White noise Continuum partition functions The continuum DPRE Pinning models Multiple stochastic integrals We can define � W ⊗ k ( g ) = ( R d ) k g ( x 1 , . . . , x k ) W (d x 1 ) · · · W (d x k ) For d = 1 we can restrict x 1 < x 2 < . . . < x k � iterated Ito integrals For symmetric functions we have E [ W ⊗ k ( g )] = 0 E [ W ⊗ k ( g ) 2 ] = k ! � g � 2 L 2 (( R d ) k ) C ov[ W ⊗ k ( f ) , W ⊗ k ′ ( g )] = 0 ∀ k � = k ′ Wiener chaos expansion Any r.v. X ∈ L 2 (Ω W ) measurable w.r.t. σ ( W ) can be written as ∞ 1 � k ! W ⊗ k ( f k ) f k ∈ L 2 sym (( R d ) k ) X = with k =0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Discrete sums and stochastic integrals Consider a lattice T δ ⊆ R d whose cells have volume v δ → 0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Discrete sums and stochastic integrals Consider a lattice T δ ⊆ R d whose cells have volume v δ → 0 Take i.i.d. random variables ( X z ) z ∈ T δ with zero mean and unit variance Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Discrete sums and stochastic integrals Consider a lattice T δ ⊆ R d whose cells have volume v δ → 0 Take i.i.d. random variables ( X z ) z ∈ T δ with zero mean and unit variance Consider the “stochastic Riemann sum” (multi-linear polynomial) � Ψ δ := f ( z 1 , . . . , z k ) X z 1 X z 2 · · · X z k ( z 1 ,..., z k ) ∈ ( T δ ) k z i � = z i ∀ i � = j where f ∈ L 2 ( R d ) is (say) continuous. Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Discrete sums and stochastic integrals Consider a lattice T δ ⊆ R d whose cells have volume v δ → 0 Take i.i.d. random variables ( X z ) z ∈ T δ with zero mean and unit variance Consider the “stochastic Riemann sum” (multi-linear polynomial) � Ψ δ := f ( z 1 , . . . , z k ) X z 1 X z 2 · · · X z k ( z 1 ,..., z k ) ∈ ( T δ ) k z i � = z i ∀ i � = j where f ∈ L 2 ( R d ) is (say) continuous. ( √ v δ ) k Ψ δ � d − − − → ( R d ) k g ( z 1 , . . . , z k ) W (d z 1 ) · · · W (d z k ) δ → 0 (Check the variance!) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Outline 1. White noise and Wiener chaos 2. Continuum partition functions 3. The continuum DPRE 4. Pinning models Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 9 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z � � N �� δ = E ref � � H ω �� Z ω = E ref � � � exp exp βω ( n , S n ) − λ ( β ) n =1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z � � N �� δ = E ref � � H ω �� Z ω = E ref � � � exp exp βω ( n , S n ) − λ ( β ) n =1 � P ref ( S δ = 1 + t = x ) X t , x ( t , x ) ∈ T δ + 1 � P ref ( S δ t = x , S δ t ′ = x ′ ) X t , x X t ′ , x ′ + . . . 2 ( t , x ) � =( t ′ , x ′ ) ∈ T δ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z � � N �� δ = E ref � � H ω �� Z ω = E ref � � � exp exp βω ( n , S n ) − λ ( β ) n =1 � P ref ( S δ = 1 + t = x ) X t , x ( t , x ) ∈ T δ + 1 � P ref ( S δ t = x , S δ t ′ = x ′ ) X t , x X t ′ , x ′ + . . . 2 ( t , x ) � =( t ′ , x ′ ) ∈ T δ with g ( z ) = e − z 2 � x Recall the LLT: P ref ( S n = x ) ∼ 1 � 2 √ n g √ n √ 2 π Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z � � N �� δ = E ref � � H ω �� Z ω = E ref � � � exp exp βω ( n , S n ) − λ ( β ) n =1 � P ref ( S δ = 1 + t = x ) X t , x ( t , x ) ∈ T δ + 1 � P ref ( S δ t = x , S δ t ′ = x ′ ) X t , x X t ′ , x ′ + . . . 2 ( t , x ) � =( t ′ , x ′ ) ∈ T δ with g ( z ) = e − z 2 � x Recall the LLT: P ref ( S n = x ) ∼ 1 � 2 √ n g √ n √ 2 π g t ( x ) = e − x 2 √ 2 t P ref ( S δ t = x ) = P ref ( S t x √ δ = δ ) ∼ δ g t ( x ) √ 2 π t Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ √ 1d rescaled RW S δ � � t := δ S t /δ lives on T δ = [0 , 1] ∩ δ N 0 × δ Z � � N �� δ = E ref � � H ω �� Z ω = E ref � � � exp exp βω ( n , S n ) − λ ( β ) n =1 � P ref ( S δ = 1 + t = x ) X t , x ( t , x ) ∈ T δ + 1 � P ref ( S δ t = x , S δ t ′ = x ′ ) X t , x X t ′ , x ′ + . . . 