Scaling limit of the two-dimensional dynamic Ising-Kac model

Transcription

1 Scaling limit of the two-dimensional dynamic Ising-Kac model Jean-Christophe Mourrat Hendrik Weber Mathematics Institute University of Warwick Statistical Mechanics Seminar, Warwick

2 A famous result Theorem (Bertini-Giacomin 97) ζ(t, x) = height field associated to a weakly asymmetric simple exclusion process. p.2

3 A famous result Theorem (Bertini-Giacomin 97) ζ(t, x) = height field associated to a weakly asymmetric simple exclusion process. Rescaled field: X ε (t, x) = εζ(ε 2 t, ε 1 x) v ε. Average speed: v ε = 1 2ε p.2

4 A famous result Theorem (Bertini-Giacomin 97) ζ(t, x) = height field associated to a weakly asymmetric simple exclusion process. Rescaled field: X ε (t, x) = εζ(ε 2 t, ε 1 x) v ε. Average speed: v ε = 1 2ε X ε solution of the KPZ equation (w.r.t D(R +, C(R)) topology). t h = x h 1 ( x h) 2( 2 ) + ξ. p.2

5 A famous result Theorem (Bertini-Giacomin 97) ζ(t, x) = height field associated to a weakly asymmetric simple exclusion process. Rescaled field: X ε (t, x) = εζ(ε 2 t, ε 1 x) v ε. Average speed: v ε = 1 2ε X ε solution of the KPZ equation (w.r.t D(R +, C(R)) topology). t h = x h 1 ( x h) 2( 2 ) + ξ. Scaling leaves stochastic heat equation invariant. Gives interpretation for infinite renormalisation in KPZ. Particle model changes with the scaling (asymmetry). p.2

6 Kac-Ising model I Grid: Spin configurations: Λ = Z d /(2N + 1)Z d { N,..., N} d, σ(k) { 1, 1} for k Λ, p.3

7 Kac-Ising model I Grid: Spin configurations: Kac-interaction: Hamiltonian: Λ = Z d /(2N + 1)Z d { N,..., N} d, σ(k) { 1, 1} for k Λ, κ γ (k) γ d K(γk), H (σ) = 1 2 k,l κ γ(k l)σ(k) σ(l), p.3

8 Kac-Ising model I Grid: Spin configurations: Kac-interaction: Hamiltonian: Gibbs measure: Λ = Z d /(2N + 1)Z d { N,..., N} d, σ(k) { 1, 1} for k Λ, κ γ (k) γ d K(γk), H (σ) = 1 2 k,l κ γ(k l)σ(k) σ(l), λ β (σ) exp ( βh (σ) ). p.3

9 Kac-Ising model II - Glauber dynamics Gibbs measure: λ β (σ) exp ( βh (σ) ), p.4

10 Kac-Ising model II - Glauber dynamics Gibbs measure: λ β (σ) exp ( βh (σ) ), Jump rate: c γ (σ, j) := λ β (σ j ) λ β (σ)+λ β (σ j ), p.4

11 Kac-Ising model II - Glauber dynamics Gibbs measure: λ β (σ) exp ( βh (σ) ), Jump rate: Generator: c γ (σ, j) := λ β (σ j ) λ β (σ)+λ β (σ j ), L γ f (σ) = j Λ N c γ (σ, j) ( f (σ j ) f (σ) ). p.4

12 Kac-Ising model II - Glauber dynamics Gibbs measure: λ β (σ) exp ( βh (σ) ), Jump rate: Generator: c γ (σ, j) := λ β (σ j ) λ β (σ)+λ β (σ j ), L γ f (σ) = j Λ N c γ (σ, j) ( f (σ j ) f (σ) ). Defines reversible Markov process. p.4

13 Kac -Ising III Introduced in 60s. (e.g. Kac 1963, Lebowitz-Penrose 1966) Very detailed study by Presutti et al. in 90s. Study hydrodynamic limit, linear fluctuations,... p.5

