Polynomial Chaos and Scaling Limits of Disordered Systems 3. - PowerPoint PPT Presentation

Polynomial Chaos and Scaling Limits of Disordered Systems 3. Marginally relevant models Francesco Caravenna Universit` a degli Studi di Milano-Bicocca YEP XIII, Eurandom ∼ March 7-11, 2016 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 1 / 17

Overview In the previous lectures we focused on systems that are disorder relevant (in particular DPRE with d = 1 and Pinning model with α > 1 2 ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 17

Overview In the previous lectures we focused on systems that are disorder relevant (in particular DPRE with d = 1 and Pinning model with α > 1 2 ) ◮ We constructed continuum partition functions Z W ◮ We used Z W to build continuum disordered models P W ◮ We used Z W to get estimates on the free energy F ( β, h ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 17

Overview In the previous lectures we focused on systems that are disorder relevant (in particular DPRE with d = 1 and Pinning model with α > 1 2 ) ◮ We constructed continuum partition functions Z W ◮ We used Z W to build continuum disordered models P W ◮ We used Z W to get estimates on the free energy F ( β, h ) In this last lecture we consider the subtle marginally relevant regime (in particular DPRE with d = 2, Pinning model with α = 1 2 , 2d SHE) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 17

Overview In the previous lectures we focused on systems that are disorder relevant (in particular DPRE with d = 1 and Pinning model with α > 1 2 ) ◮ We constructed continuum partition functions Z W ◮ We used Z W to build continuum disordered models P W ◮ We used Z W to get estimates on the free energy F ( β, h ) In this last lecture we consider the subtle marginally relevant regime (in particular DPRE with d = 2, Pinning model with α = 1 2 , 2d SHE) We present some results on the the continuum partition function Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 17

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) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 17

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) ◮ Stochastic Heat Equation in d = 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 17

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) ◮ Stochastic Heat Equation in d = 2 ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 17

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) ◮ Stochastic Heat Equation in d = 2 ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n 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 March 7-11, 2016 3 / 17

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) ◮ Stochastic Heat Equation in d = 2 ◮ DPRE with d = 1 (RW with Cauchy tails: P( | S 1 | > n ) ∼ c n 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 For simplicity, we will perform our computations on the pinning model Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 17

The 2 d Stochastic Heat Equation  ∂ t u ( t , x ) = 1 2 ∆ x u ( t , x ) + β W ( t , x ) u ( t , x )  ( t , x ) ∈ [0 , ∞ ) × R 2 u (0 , x ) ≡ 1  where W ( t , x ) is (space-time) white noise on [0 , ∞ ) × R 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 17

The 2 d Stochastic Heat Equation  ∂ t u ( t , x ) = 1 2 ∆ x u ( t , x ) + β W ( t , x ) u ( t , x )  ( t , x ) ∈ [0 , ∞ ) × R 2 u (0 , x ) ≡ 1  where W ( t , x ) is (space-time) white noise on [0 , ∞ ) × R 2 Mollification in space: fix j ∈ C ∞ 0 ( R d ) with � j � L 2 = 1 � � x − y � √ W δ ( t , x ) := R 2 δ j W ( t , y ) d y δ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 17

The 2 d Stochastic Heat Equation  ∂ t u ( t , x ) = 1 2 ∆ x u ( t , x ) + β W ( t , x ) u ( t , x )  ( t , x ) ∈ [0 , ∞ ) × R 2 u (0 , x ) ≡ 1  where W ( t , x ) is (space-time) white noise on [0 , ∞ ) × R 2 Mollification in space: fix j ∈ C ∞ 0 ( R d ) with � j � L 2 = 1 � � x − y � √ W δ ( t , x ) := R 2 δ j W ( t , y ) d y δ � �� �� t δ d 2 β 2 � � 1 Then u δ ( t , x ) = E exp β W 1 ( s , B s ) − d s x √ δ 0 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 17

The 2 d Stochastic Heat Equation  ∂ t u ( t , x ) = 1 2 ∆ x u ( t , x ) + β W ( t , x ) u ( t , x )  ( t , x ) ∈ [0 , ∞ ) × R 2 u (0 , x ) ≡ 1  where W ( t , x ) is (space-time) white noise on [0 , ∞ ) × R 2 Mollification in space: fix j ∈ C ∞ 0 ( R d ) with � j � L 2 = 1 � � x − y � √ W δ ( t , x ) := R 2 δ j W ( t , y ) d y δ � �� �� t δ d 2 β 2 � � 1 Then u δ ( t , x ) = E exp β W 1 ( s , B s ) − d s x √ δ 0 d ≈ Z ω By soft arguments u δ (1 , x ) N (partition function of 2d DPRE) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e � N � = E ref � e ( βω n − λ ( β )) 1 { n ∈ τ } n =1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e � N � N � �� = E ref � e ( βω n − λ ( β )) 1 { n ∈ τ } = E ref � � 1 + X n 1 { n ∈ τ } n =1 n =1 ◮ X n = e βω n − λ ( β ) − 1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e � N � N � �� = E ref � e ( βω n − λ ( β )) 1 { n ∈ τ } = E ref � � 1 + X n 1 { n ∈ τ } n =1 n =1 N � P ref ( n ∈ τ ) X n + � P ref ( n ∈ τ, m ∈ τ ) X n X m + . . . = 1 + n =1 0 < n < m ≤ N ◮ X n = e βω n − λ ( β ) − 1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e � N � N � �� = E ref � e ( βω n − λ ( β )) 1 { n ∈ τ } = E ref � � 1 + X n 1 { n ∈ τ } n =1 n =1 N � P ref ( n ∈ τ ) X n + � P ref ( n ∈ τ, m ∈ τ ) X n X m + . . . = 1 + n =1 0 < n < m ≤ N ◮ X n = e βω n − λ ( β ) − 1 ≈ β Y n with Y n ∼ N (0 , 1) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Recall what we did for α > 1 2 (for simplicity h = 0) e H ω � N Z ω N = E ref � = E ref � n =1 ( βω n − λ ( β )) 1 { n ∈ τ } � N � e � N � N � �� = E ref � e ( βω n − λ ( β )) 1 { n ∈ τ } = E ref � � 1 + X n 1 { n ∈ τ } n =1 n =1 N � P ref ( n ∈ τ ) X n + � P ref ( n ∈ τ, m ∈ τ ) X n X m + . . . = 1 + n =1 0 < n < m ≤ N ◮ X n = e βω n − λ ( β ) − 1 ≈ β Y n with Y n ∼ N (0 , 1) c ◮ P ref ( n ∈ τ ) ∼ n 1 − α Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 17

Pinning in the relevant regime α > 1 2 Y n Y n Y m Z ω � n 1 − α + β 2 � N = 1 + β n 1 − α ( m − n ) 1 − α + . . . 0 < n ≤ N 0 < n < m ≤ N Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 17