Polynomial Chaos and Scaling Limits of Disordered Systems Rongfeng - PowerPoint PPT Presentation

Polynomial Chaos and Scaling Limits of Disordered Systems Rongfeng Sun National University of Singapore Joint work with Francesco Caravenna (Milano-Bicocca) Nikos Zygouras (Warwick)

Outline 1. Disordered Systems (Disorder Relevance vs Irrelevance) Disordered Pinning Model Long-range Directed Polymer Model Random Field Ising Model 2. Disorder Relevance via Continuum and Weak Disorder Limits Polynomial chaos expansions for partition functions Lindeberg Principle for polynomial chaos expansions Convergence of polynomial chaos to Wiener chaos expansions From partition functions to disordered continuum models 3. Some Open Questions

1.1 The Homogeneous Pinning Model 0 = τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Let τ := { τ 0 = 0 < τ 1 < τ 2 · · · } ⊂ N 0 be a recurrent renewal process, with law P , and C P ( τ 1 = n ) ∼ for some exponent α > 0 . n 1+ α The Pinning Model is defined by the family of Gibbs measures: 1 e h � N n =1 1 { n ∈ τ } P ( τ ) P N,h ( τ ) = (expectation E N,h [ · ]) , Z N,h where N is the system size, h ∈ R determines the interaction strength, and Z N,h = E [ e h � N n =1 1 { n ∈ τ } ] is the partition function.

1.1 The Homogeneous Pinning Model 0 = τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Let τ := { τ 0 = 0 < τ 1 < τ 2 · · · } ⊂ N 0 be a recurrent renewal process, with law P , and C P ( τ 1 = n ) ∼ for some exponent α > 0 . n 1+ α The Pinning Model is defined by the family of Gibbs measures: 1 e h � N n =1 1 { n ∈ τ } P ( τ ) P N,h ( τ ) = (expectation E N,h [ · ]) , Z N,h where N is the system size, h ∈ R determines the interaction strength, and Z N,h = E [ e h � N n =1 1 { n ∈ τ } ] is the partition function.

1.2 Phase Transition for the Pinning Model As h varies, the pinning model undergoes a localization-delocalization transition. More precisely, there is a critical h c (= 0 in this case) such that For h < h c , the limiting contact fraction � 1 N � � g ( h ) := lim 1 { n ∈ τ } = 0; N →∞ E N,h N n =1 For h > h c , the limiting contact fraction g ( h ) > 0. Furthermore, g ( h ) = F ′ ( h ), where the free energy � = 0 if h ≤ h c , 1 F ( h ) = lim N log Z N,h ≈ C ( h − h c ) γ as h ↓ h c . N →∞ 1 The exponent, γ = min { 1 ,α } , is known as a critical exponent.

1.2 Phase Transition for the Pinning Model As h varies, the pinning model undergoes a localization-delocalization transition. More precisely, there is a critical h c (= 0 in this case) such that For h < h c , the limiting contact fraction � 1 N � � g ( h ) := lim 1 { n ∈ τ } = 0; N →∞ E N,h N n =1 For h > h c , the limiting contact fraction g ( h ) > 0. Furthermore, g ( h ) = F ′ ( h ), where the free energy � = 0 if h ≤ h c , 1 F ( h ) = lim N log Z N,h ≈ C ( h − h c ) γ as h ↓ h c . N →∞ 1 The exponent, γ = min { 1 ,α } , is known as a critical exponent.

1.2 Phase Transition for the Pinning Model As h varies, the pinning model undergoes a localization-delocalization transition. More precisely, there is a critical h c (= 0 in this case) such that For h < h c , the limiting contact fraction � 1 N � � g ( h ) := lim 1 { n ∈ τ } = 0; N →∞ E N,h N n =1 For h > h c , the limiting contact fraction g ( h ) > 0. Furthermore, g ( h ) = F ′ ( h ), where the free energy � = 0 if h ≤ h c , 1 F ( h ) = lim N log Z N,h ≈ C ( h − h c ) γ as h ↓ h c . N →∞ 1 The exponent, γ = min { 1 ,α } , is known as a critical exponent.

1.3 The Disordered Pinning Model We now add disorder. Let ω := ( ω n ) n ∈ N be i.i.d. with E [ ω 1 ] = 0 and E [ e λω 1 ] < ∞ for all λ close to 0. Given disorder ω , the Disordered Pinning Model is defined by the family of Gibbs measures: 1 � N P ω n =1 ( βω n + h )1 { n ∈ τ } P ( τ ) , N,β,h ( τ ) = e Z ω N,β,h where β ≥ 0 determines the disorder strength, h ∈ R determines the bias, and Z ω N,β,h is the disordered partition function.

1.3 The Disordered Pinning Model We now add disorder. Let ω := ( ω n ) n ∈ N be i.i.d. with E [ ω 1 ] = 0 and E [ e λω 1 ] < ∞ for all λ close to 0. Given disorder ω , the Disordered Pinning Model is defined by the family of Gibbs measures: 1 � N P ω n =1 ( βω n + h )1 { n ∈ τ } P ( τ ) , N,β,h ( τ ) = e Z ω N,β,h where β ≥ 0 determines the disorder strength, h ∈ R determines the bias, and Z ω N,β,h is the disordered partition function.

1.3 The Disordered Pinning Model We now add disorder. Let ω := ( ω n ) n ∈ N be i.i.d. with E [ ω 1 ] = 0 and E [ e λω 1 ] < ∞ for all λ close to 0. Given disorder ω , the Disordered Pinning Model is defined by the family of Gibbs measures: 1 � N P ω n =1 ( βω n + h )1 { n ∈ τ } P ( τ ) , N,β,h ( τ ) = e Z ω N,β,h where β ≥ 0 determines the disorder strength, h ∈ R determines the bias, and Z ω N,β,h is the disordered partition function.

