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

Disordered Systems Partition Function CDPM Further Developments Polynomial Chaos and Scaling Limits of Disordered Systems Francesco Caravenna Universit` a degli Studi di Milano-Bicocca Nantes ∼ June 6, 2014 Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 1 / 28

Disordered Systems Partition Function CDPM Further Developments Coworkers Joint work with Nikos Zygouras (Warwick) and Rongfeng Sun (NUS) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 2 / 28

Disordered Systems Partition Function CDPM Further Developments Summary We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments Summary We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments Summary We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime Very general framework, illustrated by 3 concrete examples 1. Disordered pinning models (Pinning) 2. Directed polymer in random environment (DPRE) 3. Random-field Ising model (Ising) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments Summary We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime Very general framework, illustrated by 3 concrete examples 1. Disordered pinning models (Pinning) 2. Directed polymer in random environment (DPRE) 3. Random-field Ising model (Ising) Inspired by recent work of Alberts, Quastel and Khanin on DPRE Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments Outline 1. Disordered Systems and their Scaling Limits 2. Main Results (I): Partition Function 3. Main Results (II): Continuum Disordered Pinning Model 4. Further Developments Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 4 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d ◮ Reference law P ref Ω ( σ ) on “spin configurations” (non trivial!) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d ◮ Reference law P ref Ω ( σ ) on “spin configurations” (non trivial!) ◮ Disorder ( ω x ) x ∈ Z d i.i.d. random variables, independent of σ E [ e t ω x ] < ∞ for small | t | E [ ω x ] = 0 V ar[ ω x ] = 1 Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d ◮ Reference law P ref Ω ( σ ) on “spin configurations” (non trivial!) ◮ Disorder ( ω x ) x ∈ Z d i.i.d. random variables, independent of σ E [ e t ω x ] < ∞ for small | t | E [ ω x ] = 0 V ar[ ω x ] = 1 ( λω x + h ) x ∈ Z d disorder with strength λ > 0 and bias h ∈ R Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d ◮ Reference law P ref Ω ( σ ) on “spin configurations” (non trivial!) ◮ Disorder ( ω x ) x ∈ Z d i.i.d. random variables, independent of σ E [ e t ω x ] < ∞ for small | t | E [ ω x ] = 0 V ar[ ω x ] = 1 ( λω x + h ) x ∈ Z d disorder with strength λ > 0 and bias h ∈ R Disordered law Random Gibbs measure on spin configurations σ , indexed by disorder ω � � � P ω P ref Ω ,λ, h ( σ ) ∝ exp ( λω x + h ) σ x Ω ( σ ) x ∈ Ω Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments General Framework “spins” σ = ( σ x ) x ∈ Ω ∈ { 0 , 1 } Ω or {− 1 , +1 } Ω Lattice Ω ⊆ R d ◮ Reference law P ref Ω ( σ ) on “spin configurations” (non trivial!) ◮ Disorder ( ω x ) x ∈ Z d i.i.d. random variables, independent of σ E [ e t ω x ] < ∞ for small | t | E [ ω x ] = 0 V ar[ ω x ] = 1 ( λω x + h ) x ∈ Z d disorder with strength λ > 0 and bias h ∈ R Disordered law Random Gibbs measure on spin configurations σ , indexed by disorder ω � � � 1 P ω P ref Ω ,λ, h ( σ ) := exp ( λω x + h ) σ x Ω ( σ ) Z ω Ω ,λ, h x ∈ Ω Partition function Z ω Ω ,λ, h = E ref � x ∈ Ω ( λω x + h ) σ x ] Ω [ e Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments 1. Disordered pinning model 0 = τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Reference law: renewal process τ = { 0 = τ 0 < τ 1 < τ 2 < . . . } ⊆ N 0 C P ref � � ( τ i +1 − τ i ) = n ∼ n 1+ α , tail exponent α ∈ (0 , 1) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments 1. Disordered pinning model 0 = τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Reference law: renewal process τ = { 0 = τ 0 < τ 1 < τ 2 < . . . } ⊆ N 0 C P ref � � ( τ i +1 − τ i ) = n ∼ n 1+ α , tail exponent α ∈ (0 , 1) “spins” σ n := 1 { n ∈ τ } ∈ { 0 , 1 } (long-range correlations) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments 1. Disordered pinning model 0 = τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Reference law: renewal process τ = { 0 = τ 0 < τ 1 < τ 2 < . . . } ⊆ N 0 C P ref � � ( τ i +1 − τ i ) = n ∼ n 1+ α , tail exponent α ∈ (0 , 1) “spins” σ n := 1 { n ∈ τ } ∈ { 0 , 1 } (long-range correlations) Lattice Ω := { 1 , . . . , N } Disordered law: disordered pinning model 1 � N P ω n =1 ( λω n + h ) 1 { n ∈ τ } P ref ( τ ) Ω ,λ, h ( τ ) = e Z ω Ω ,λ, h Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments 2. Directed polymer in random environment Reference law: symmetric random walk X = ( X n ) n ≥ 0 on Z , in the domain of attraction of a stable L´ evy process with index α ∈ (0 , 2] ∼ C V ar ref ( X 1 ) < ∞ if α = 2 P ref � � | X 1 | > x if α ∈ (0 , 2) x α Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28

Disordered Systems Partition Function CDPM Further Developments 2. Directed polymer in random environment Reference law: symmetric random walk X = ( X n ) n ≥ 0 on Z , in the domain of attraction of a stable L´ evy process with index α ∈ (0 , 2] ∼ C V ar ref ( X 1 ) < ∞ if α = 2 P ref � � | X 1 | > x if α ∈ (0 , 2) x α “spins” σ n , x := 1 { X n = x } ∈ { 0 , 1 } (long-range correlations) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28

Disordered Systems Partition Function CDPM Further Developments 2. Directed polymer in random environment Reference law: symmetric random walk X = ( X n ) n ≥ 0 on Z , in the domain of attraction of a stable L´ evy process with index α ∈ (0 , 2] ∼ C V ar ref ( X 1 ) < ∞ if α = 2 P ref � � | X 1 | > x if α ∈ (0 , 2) x α “spins” σ n , x := 1 { X n = x } ∈ { 0 , 1 } (long-range correlations) Lattice Ω := { 1 , . . . , N } × Z Disordered law: directed polymer in random environment 1 � N P ω n =1 λω n , x 1 { Xn = x } P ref ( X ) Ω ,λ ( X ) = e Z ω Ω ,λ Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28