N k Z ω � β k � � P ref ( n i − n i − 1 ∈ τ ) X n i N = k =0 0 < n 1 <...< n k ≤ N i =1 N k X n i � � � = 1 + ( n i − n i − 1 ) 1 − α k =1 0 < n 1 <...< n k ≤ N i =1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  68. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 n 1 − α ) N k Z ω � β k � � P ref ( n i − n i − 1 ∈ τ ) X n i N = k =0 0 < n 1 <...< n k ≤ N i =1 N X n 1 X n 2 · · · X n k � � β k = ( n 2 − n 1 ) 1 − α · · · ( n k − n k − 1 ) 1 − α n 1 − α 1 k =0 0 < n 1 <...< n k ≤ N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  69. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) ˆ Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 β n 1 − α ) β ∼ N α − 1 2 N k Z ω � β k � � P ref ( n i − n i − 1 ∈ τ ) X n i N = k =0 0 < n 1 <...< n k ≤ N i =1 N X n 1 X n 2 · · · X n k � � β k = ( n 2 − n 1 ) 1 − α · · · ( n k − n k − 1 ) 1 − α n 1 − α 1 k =0 0 < n 1 <...< n k ≤ N √ � β 1 1 1 N � k N X n 1 N X n 2 · · · N X n k √ √ √ N � � = N ) 1 − α · · · ( n k N − n k − 1 N 1 − α ( n 1 N ) 1 − α ( n 2 N − n 1 N ) 1 − α k =0 0 < n 1 <...< n k ≤ N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  70. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime For computation we focus on Pinning model (for simplicity h = 0) ˆ Look at relevant case α > 1 2 ( P ref ( n ∈ τ ) ∼ 1 β n 1 − α ) β ∼ N α − 1 2 N k Z ω � β k � � P ref ( n i − n i − 1 ∈ τ ) X n i N = k =0 0 < n 1 <...< n k ≤ N i =1 N X n 1 X n 2 · · · X n k � � β k = ( n 2 − n 1 ) 1 − α · · · ( n k − n k − 1 ) 1 − α n 1 − α 1 k =0 0 < n 1 <...< n k ≤ N √ � β 1 1 1 N � k N X n 1 N X n 2 · · · N X n k √ √ √ N � � = N ) 1 − α · · · ( n k N − n k − 1 N 1 − α ( n 1 N ) 1 − α ( n 2 N − n 1 N ) 1 − α k =0 0 < n 1 <...< n k ≤ N ∞ � d W t 1 d W t 2 · · · d W t k d ˆ � β k − − − − → t 1 − α ( t 2 − t 1 ) 1 − α · · · ( t k − t k − 1 ) 1 − α N →∞ 0 < t 1 <...< t k ≤ 1 1 k =0 For α = 1 √ t �∈ L 2 ([0 , 1]) 1 2 last step breaks down How to make sense? Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 23 / 26

  71. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  72. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 ∞ Γ(1 − χ ) k +1 � β k ˆ ≤ Γ((1 − χ )( k + 1)) k =0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  73. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 ∞ ∞ Γ(1 − χ ) k +1 c k � β k ˆ � β k ˆ 1 ≤ Γ((1 − χ )( k + 1)) ≤ ( c 2 k )! < ∞ k =0 k =0 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  74. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 ∞ ∞ Γ(1 − χ ) k +1 c k � β k ˆ � β k ˆ 1 ≤ Γ((1 − χ )( k + 1)) ≤ ( c 2 k )! < ∞ k =0 k =0 The k ! makes the series converge for all ˆ β > 0 It arises from the constraint 0 < t 1 < . . . < t k ≤ 1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  75. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 ∞ ∞ Γ(1 − χ ) k +1 c k � β k ˆ � β k ˆ 1 ≤ Γ((1 − χ )( k + 1)) ≤ ( c 2 k )! < ∞ k =0 k =0 The k ! makes the series converge for all ˆ β > 0 It arises from the constraint 0 < t 1 < . . . < t k ≤ 1 Exercise Prove “by bare hands” that � d t 1 d t 2 · · · d t k 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ ≤ e − Ck log k t χ 0 < t 1 <...< t k ≤ 1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  76. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime Always in the relevant case α > 1 (setting χ = 2(1 − α ) < 1 ) 2 ∞ � d t 1 d t 2 · · · d t k V ar[ Z ω � ˆ β k N ] − − − − → 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ t χ N →∞ 0 < t 1 <...< t k ≤ 1 k =0 ∞ ∞ Γ(1 − χ ) k +1 c k � β k ˆ � β k ˆ 1 ≤ Γ((1 − χ )( k + 1)) ≤ ( c 2 k )! < ∞ k =0 k =0 The k ! makes the series converge for all ˆ β > 0 It arises from the constraint 0 < t 1 < . . . < t k ≤ 1 Exercise Prove “by bare hands” that (probabilistic argument!) � d t 1 d t 2 · · · d t k 1 ( t 2 − t 1 ) χ · · · ( t k − t k − 1 ) χ ≤ e − Ck log k t χ 0 < t 1 <...< t k ≤ 1 Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 24 / 26

  77. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime In the marginal regime α = 1 2 N X n 1 X n 2 · · · X n k Z ω � β k � √ n 2 − n 1 · · · √ n k − n k − 1 N = √ n 1 k =0 0 < n 1 <...< n k ≤ N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 25 / 26

  78. Pinning models Weak disorder regime The marginally relevant regime Relevant vs. marginal regime In the marginal regime α = 1 2 N X n 1 X n 2 · · · X n k Z ω � β k � √ n 2 − n 1 · · · √ n k − n k − 1 N = √ n 1 k =0 0 < n 1 <...< n k ≤ N X n X n X n ′ � √ n + β 2 � √ = 1 + β √ n n ′ − n + . . . 0 < n ≤ N 0 < n < n ′ ≤ N Francesco Caravenna Scaling Limits of Disordered Systems Sep 30 - Oct 2, 2015 25 / 26

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