14 Kac -Ising III Introduced in 60s. (e.g. Kac 1963, Lebowitz-Penrose 1966) Very detailed study by Presutti et al. in 90s. Study hydrodynamic limit, linear fluctuations,... Aim: Describe non-linear fluctuations dynamical near criticality in dimension d = 2! Local averages described by SPDE t X = X X 3 + AX + ξ. p.5

15 Coarse-grained field and rescaling I Local averages: h γ (σ(t), k) = l Λ κ γ(l k) σ(t, l). Dynamic can be described in terms of h γ : H γ (σ) = 1 σ(k) h γ (σ, k), 2 k Λ N c γ (σ, j) = 1 (1 σ(j) tanh ( βh γ (σ, j) ) ). 2 }{{} βh γ(σ,j) 1 3 β3 h γ(σ,j) 3 p.6

16 Coarse-grained field and rescaling II Evolution equation: t h γ (t, k) = h γ (0, k) + L γ h γ (s, k) ds + m γ (s, k), 0 where m γ (, k) martingale. p.7

17 Coarse-grained field and rescaling II Evolution equation: t h γ (t, k) = h γ (0, k) + L γ h γ (s, k) ds + m γ (s, k), 0 where m γ (, k) martingale. Generator: L γ h γ (σ, ) = ( ) κ γ h γ (σ, ) h γ (σ, ) + (β 1)κ γ h γ (σ, ) β3 ( κ γ h γ (σ, ) 3) +..., 3 p.7

18 Coarse-grained field and rescaling II Evolution equation: t h γ (t, k) = h γ (0, k) + L γ h γ (s, k) ds + m γ (s, k), 0 where m γ (, k) martingale. Generator: L γ h γ (σ, ) = ( ) κ γ h γ (σ, ) h γ (σ, ) + (β 1)κ γ h γ (σ, ) β3 ( κ γ h γ (σ, ) 3) +..., 3 Predictable quadratic variation: t ( ) m γ (, k), m γ (, j) t = 4 κ γ (k l) κ γ (j l) c γ σ(s), l ds. 0 }{{} l Λ N = p.7

19 Coarse-grained field and rescaling III Rescaled dynamics: X γ (t, x) = 1 δ h γ( t α, x ε ). p.8

20 Coarse-grained field and rescaling III Rescaled dynamics: X γ (t, x) = 1 δ h γ( t α, x ε Evolution equation: X γ (t, x) =X γ (0, x) + t 0 ( ε 2 γ 2 1 α γx γ (s, x) + 1 β 3 δ 2 3 α K γ ε Xγ 3 (s, x) +... ). (β 1) K γ ε X γ (s, x) α ) ds + M γ (t, x). p.8

21 Coarse-grained field and rescaling III Rescaled dynamics: X γ (t, x) = 1 δ h γ( t α, x ε Evolution equation: X γ (t, x) =X γ (0, x) + t 0 ( ε 2 γ 2 1 α γx γ (s, x) + 1 β 3 δ 2 3 α K γ ε Xγ 3 (s, x) +... Quadratic variation: M γ (, x),m γ (, y) t = 4 εd δ 2 α t 0 ). (β 1) K γ ε X γ (s, x) α ) ds + M γ (t, x). ε d ( ) K γ (x z) K γ (y z) C γ s, z ds. z Λ ε p.8

22 Coarse-grained field and rescaling III Rescaled dynamics: X γ (t, x) = 1 δ h γ( t α, x ε Evolution equation: X γ (t, x) =X γ (0, x) + t 0 ( ε 2 γ 2 1 α γx γ (s, x) + 1 β 3 δ 2 3 α K γ ε Xγ 3 (s, x) +... Quadratic variation: ). (β 1) K γ ε X γ (s, x) α ) ds + M γ (t, x). M γ (, x),m γ (, y) t = 4 εd δ 2 α Non-trivial scaling: ε = γ 4 4 d t 0 δ = γ d 4 d ε d ( ) K γ (x z) K γ (y z) C γ s, z ds. z Λ ε α = γ 2d 4 d p.8