1.4 Phase Transition for the Disordered Pinning Model For each β > 0, as h varies, the disordered pinning model also undergoes a localization-delocalization transition. There exists ˆ h c ( β ) < 0, s.t. for P -a.e. ω , the contact fraction � 1 if h < ˆ N � � = 0 h c ( β ) , N →∞ E E ω � g ( β, h ) := lim ˆ 1 { n ∈ τ } N,β,h if h > ˆ N > 0 h c ( β ) . n =1 g ( β, h ) = ∂ ˆ F Furthermore, ˆ ∂h ( β, h ), where the disordered free energy  if h ≤ ˆ = 0 h c ( β ) , 1  ˆ N E [log Z ω F ( β, h ) = lim N,β,h ] Conj . C ( h − ˆ as h ↓ ˆ N →∞ h c ( β )) ˆ γ ( β ) ≈ h c ( β )  for some critical exponent ˆ γ ( β ).

1.4 Phase Transition for the Disordered Pinning Model For each β > 0, as h varies, the disordered pinning model also undergoes a localization-delocalization transition. There exists ˆ h c ( β ) < 0, s.t. for P -a.e. ω , the contact fraction � 1 if h < ˆ N � � = 0 h c ( β ) , N →∞ E E ω � g ( β, h ) := lim ˆ 1 { n ∈ τ } N,β,h if h > ˆ N > 0 h c ( β ) . n =1 g ( β, h ) = ∂ ˆ F Furthermore, ˆ ∂h ( β, h ), where the disordered free energy  if h ≤ ˆ = 0 h c ( β ) , 1  ˆ N E [log Z ω F ( β, h ) = lim N,β,h ] Conj . C ( h − ˆ as h ↓ ˆ N →∞ h c ( β )) ˆ γ ( β ) ≈ h c ( β )  for some critical exponent ˆ γ ( β ).

1.4 Phase Transition for the Disordered Pinning Model For each β > 0, as h varies, the disordered pinning model also undergoes a localization-delocalization transition. There exists ˆ h c ( β ) < 0, s.t. for P -a.e. ω , the contact fraction � 1 if h < ˆ N � � = 0 h c ( β ) , N →∞ E E ω � g ( β, h ) := lim ˆ 1 { n ∈ τ } N,β,h if h > ˆ N > 0 h c ( β ) . n =1 g ( β, h ) = ∂ ˆ F Furthermore, ˆ ∂h ( β, h ), where the disordered free energy  if h ≤ ˆ = 0 h c ( β ) , 1  ˆ N E [log Z ω F ( β, h ) = lim N,β,h ] Conj . C ( h − ˆ as h ↓ ˆ N →∞ h c ( β )) ˆ γ ( β ) ≈ h c ( β )  for some critical exponent ˆ γ ( β ).

1.5 Disorder Relevance/Irrelevance Basic Question: Does disorder modify the qualitative nature of the homogeneous model (without disorder)? For the pinning model, we say that disorder is relevant if the critical exponents ˆ γ ( β ) � = γ for all β > 0 (no matter how weak is the disorder strength); irrelevant if ˆ γ ( β ) = γ for β > 0 sufficiently small. For the pinning model with renewal exponent α , it has been shown: Disorder is relevant for α > 1 2 ; Disorder is irrelevant for α < 1 2 ; Disorder is marginally relevant for α = 1 2 . Alexander, Zygouras; Derrida, Giacomin, Lacoin, Toninelli; Cheliotis, den Hollander ...

1.5 Disorder Relevance/Irrelevance Basic Question: Does disorder modify the qualitative nature of the homogeneous model (without disorder)? For the pinning model, we say that disorder is relevant if the critical exponents ˆ γ ( β ) � = γ for all β > 0 (no matter how weak is the disorder strength); irrelevant if ˆ γ ( β ) = γ for β > 0 sufficiently small. For the pinning model with renewal exponent α , it has been shown: Disorder is relevant for α > 1 2 ; Disorder is irrelevant for α < 1 2 ; Disorder is marginally relevant for α = 1 2 . Alexander, Zygouras; Derrida, Giacomin, Lacoin, Toninelli; Cheliotis, den Hollander ...

1.5 Disorder Relevance/Irrelevance Basic Question: Does disorder modify the qualitative nature of the homogeneous model (without disorder)? For the pinning model, we say that disorder is relevant if the critical exponents ˆ γ ( β ) � = γ for all β > 0 (no matter how weak is the disorder strength); irrelevant if ˆ γ ( β ) = γ for β > 0 sufficiently small. For the pinning model with renewal exponent α , it has been shown: Disorder is relevant for α > 1 2 ; Disorder is irrelevant for α < 1 2 ; Disorder is marginally relevant for α = 1 2 . Alexander, Zygouras; Derrida, Giacomin, Lacoin, Toninelli; Cheliotis, den Hollander ...

2.1 Directed Polymer Model Let X := ( X n ) n ∈ N 0 be a mean-zero random walk on Z d with law P . Let ω := ( ω ( n, x )) n ∈ N 0 ,x ∈ Z d be i.i.d. with E [ ω (0 , o )] = 0, and E [ e λω (0 ,o ) ] < ∞ for all λ close to 0. Given disorder ω , the Directed Polymer Model on Z d +1 is defined by the family of Gibbs measures 1 e β � N P ω n =1 ω ( n,X n ) P ( X ) , N,β ( X ) = Z ω N,β where β ≥ 0 is the disorder strength, Z ω N,β is the partition function.