2 ( t , x ) � =( t ′ , x ′ ) ∈ T δ with g ( z ) = e − z 2 � x Recall the LLT: P ref ( S n = x ) ∼ 1 � 2 √ n g √ n √ 2 π g t ( x ) = e − x 2 √ 2 t P ref ( S δ t = x ) = P ref ( S t x √ δ = δ ) ∼ δ g t ( x ) √ 2 π t Replacing X t , x = e ( βω ( t , x ) − λ ( β )) − 1 ≈ β Y t , x with Y t , x i.i.d. N (0 , 1) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ � Z ω N = 1 + β δ g t ( x ) Y t , x ( t , x ) ∈ T δ √ + 1 g t ( x ) g t ′ − t ( x ′ − x ) Y t , x Y t ′ , x ′ + . . . δ ) 2 � 2 ( β ( t , x ) � =( t ′ , x ′ ) ∈ T δ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ � Z ω N = 1 + β δ g t ( x ) Y t , x ( t , x ) ∈ T δ √ + 1 g t ( x ) g t ′ − t ( x ′ − x ) Y t , x Y t ′ , x ′ + . . . δ ) 2 � 2 ( β ( t , x ) � =( t ′ , x ′ ) ∈ T δ √ 3 Cells in T δ have volume v δ = δ δ = δ 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ � Z ω N = 1 + β δ g t ( x ) Y t , x ( t , x ) ∈ T δ √ + 1 g t ( x ) g t ′ − t ( x ′ − x ) Y t , x Y t ′ , x ′ + . . . δ ) 2 � 2 ( β ( t , x ) � =( t ′ , x ′ ) ∈ T δ √ 3 Cells in T δ have volume v δ = δ δ = δ � “Stochastic Riemann sums” 2 √ δ ≈ √ v δ converge to stochastic integrals if β Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ � Z ω N = 1 + β δ g t ( x ) Y t , x ( t , x ) ∈ T δ √ + 1 g t ( x ) g t ′ − t ( x ′ − x ) Y t , x Y t ′ , x ′ + . . . δ ) 2 � 2 ( β ( t , x ) � =( t ′ , x ′ ) ∈ T δ √ 3 Cells in T δ have volume v δ = δ δ = δ � “Stochastic Riemann sums” 2 √ δ ≈ √ v δ converge to stochastic integrals if β ˆ β 1 β ∼ ˆ 4 = β δ 1 N 4 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Continuum partition function for DPRE √ � Z ω N = 1 + β δ g t ( x ) Y t , x ( t , x ) ∈ T δ √ + 1 g t ( x ) g t ′ − t ( x ′ − x ) Y t , x Y t ′ , x ′ + . . . δ ) 2 � 2 ( β ( t , x ) � =( t ′ , x ′ ) ∈ T δ √ 3 Cells in T δ have volume v δ = δ δ = δ � “Stochastic Riemann sums” 2 √ δ ≈ √ v δ converge to stochastic integrals if β ˆ β 1 β ∼ ˆ 4 = β δ 1 N 4 � Z W = 1 + ˆ d Z ω − − − → β g t ( x ) W (d t d x ) N δ → 0 [0 , 1] × R ˆ β 2 � ([0 , 1] × R ) 2 g t ( x ) g t ′ − t ( x ′ − x ) W (d t d x ) W (d t ′ d x ′ ) + 2 + . . . Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Constrained partition functions We have constructed Z W = “free” partition function on [0 , 1] × R RW paths starting at (0 , 0) with no constraint on right endpoint Z W = Z W � Z W � � � (0 , 0) , (1 , ⋆ ) E = 1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Constrained partition functions We have constructed Z W = “free” partition function on [0 , 1] × R RW paths starting at (0 , 0) with no constraint on right endpoint Z W = Z W � Z W � � � (0 , 0) , (1 , ⋆ ) E = 1 Consider now constrained partition functions: for ( s , y ) , ( t , x ) ∈ [0 , 1] × R � = E ref � � H ω � � Z ω � S δ � � Discrete: ( s , y ) , ( t , x ) exp s = y 1 { S δ � δ t = x } Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Constrained partition functions We have constructed Z W = “free” partition function on [0 , 1] × R RW paths starting at (0 , 0) with no constraint on right endpoint Z W = Z W � Z W � � � (0 , 0) , (1 , ⋆ ) E = 1 Consider now constrained partition functions: for ( s , y ) , ( t , x ) ∈ [0 , 1] × R � = E ref � � H ω � � Z ω � S δ � � Discrete: ( s , y ) , ( t , x ) exp s = y 1 { S δ � δ t = x } √ Divided by δ , they converge to a continuum limit: Z W � � � Z W � �� ( s , y ) , ( t , x ) ( s , y ) , ( t , x ) = g t − s ( x − y ) E Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Constrained partition functions We have constructed Z W = “free” partition function on [0 , 1] × R RW paths starting at (0 , 0) with no constraint on right endpoint Z W = Z W � Z W � � � (0 , 0) , (1 , ⋆ ) E = 1 Consider now constrained partition functions: for ( s , y ) , ( t , x ) ∈ [0 , 1] × R � = E ref � � H ω � � Z ω � S δ � � Discrete: ( s , y ) , ( t , x ) exp s = y 1 { S δ � δ t = x } √ Divided by δ , they converge to a continuum limit: Z W � � � Z W � �� ( s , y ) , ( t , x ) ( s , y ) , ( t , x ) = g t − s ( x − y ) E This is a function of white noise in the stripe W ([ s , t ] × R ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Constrained partition functions We have constructed Z W = “free” partition function on [0 , 1] × R RW paths starting at (0 , 0) with no constraint on right endpoint Z W = Z W � Z W � � � (0 , 0) , (1 , ⋆ ) E = 1 Consider now constrained partition functions: for ( s , y ) , ( t , x ) ∈ [0 , 1] × R � = E ref � � H ω � � Z ω � S δ � � Discrete: ( s , y ) , ( t , x ) exp s = y 1 { S δ � δ t = x } √ Divided by δ , they converge to a continuum limit: Z W � � � Z W � �� ( s , y ) , ( t , x ) ( s , y ) , ( t , x ) = g t − s ( x − y ) E This is a function of white noise in the stripe W ([ s , t ] × R ) Four-parameter random process Z W � � ( s , y ) , ( t , x ) � regularity? Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Key properties Key properties For a.e. realization of W the following properties hold: ◮ Continuity: Z W (( s , y ) , ( t , x )) is jointly continuous in ( s , y , t , x ) (on the domain s < t ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Key properties Key properties For a.e. realization of W the following properties hold: ◮ Continuity: Z W (( s , y ) , ( t , x )) is jointly continuous in ( s , y , t , x ) (on the domain s < t ) ◮ Positivity: Z W (( s , y ) , ( t , x )) > 0 for all ( s , y , t , x ) satisfying s < t Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Key properties Key properties For a.e. realization of W the following properties hold: ◮ Continuity: Z W (( s , y ) , ( t , x )) is jointly continuous in ( s , y , t , x ) (on the domain s < t ) ◮ Positivity: Z W (( s , y ) , ( t , x )) > 0 for all ( s , y , t , x ) satisfying s < t ◮ Semigroup (Chapman-Kolmogorov): for all s < r < t and x , y ∈ R � Z W (( s , y ) , ( t , x )) = Z W (( s , y ) , ( r , z )) Z W (( r , z ) , ( t , x )) d z R (Inherited from discrete partition functions: drawing!) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Key properties Key properties For a.e. realization of W the following properties hold: ◮ Continuity: Z W (( s , y ) , ( t , x )) is jointly continuous in ( s , y , t , x ) (on the domain s < t ) ◮ Positivity: Z W (( s , y ) , ( t , x )) > 0 for all ( s , y , t , x ) satisfying s < t ◮ Semigroup (Chapman-Kolmogorov): for all s < r < t and x , y ∈ R � Z W (( s , y ) , ( t , x )) = Z W (( s , y ) , ( r , z )) Z W (( r , z ) , ( t , x )) d z R (Inherited from discrete partition functions: drawing!) How to prove these properties? Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The 1d Stochastic Heat Equation The four-parameter field Z W (( s , y ) , ( t , x )) solves the 1d SHE ∂ t Z W = 1 2 ∆ x Z W + ˆ β W Z W lim t ↓ s Z W (( s , y ) , ( t , x )) = δ ( y − x ) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The 1d Stochastic Heat Equation The four-parameter field Z W (( s , y ) , ( t , x )) solves the 1d SHE ∂ t Z W = 1 2 ∆ x Z W + ˆ β W Z W lim t ↓ s Z W (( s , y ) , ( t , x )) = δ ( y − x ) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016]) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The 1d Stochastic Heat Equation The four-parameter field Z W (( s , y ) , ( t , x )) solves the 1d SHE ∂ t Z W = 1 2 ∆ x Z W + ˆ β W Z W lim t ↓ s Z W (( s , y ) , ( t , x )) = δ ( y − x ) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016]) ◮ Prove continuity by Kolmogorov criterion, showing that Z W (( s , y ) , ( t , x )) is continuous also for t = s g t − s ( x − y ) ◮ Use continuity to prove semigroup for all times ◮ Use continuity to deduce positivity for close times, then bootstrap to arbitrary times using semigroup Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Outline 1. White noise and Wiener chaos 2. Continuum partition functions 3. The continuum DPRE 4. Pinning models Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 15 / 43