23 Coarse-grained field and rescaling III Rescaled dynamics: X γ (t, x) = 1 δ h γ( t α, x ε Evolution equation: X γ (t, x) =X γ (0, x) + t 0 ( ε 2 γ 2 1 α γx γ (s, x) + 1 β 3 δ 2 3 α K γ ε Xγ 3 (s, x) +... Quadratic variation: ). (β 1) K γ ε X γ (s, x) α ) ds + M γ (t, x). M γ (, x),m γ (, y) t = 4 εd δ 2 α Non-trivial scaling: ε = γ 4 4 d t 0 δ = γ d 4 d ε d ( ) K γ (x z) K γ (y z) C γ s, z ds. z Λ ε α = γ 2d 4 d (β 1) α??? p.8

24 Scaling limit For d = 1 result known: Theorem (Bertini-Presutti-Rüdiger-Saada 93, Fritz-Rüdiger 95) If d = 1 and ε = γ 4 3 δ = γ 1 3 α = γ 2 3 (β 1) = Cα, then X γ converges in law to the solution of the stochastic PDE t X = X 1 3 X 3 + CX + 2ξ, where ξ = space-time white noise. p.9

25 Main result Theorem (Mourrat, W ) If d = 2 and ε = γ 2 δ = γ α = γ 2 (β 1) α = c γ + A p.10

26 Main result Theorem (Mourrat, W ) If d = 2 and ε = γ 2 δ = γ α = γ 2 (β 1) = c γ + A α c γ = 1 4π 2 k 2 + O(1). k Z 2 0< k <γ 1 p.10

27 Main result Theorem (Mourrat, W ) If d = 2 and ε = γ 2 δ = γ α = γ 2 (β 1) = c γ + A α c γ = 1 4π 2 k 2 + O(1). k Z 2 0< k <γ 1 if X 0 γ converges in B ν,, ν > 0 small enough, p.10

28 Main result Theorem (Mourrat, W ) If d = 2 and ε = γ 2 δ = γ α = γ 2 (β 1) = c γ + A α c γ = 1 4π 2 k 2 + O(1). k Z 2 0< k <γ 1 if X 0 γ converges in B ν,, ν > 0 small enough, then X γ converges in law (w.r.t. D(R +, B ν κ, ) topology) to the solution of the stochastic PDE t X = X 1 3 (X 3 X ) + AX + 2ξ. p.10

29 Discussion I Infinite normalisation constant is shift of temperature away from critical temperature for mean field. p.11

30 Discussion I Infinite normalisation constant is shift of temperature away from critical temperature for mean field. Kac-interaction plays role of weak asymmetry in Bertini-Giacomin s result. p.11

31 Discussion I Infinite normalisation constant is shift of temperature away from critical temperature for mean field. Kac-interaction plays role of weak asymmetry in Bertini-Giacomin s result. Coarse graining h γ defined on the same scale as interaction. A posteriori X γ (t, ϕ) = x Λ ε ε 2 ϕ(x) δ 1 σ(t/α, x/ε), converges to same limit with respect to D(R +, S ) topology. p.11

32 Discussion I Infinite normalisation constant is shift of temperature away from critical temperature for mean field. Kac-interaction plays role of weak asymmetry in Bertini-Giacomin s result. Coarse graining h γ defined on the same scale as interaction. A posteriori X γ (t, ϕ) = x Λ ε ε 2 ϕ(x) δ 1 σ(t/α, x/ε), converges to same limit with respect to D(R +, S ) topology. Higher dimensions: In principle scaling works also for d = 3. Hairer / Catellier-Chouk have stable notion of solutions for SPDE. For d = 4 scaling and SPDE theory break down. p.11

33 Discussion II Literature: Conjectured by Giacomin-Lebowitz-Presutti 99. In static case: trivial limit without renormalisation Cassandro-Marra-Presutti 95, 97. Glimm-Jaffe-Spencer: Phase transition for static Φ Mass parameter A plays role of inverse temperature. Construction of da Prato-Debussche 03 crucial for our argument. p.12

