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 ) ◮ 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) ◮ 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 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 � 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 � 2 β Y t � β Y s Y t � � = 1 + t 1 − α + s 1 − α ( t − s ) 1 − α + . N 1 − α N 1 − α t ∈ (0 , 1] ∩ Z s < t ∈ (0 , 1] ∩ Z N N � N has cells with volume 1 β 1 Lattice Z N , hence if N 1 − α ≈ N that is ˆ β β = N α − 1 2 We obtain � 1 d W t � d W s d W t d 1 + ˆ t 1 − α + ˆ Z ω β 2 − − − − → β s 1 − α ( t − s ) 1 − α + . . . N N →∞ 0 0 < s < t < 1 What happens for α = 1 √ t �∈ L 2 1 2 ? Stochastic integrals ill-defined: loc . . . Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 17
The marginal regime α = 1 2 Y n Y n Y m � � Z ω √ n + β 2 √ n √ m − n + . . . N = 1 + β 0 < n ≤ N 0 < n < m ≤ N Goal: find the joint limit in distribution of all these sums Linear term is easy ( Y n ∼ N (0 , 1) by Lindeberg): asympt. N (0 , σ 2 ) 1 σ 2 = β 2 � n ∼ β 2 log N 0 < n ≤ N ˆ β We then rescale β = β N ∼ √ log N Other terms converge? Interestingly, every sum gives contribution 1 to the variance! 1 β 2 + ˆ β 4 + . . . = = 1 + ˆ ˆ Z ω � � V ar blows up at β = 1! N 1 − ˆ β 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 17
Scaling limit of marginal partition function Theorem 1. [C., Sun, Zygouras ’15b] Pinning α = 1 Consider DPRE d = 2 or or 2d SHE 2 (or long-range DPRE with d = 1 and Cauchy tails) ˆ β Rescaling β := √ log N (and h ≡ 0) the partition function converges in � if ˆ log-normal β < 1 Z W = d Z ω law to an explicit limit: − − − − → N if ˆ β ≥ 1 0 N →∞ � β W 1 − 1 � 1 d Z W 2 σ 2 = exp σ ˆ with σ ˆ β = log ˆ β 1 − ˆ β 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 17
Multi-scale correlations for ˆ β < 1 Define Z ω N ( t , x ) as partition function for rescaled RW starting at ( t , x ) e H ω ( S ) � Z ω N ( t , x ) = E ref � � S δ � t = x √ [ δ = 1 where { S δ t = x } = { S Nt = Nx } N ] Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 9 / 17
Multi-scale correlations for ˆ β < 1 Theorem 2. [C., Sun, Zygouras ’15b] (fix ˆ Consider DPRE with d = 2 or 2d SHE β < 1) Z ω N (X) and Z ω N (X ′ ) are asymptotically independent for fixed X � = X ′ More generally, if X = ( t N , x N ) and X ′ = ( t ′ N , x ′ N ) are such that 1 N | 2 ∼ d ( X , X ′ ) := | t N − t ′ N | + | x N − x ′ ζ ∈ [0 , 1] N 1 − ζ 2 V ar[ Y ] , e Y ′ − 1 d e Y − 1 2 V ar[ Y ′ ] � N (X ′ ) � Z ω N (X) , Z ω � − − − − → � then N →∞ = log 1 − ζ ˆ β 2 Y , Y ′ joint N (0 , σ 2 Y , Y ′ � � β ) with C ov ˆ 1 − ˆ β 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 17
Multi-scale correlations for ˆ β < 1 N against a test function φ ∈ C 0 ([0 , 1] × R 2 ) We can integrate Z ω � � Z ω [0 , 1] × R 2 φ ( t , x ) Z ω N , φ � := N ( t , x ) d t d x ≃ 1 � φ ( t , x ) Z ω N ( t , x ) N 2 t ∈ [0 , 1] ∩ Z N , x ∈ ( Z N ) 2 √ Corollary � Z ω N , φ � → � 1 , φ � in probability as N → ∞ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 17
Fluctuations for ˆ β < 1 Theorem 3. [C., Sun, Zygouras ’15b] (fix ˆ Consider DPRE with d = 2 or 2d SHE β < 1) 1 Z ω (in S ′ ) N ( t , x ) ≈ 1 + √ log N G ( t , x ) where G(t,x) is a generalized Gaussian field on [0 , 1] × R 2 with 1 G (X) , G (X ′ ) � � ∼ C log C ov � X − X ′ � Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 17
The regime ˆ β = 1 (in progress) For ˆ Z ω V ar[ Z ω β = 1 : N ( t , x ) → 0 in law N ( t , x )] → ∞ However, covariances are finite: cf. [Bertini, Cancrini 95] C ov[ Z ω N ( t , x ) , Z ω N ( t ′ , x ′ )] ( t , x ) , ( t ′ , x ′ ) ∼ � � < ∞ N →∞ K where 1 ( t , x ) , ( t ′ , x ′ ) � � K ≈ log | ( t , x ) − ( t ′ , x ′ ) | Then � Z ω � � V ar N , φ � → ( φ, K φ ) < ∞ Conjecture For ˆ β = 1 the partition function Z ω N ( t , x ) has a non-trivial limit in law, viewed as a random Schwartz distribution in ( t , x ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 17
Proof of Theorem 1. for pinning N Y n 1 Y n 2 · · · Y 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 � 2 � ˆ ˆ β Y n β Y n Y n ′ � � √ log N √ n √ log N √ = 1 + + √ n + . . . n ′ − n 0 < n ≤ N 0 < n < n ′ ≤ N Goal: find the joint limit in distribution of all these sums � blackboard! Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 17
Fourth moment theorem 4th Moment Theorem [de Jong 90] [Nualart, Peccati, Reinert 10] � � Consider homogeneous (deg. ℓ ) polynomial chaos Y N = ψ N ( I ) Y i | I | = ℓ i ∈ I ◮ max i ψ N ( i ) − − − − → 0 (in case ℓ = 1) [Small influences!] N →∞ ◮ E [( Y N ) 2 ] − σ 2 − − − → N →∞ ◮ E [( Y N ) 4 ] − 3 σ 4 − − − → N →∞ d N (0 , σ 2 ) − − − − → Then Y N N →∞ Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 15 / 17
References ◮ L. Bertini, N. Cancrini The two-dimensional stochastic heat equation: renormalizing a multiplicative noise J. Phys. A: Math. Gen. 31 (1998) 615–622 ◮ F. Caravenna, R. Sun, N. Zygouras Universality in marginally relevant disordered systems preprint (2015) ◮ P. de Jong A central limit theorem for generalized multilinear forms J. Multivariate Anal. 34 (1990), 275–289 ◮ I. Nourdin, G. Peccati, G. Reinert Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos Ann. Probab. 38 (2010) 1947–1985 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 17
Collaborators Nikos Zygouras (Warwick) and Rongfeng Sun (NUS) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 17
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