White noise Continuum partition functions The continuum DPRE Pinning models A naive approach Consider DPRE in d = 1 (random walk + disorder) � N n =1 β ω ( n , S n ) P ref ( S ) P ω ( S ) ∝ e Can we define its continuum analogue (BM + disorder)? Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43
White noise Continuum partition functions The continuum DPRE Pinning models A naive approach Consider DPRE in d = 1 (random walk + disorder) � N n =1 β ω ( n , S n ) P ref ( S ) P ω ( S ) ∝ e Can we define its continuum analogue (BM + disorder)? Naively � 1 P W (d B ) ∝ e β W ( t , B t ) d t P ref (d B ) 0 ˆ P ref = law of BM W ( t , x ) = white noise on R 2 (space-time) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43
White noise Continuum partition functions The continuum DPRE Pinning models A naive approach Consider DPRE in d = 1 (random walk + disorder) � N n =1 β ω ( n , S n ) P ref ( S ) P ω ( S ) ∝ e Can we define its continuum analogue (BM + disorder)? Naively � 1 P W (d B ) ∝ e β W ( t , B t ) d t P ref (d B ) 0 ˆ P ref = law of BM W ( t , x ) = white noise on R 2 (space-time) ◮ � 1 0 W ( t , B t ) d t ill-defined. Regularization? Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43
White noise Continuum partition functions The continuum DPRE Pinning models A naive approach Consider DPRE in d = 1 (random walk + disorder) � N n =1 β ω ( n , S n ) P ref ( S ) P ω ( S ) ∝ e Can we define its continuum analogue (BM + disorder)? Naively � 1 P W (d B ) ∝ e β W ( t , B t ) d t P ref (d B ) 0 ˆ P ref = law of BM W ( t , x ) = white noise on R 2 (space-time) ◮ � 1 0 W ( t , B t ) d t ill-defined. Regularization? NO! The problem is more subtle (and interesting!) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times δ � Z W to define the law of continuum DPRE Idea: Replace Z ω Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times δ � Z W to define the law of continuum DPRE Idea: Replace Z ω Recall: to define a process ( X t ) t ∈ [0 , 1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µ t 1 ,..., t k ( A 1 , . . . , A k ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times δ � Z W to define the law of continuum DPRE Idea: Replace Z ω Recall: to define a process ( X t ) t ∈ [0 , 1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µ t 1 ,..., t k ( A 1 , . . . , A k ) “ = P( X t 1 ∈ A 1 , . . . , X t k ∈ A k ) ” Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times δ � Z W to define the law of continuum DPRE Idea: Replace Z ω Recall: to define a process ( X t ) t ∈ [0 , 1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µ t 1 ,..., t k ( A 1 , . . . , A k ) “ = P( X t 1 ∈ A 1 , . . . , X t k ∈ A k ) ” that are consistent µ t 1 ,..., t j ,..., t k ( A 1 , . . . , R , . . . , A k ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Partition functions and f.d.d. Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1 t ′ = x ′ ) = Z ω � � Z ω � ( t , x ) , ( t ′ , x ′ ) � Z ω � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) δ δ δ P ω δ ( S δ t = x , S δ Z ω � � (0 , 0) , (1 , ⋆ ) δ (drawing!) Analogous formula for any finite number of times δ � Z W to define the law of continuum DPRE Idea: Replace Z ω Recall: to define a process ( X t ) t ∈ [0 , 1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µ t 1 ,..., t k ( A 1 , . . . , A k ) “ = P( X t 1 ∈ A 1 , . . . , X t k ∈ A k ) ” that are consistent µ t 1 ,..., t j ,..., t k ( A 1 , . . . , R , . . . , A k ) = µ t 1 ,..., t j − 1 , t j +1 ,..., t k ( A 1 , . . . , A j − 1 , A j +1 , . . . , A k ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The continuum 1d DPRE β ∈ (0 , ∞ ) (on which Z W depend) ˆ [recall that β ∼ ˆ 1 ◮ Fix 4 ] βδ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The continuum 1d DPRE β ∈ (0 , ∞ ) (on which Z W depend) ˆ [recall that β ∼ ˆ 1 ◮ Fix 4 ] βδ ◮ Fix space-time white noise W on [0 , 1] × R and a realization of continuum partition functions Z W satisfying the key properties (continuity, strict positivity, semigroup) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The continuum 1d DPRE β ∈ (0 , ∞ ) (on which Z W depend) ˆ [recall that β ∼ ˆ 1 ◮ Fix 4 ] βδ ◮ Fix space-time white noise W on [0 , 1] × R and a realization of continuum partition functions Z W satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process ( X t ) t ∈ [0 , 1] with f.d.d. P W ( X t ∈ d x , X t ′ ∈ d x ′ ) d x d x ′ := Z W � � Z W � ( t , x ) , ( t ′ , x ′ ) � Z W � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) Z W � � (0 , 0) , (1 , ⋆ ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The continuum 1d DPRE β ∈ (0 , ∞ ) (on which Z W depend) ˆ [recall that β ∼ ˆ 1 ◮ Fix 4 ] βδ ◮ Fix space-time white noise W on [0 , 1] × R and a realization of continuum partition functions Z W satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process ( X t ) t ∈ [0 , 1] with f.d.d. P W ( X t ∈ d x , X t ′ ∈ d x ′ ) d x d x ′ := Z W � � Z W � ( t , x ) , ( t ′ , x ′ ) � Z W � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) Z W � � (0 , 0) , (1 , ⋆ ) ◮ Well-defined by strict positivity of Z W Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models The continuum 1d DPRE β ∈ (0 , ∞ ) (on which Z W depend) ˆ [recall that β ∼ ˆ 1 ◮ Fix 4 ] βδ ◮ Fix space-time white noise W on [0 , 1] × R and a realization of continuum partition functions Z W satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process ( X t ) t ∈ [0 , 1] with f.d.d. P W ( X t ∈ d x , X t ′ ∈ d x ′ ) d x d x ′ := Z W � � Z W � ( t , x ) , ( t ′ , x ′ ) � Z W � ( t ′ , x ′ ) , (1 , ⋆ ) � (0 , 0) , ( t , x ) Z W � � (0 , 0) , (1 , ⋆ ) ◮ Well-defined by strict positivity of Z W ◮ Consistent by semigroup property Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Relation with Wiener measure The law of the continuum DPRE is a random probability P W ( X ∈ · ) (quenched law) [ Probab. kernel S ′ ( R ) → R [0 , 1] ] for the process X = ( X t ) t ∈ [0 , 1] Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Relation with Wiener measure The law of the continuum DPRE is a random probability P W ( X ∈ · ) (quenched law) [ Probab. kernel S ′ ( R ) → R [0 , 1] ] for the process X = ( X t ) t ∈ [0 , 1] Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P d P ( W ) = Z W � � (0 , 0) , (1 , ⋆ ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Relation with Wiener measure The law of the continuum DPRE is a random probability P W ( X ∈ · ) (quenched law) [ Probab. kernel S ′ ( R ) → R [0 , 1] ] for the process X = ( X t ) t ∈ [0 , 1] Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P d P ( W ) = Z W � � (0 , 0) , (1 , ⋆ ) Key Lemma � P W ( X ∈ · ) ˜ P ann ( X ∈ · ) := P (d W ) S ′ ( R ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Relation with Wiener measure The law of the continuum DPRE is a random probability P W ( X ∈ · ) (quenched law) [ Probab. kernel S ′ ( R ) → R [0 , 1] ] for the process X = ( X t ) t ∈ [0 , 1] Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P d P ( W ) = Z W � � (0 , 0) , (1 , ⋆ ) Key Lemma � P W ( X ∈ · ) ˜ P ann ( X ∈ · ) := P (d W ) = P( BM ∈ · ) S ′ ( R ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Relation with Wiener measure The law of the continuum DPRE is a random