34 Renormalisation of Φ 4 2 t X δ = X δ ( X 3 δ C δx δ ) + ξδ. ξ δ (t, x) := k 1 e ik x β(t, k). Noise white in time, spatial δ corellations δ. p.13

35 Renormalisation of Φ 4 2 t X δ = X δ ( X 3 δ C δx δ ) + ξδ. ξ δ (t, x) := k 1 e ik x β(t, k). Noise white in time, spatial δ corellations δ. Theorem (da Prato/Debussche 03) d = 2 and C δ = C 1 log(δ 1 ) Then X δ converge to a limit X. Formally t X = X ( X 3 X ) + ξ, or dynamic φ 4 2 model. p.13

36 Renormalisation of Φ 4 2 t X δ = X δ ( X 3 δ C δx δ ) + ξδ. ξ δ (t, x) := k 1 e ik x β(t, k). Noise white in time, spatial δ corellations δ. Theorem (da Prato/Debussche 03) d = 2 and C δ = C 1 log(δ 1 ) Then X δ converge to a limit X. Formally t X = X ( X 3 X ) + ξ, or dynamic φ 4 2 model. In d = 3 recently by Hairer. Then C δ = C 1 δ + C 2 log(δ 1 ). p.13

37 Regularity of the linear part Stochastic heat equation: t Z δ = Z δ + 2ξ δ on [0, ) T d. Solution Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). p.14

38 Regularity of the linear part Stochastic heat equation: t Z δ = Z δ + 2ξ δ on [0, ) T d. Solution Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). Sλ x η = scaled testfunction E Z δ (t, ), S λ x η 2 = t 0 T d unif. in δ ( t ) 2 K δ (t s, x y)s λ x η(x) ds dy 0 T d log(λ 1 ) if d = 2, λ d 2 if d 3. p.14

39 Regularity of the linear part Stochastic heat equation: t Z δ = Z δ + 2ξ δ on [0, ) T d. Solution Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). Sλ x η = scaled testfunction E Z δ (t, ), S λ x η 2 = t 0 T d unif. in δ ( t ) 2 K δ (t s, x y)s λ x η(x) ds dy 0 T d log(λ 1 ) if d = 2, λ d 2 if d 3. Gaussian equivalent bounds in L p regularity estimates. p.14

40 Powers of the solutions Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). Itô formula: Z δ (t, x) 3 = W (3) ([0,t] T) 3 δ (t, x; z 1, z 2, z 3 ) ξ(dz 1 ) ξ(dz 2 ) ξ(dz 3 ) + 3Z δ (t, x) K δ (t s, x y) 2 ds dy. ([0,t] T) } {{ } C δ p.15

41 Powers of the solutions Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). Itô formula: Z δ (t, x) 3 = W (3) ([0,t] T) 3 δ (t, x; z 1, z 2, z 3 ) ξ(dz 1 ) ξ(dz 2 ) ξ(dz 3 ) + 3Z δ (t, x) K δ (t s, x y) 2 ds dy. ([0,t] T) } {{ } C δ Expectations of first term tested against scaled test functions can be bounded uniformly in δ for d = 1, 2, (3). Second term diverges for d = 2, 3. p.15

42 Powers of the solutions Z δ (t, x) = t 0 T d K δ (t s, x y) ξ(ds, dy). Itô formula: Z δ (t, x) 3 = W (3) ([0,t] T) 3 δ (t, x; z 1, z 2, z 3 ) ξ(dz 1 ) ξ(dz 2 ) ξ(dz 3 ) + 3Z δ (t, x) K δ (t s, x y) 2 ds dy. ([0,t] T) } {{ } C δ Expectations of first term tested against scaled test functions can be bounded uniformly in δ for d = 1, 2, (3). Second term diverges for d = 2, 3. Nelson estimate equivalent bounds in L p regularity estimates. p.15

43 Construction of the model": For d = 2 Lemma For every p 1, ν > 0 Z δ Z, Z 2 δ C δ Z : 2:, Z 3 δ 3C δz δ Z : 3:, in L p (C([0, T ], B ν, (T 2 ))). Z : n : are called Wick powers. Key ingredient: Equivalence of moments in fixed Wiener chaos - Nelson estimate. Regularity measured in Besov spaces. p.16

44 Analysis of linearised particle system I Evolution equation: X γ (t, x) X γ (0, x) + M γ (t, x) + t 0 [ γ X γ (s, x) 1 )] 3 K γ ε (X γ 3 (s, x) C 1 log(γ 1 )X γ (s, x) + Err ds. Linearised system: R γ,t (s, x) := s r=0 Pγ t r dm γ(r, x) where P γ t = e t γ. Approximate Wick powers: R :n: γ,t (s, x) = n s r=0 R:n 1: γ,t (r, x) dr γ,t (r, x). p.17

45 Analysis of linearised particle system II R γ,t (s, x) := s r=0 Pγ t r dm γ(r, x) R :n: γ,t (s, x) = n s Lemma r=0 R:n 1: γ,t (r, x) dr γ,t (r, x). For every n, the process Z γ :n: (t) := R γ,t :n: (t, ) converges in distribution to Z : n : with respect to D(R +, B, (T ν 2 )) topology. In tightness proof Nelson estimate replaced by martingale inequality (BDG). D(...) = space of càdlàg functions. p.18

46 A Lemma for iterated integrals ( E sup 0 r t t r 1 =0 (R γ,t(r, :n: x), f (x)) ) 2 p p rn 1... r n=0 ε 2n (W (n) (, r, z), f ( )) 2 dr + Err, z Λ n ε p.19

47 A Lemma for iterated integrals where ( E sup 0 r t t r 1 =0 (R γ,t(r, :n: x), f (x)) ) 2 p p rn 1... r n=0 ε 2n (W (n) (, r, z), f ( )) 2 dr + Err, z Λ n ε W (n) (, r, z) = n P γ t r j K γ ( z j ), j=1 p.19

48 A Lemma for iterated integrals where and ( E sup 0 r t t r 1 =0 Err =C ε 4 δ 2 ( E sup (R γ,t(r, :n: x), f (x)) ) 2 p p rn 1... r n=0 ε 2n (W (n) (, r, z), f ( )) 2 dr + Err, z Λ n ε W (n) (, r, z) = n t l=1 0 r l r l 1 z l Λ ε r 1 =0 n P γ t r j K γ ( z j ), j=1 rl 2... r l 1 =0 ( f ( ), R γ,t :n l: (r l, ) z 1,...,z l 1 Λ ε ε 2(l 1) l P γ t r j K γ ( z j ) ) ) 2 p p drl 1. j=1 p.19

49 Da Prato-Debussche Strategy II 2.) Non-linear evolution as continuous function of lifted Gaussian process Regularisation trick: v δ = X δ Z δ. t v = v δ ( (Z δ + v δ ) 3 3 C δ (Z δ + v δ ) ) = v δ ( Z δ : 3: + 3Z δ : 2: v δ + 3Z δ vδ 2 + vδ 3 ). Multiplicative inequality: If α < 0 < β with α + β > 0 u v B α, u B α, v B β,. Used to deal with nonlinearity. Can be repeated on level of X γ. p.20

50 Conclusion Prove conjecture about two dimensional Kac-Ising model near criticality. p.21

51 Conclusion Prove conjecture about two dimensional Kac-Ising model near criticality. The need for renormalisation for the SPDE is reflected in choice of scaling. p.21

52 Conclusion Prove conjecture about two dimensional Kac-Ising model near criticality. The need for renormalisation for the SPDE is reflected in choice of scaling. Modern approach to irregular SPDE useful 1.) Treat a linearised version with probabilistic techniques. 2.) Treat the non-linear system as a perturbation. p.21

53 Conclusion Prove conjecture about two dimensional Kac-Ising model near criticality. The need for renormalisation for the SPDE is reflected in choice of scaling. Modern approach to irregular SPDE useful 1.) Treat a linearised version with probabilistic techniques. 2.) Treat the non-linear system as a perturbation. Core part of proof: Convergence of a linearised particle system and nonlinear functions thereof. p.21