probability P W ( X ∈ · ) (quenched law) [ Probab. kernel S ′ ( R ) → R [0 , 1] ] for the process X = ( X t ) t ∈ [0 , 1] Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P d P ( W ) = Z W � � (0 , 0) , (1 , ⋆ ) Key Lemma � P W ( X ∈ · ) ˜ P ann ( X ∈ · ) := P (d W ) = P( BM ∈ · ) S ′ ( R ) Proof. The factor Z W in ˜ P cancels the denominator in the f.d.d. for P W � Z W � �� Since E ( s , y ) , ( t , x ) = g t − s ( x − y ) one gets f.d.d. of BM Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Absolute continuity properties Theorem ∀ A ⊆ R [0 , 1] : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 1 ⇒ for P -a.e. W Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Absolute continuity properties Theorem ∀ A ⊆ R [0 , 1] : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 1 ⇒ for P -a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Absolute continuity properties Theorem ∀ A ⊆ R [0 , 1] : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 1 ⇒ for P -a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W Corollary P W ( X has H¨ older paths with exp. 1 2 − ) = 1 for P -a.e. W We can thus realize P W as a law on C ([0 , 1] , R ), for P -a.e. W (More precisely: P W admits a modification with H¨ older paths) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Absolute continuity properties Theorem ∀ A ⊆ R [0 , 1] : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 1 ⇒ for P -a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W Corollary P W ( X has H¨ older paths with exp. 1 2 − ) = 1 for P -a.e. W We can thus realize P W as a law on C ([0 , 1] , R ), for P -a.e. W (More precisely: P W admits a modification with H¨ older paths) One is tempted to conclude that P W is absolutely continuous w.r.t. Wiener measure, for P -a.e. W . . . Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Absolute continuity properties Theorem ∀ A ⊆ R [0 , 1] : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 1 ⇒ for P -a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W Corollary P W ( X has H¨ older paths with exp. 1 2 − ) = 1 for P -a.e. W We can thus realize P W as a law on C ([0 , 1] , R ), for P -a.e. W (More precisely: P W admits a modification with H¨ older paths) One is tempted to conclude that P W is absolutely continuous w.r.t. Wiener measure, for P -a.e. W . . . NO ! “ ∀ A ” and “ for P -a.e. W ” cannot be exchanged! Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Singularity properties Theorem The law P W is singular w.r.t. Wiener measure, for P -a.e. W . Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Singularity properties Theorem The law P W is singular w.r.t. Wiener measure, for P -a.e. W . for P -a.e. W ∃ A = A W ⊆ C ([0 , 1] , R ) : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 0 vs. Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Singularity properties Theorem The law P W is singular w.r.t. Wiener measure, for P -a.e. W . for P -a.e. W ∃ A = A W ⊆ C ([0 , 1] , R ) : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 0 vs. Unlike discrete DPRE, there is no continuum Hamiltonian P W ( X ∈ · ) �∝ e H W ( · ) P( BM ∈ · ) Absolute continuity is lost in the scaling limit Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Singularity properties Theorem The law P W is singular w.r.t. Wiener measure, for P -a.e. W . for P -a.e. W ∃ A = A W ⊆ C ([0 , 1] , R ) : P W ( X ∈ A ) = 1 P( BM ∈ A ) = 0 vs. Unlike discrete DPRE, there is no continuum Hamiltonian P W ( X ∈ · ) �∝ e H W ( · ) P( BM ∈ · ) Absolute continuity is lost in the scaling limit In a sense, the laws P W are just barely not absolutely continuous w.r.t. Wiener measure (“stochastically absolutely continuous”) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Fix a typical realization of W . Setting P ref = Wiener measure n ( X ) := d P W | F n R W ( X ) d P ref | F n Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Fix a typical realization of W . Setting P ref = Wiener measure n ( X ) := d P W | F n R W ( X ) d P ref | F n n ) n ∈ N is a martingale w.r.t. P ref (exercise!) The process ( R W Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Fix a typical realization of W . Setting P ref = Wiener measure n ( X ) := d P W | F n R W ( X ) d P ref | F n n ) n ∈ N is a martingale w.r.t. P ref (exercise!) The process ( R W a.s. R W n →∞ R W Since R W ≥ 0, the martingale converges: − − − → ∞ n n Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Fix a typical realization of W . Setting P ref = Wiener measure n ( X ) := d P W | F n R W ( X ) d P ref | F n n ) n ∈ N is a martingale w.r.t. P ref (exercise!) The process ( R W a.s. R W n →∞ R W Since R W ≥ 0, the martingale converges: − − − → ∞ n n ◮ P W ≪ P ref if and only if E ref [ R W ∞ ] = 1 (the martingale is UI) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Let ( X t ) t ∈ [0 , 1] be the canonical process on C ([0 , 1] , R ) [ X t ( f ) = f ( t ) ] i : t n 2 n , 0 ≤ i ≤ 2 n ) be the dyadic filtration i Let F n := σ ( X t n i = Fix a typical realization of W . Setting P ref = Wiener measure n ( X ) := d P W | F n R W ( X ) d P ref | F n n ) n ∈ N is a martingale w.r.t. P ref (exercise!) The process ( R W a.s. R W n →∞ R W Since R W ≥ 0, the martingale converges: − − − → ∞ n n ◮ P W ≪ P ref if and only if E ref [ R W ∞ ] = 1 (the martingale is UI) ◮ P W is singular w.r.t. P ref if and only if R W ∞ = 0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity It suffices to show that R W n →∞ 0 in P ⊗ P ref -probability n ( X ) − − − → Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity It suffices to show that R W n →∞ 0 in P ⊗ P ref -probability n ( X ) − − − → Fractional moment � γ � For P ref -a.e. X �� R W n ( X ) − n →∞ 0 − − → for some γ ∈ (0 , 1) E Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity It suffices to show that R W n →∞ 0 in P ⊗ P ref -probability n ( X ) − − − → Fractional moment � γ � For P ref -a.e. X �� R W n ( X ) − n →∞ 0 − − → for some γ ∈ (0 , 1) E 2 n − 1 Z W � ( t n i ) , ( t n � i +1 ) i , X t n i +1 , X t n 1 � R W n ( X ) = Z W � � 2 n ( X t n i +1 − X t n i ) (0 , 0) , (1 , ⋆ ) g 1 i =0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity It suffices to show that R W n →∞ 0 in P ⊗ P ref -probability n ( X ) − − − → Fractional moment � γ � For P ref -a.e. X �� R W n ( X ) − n →∞ 0 − − → for some γ ∈ (0 , 1) E 2 n − 1 Z W � ( t n i ) , ( t n � i +1 ) i , X t n i +1 , X t n 1 � R W n ( X ) = Z W � � 2 n ( X t n i +1 − X t n i ) (0 , 0) , (1 , ⋆ ) g 1 i =0 ◮ Switch from E to equivalent law ˜ E to cancel the denominator Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity It suffices to show that R W n →∞ 0 in P ⊗ P ref -probability n ( X ) − − − → Fractional moment � γ � ˜ R W For P ref -a.e. X �� − − − → for some γ ∈ (0 , 1) E n ( X ) n →∞ 0 2 n − 1 Z W � ( t n i ) , ( t n � i +1 ) i , X t n i +1 , X t n 1 � R W n ( X ) = Z W � � 2 n ( X t n i +1 − X t n i ) (0 , 0) , (1 , ⋆ ) g 1 i =0 ◮ Switch from E to equivalent law ˜ E to cancel the denominator ◮ For fixed X , the Z W � ( t n i ) , ( t n � i , X t n i +1 , X t n i +1 ) ’s are independent We need to exploit translation and scale invariance of their laws Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Lemma 1 (Translation and scale invariance) i +1 − X t n X t n If we set ∆ n i i := we have � t n i +1 − t n i Z W � (0 , 0) , (1 , ∆ n � i ) Z W ( t n i ) , ( t n � � i , X t n i +1 , X t n i +1 ) ˆ β ˆ β d 2 n / 4 = g 1 (∆ n 2 n ( X t n i +1 − X t n i ) i ) g 1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 24 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity Lemma 1 (Translation and scale invariance) i +1 − X t n X t n If we set ∆ n i i := we have � t n i +1 − t n i Z W � (0 , 0) , (1 , ∆ n � i ) Z W ( t n i ) , ( t n � � i , X t n i +1 , X t n i +1 ) ˆ β ˆ β d 2 n / 4 = g 1 (∆ n 2 n ( X t n i +1 − X t n i ) i ) g 1 Lemma 2 (Expansion) For z ∈ R and ε ∈ [0 , 1] (say) Z W � � (0 , 0) , (1 , z ) = 1 + ε X z + ε 2 Y ε, z ε g 1 ( z ) X 2 Y 2 � � � � E [ X z ] = 0 E [ X ε, z ] = 0 E ≤ C E ≤ C unif. in ε, z ε, z z Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 24 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity By Taylor expansion, for fixed γ ∈ (0 , 1) � γ � �� Z W � � (0 , 0) , (1 , z ) �� � γ � ε 1 + ε X z + ε 2 Y ε, z = E E g 1 ( z ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity By Taylor expansion, for fixed γ ∈ (0 , 1) � γ � �� Z W � � (0 , 0) , (1 , z ) �� � γ � ε 1 + ε X z + ε 2 Y ε, z = E E g 1 ( z ) + γ ( γ − 1) ε E [ X z ] + ε 2 E [ Y ε, z ] ε 2 E [( X x ) 2 ] + . . . � � � � = 1 + γ + . . . 2 ( ⋆ ) First order terms vanish ( ⋆ ) γ ( γ − 1) < 0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity By Taylor expansion, for fixed γ ∈ (0 , 1) � γ � �� Z W � � (0 , 0) , (1 , z ) �� � γ � ε 1 + ε X z + ε 2 Y ε, z = E E g 1 ( z ) + γ ( γ − 1) ε E [ X z ] + ε 2 E [ Y ε, z ] ε 2 E [( X x ) 2 ] + . . . � � � � = 1 + γ + . . . 2 = 1 − c ε 2 ≤ e − c ε 2 ( ⋆ ) First order terms vanish ( ⋆ ) γ ( γ − 1) < 0 ( ⋆ ) For some c > 0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of singularity By Taylor expansion, for fixed γ ∈ (0 , 1) � γ � �� Z W � � (0 , 0) , (1 , z ) �� � γ � ε 1 + ε X z + ε 2 Y ε, z = E E g 1 ( z ) + γ ( γ − 1) ε E [ X z ] + ε 2 E [ Y ε, z ] ε 2 E [( X x ) 2 ] + . . . � � � � = 1 + γ + . . . 2 = 1 − c ε 2 ≤ e − c ε 2 ( ⋆ ) First order terms vanish ( ⋆ ) γ ( γ − 1) < 0 ( ⋆ ) For some c > 0 1 We can set z = ∆ n Estimate is uniform over z ∈ R and ε = � i 2 n / 4 � γ � 2 n − 1 �� Z W � (0 , 0) , (1 , ∆ n � i ) ≤ e − c ε 2 2 n = e − c 2 n / 2 � γ � ˜ � ε R W �� E n ( X ) = E g 1 (∆ n i ) i =0 which vanishes as n → ∞ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of Lemma 1 Introducing the dependence on ˆ β d Z W � � = Z W � � ( s , y ) , ( t , x ) (0 , 0) , ( t − s , x − y ) ˆ ˆ β β transl. invariance Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of Lemma 1 Introducing the dependence on ˆ β d Z W � � = Z W � � ( s , y ) , ( t , x ) (0 , 0) , ( t − s , x − y ) ˆ ˆ β β � � �� 1 1 , x d Z W � � √ t Z W (0 , 0) , ( t , x ) = (0 , 0) , √ t ˆ 1 β ˆ β t 4 transl. invariance + diffusive rescaling (prefactor, new ˆ β ) (drawing!) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of Lemma 1 Introducing the dependence on ˆ β d Z W � � = Z W � � ( s , y ) , ( t , x ) (0 , 0) , ( t − s , x − y ) ˆ ˆ β β � � �� 1 1 , x d Z W � � √ t Z W (0 , 0) , ( t , x ) = (0 , 0) , √ t ˆ 1 β ˆ β t 4 transl. invariance + diffusive rescaling (prefactor, new ˆ β ) (drawing!) � = g t ( x ) + ˆ Z W � � g s ( z ) g t − s ( x − z ) W (d s d z ) + . . . (0 , 0) , ( t , x ) β [0 , t ] × R Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43
White noise Continuum partition functions The continuum DPRE Pinning models Proof of Lemma 1 Introducing the dependence on ˆ β d Z W � � = Z W � � ( s , y ) , ( t , x ) (0 , 0) , ( t − s , x − y ) ˆ ˆ β β � � �� 1 1 , x d Z W � � √ t Z W (0 , 0) , ( t , x ) = (0 , 0) , √ t ˆ 1 β ˆ β t 4 transl. invariance + diffusive rescaling (prefactor, new ˆ β ) (drawing!) � = g t ( x ) + ˆ Z W � � g s ( z ) g t − s ( x − z ) W (d s d z ) + . . . (0 , 0) , ( t , x ) β [0 , t ] × R � ˆ � � 1 √ t ) + 1 β √ t ) W (d s d z ) + . . . √ t g 1 ( x t ( z t ( x − z √ t √ t = g s √ t ) g 1 − s [0 , t ] × R Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43
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