Uniqueness of random gradient states

Transcription

1 p. 1 Uniqueness of random gradient states Codina Cotar (Based on joint work with Christof Külske)

2 p. 2 Model Interface transition region that separates different phases

3 p. 2 Model Interface transition region that separates different phases Finite set Λ Z d

4 p. 2 Model Interface transition region that separates different phases Finite set Λ Z d Height Variables φ: Λ R

5 p. 2 Model Interface transition region that separates different phases Finite set Λ Z d Height Variables φ: Λ R Boundary Λ with boundary condition ψ, such that φ x = ψ x, when x Λ.

6 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 p. 3

7 p. 3 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 Hamiltonian H ψ Λ (φ) = x Λ,y Λ, x y =1 V (φ x φ y )+2 x Λ,y Λ, x y =1 V (φ x φ y )

8 p. 3 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 Hamiltonian H ψ Λ (φ) = x Λ,y Λ, x y =1 V (φ x φ y )+2 x Λ,y Λ, x y =1 V (φ x φ y ) Bonds b = (x, y), x Λ, y Λ Λ, x y = 1

9 p. 3 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 Hamiltonian H ψ Λ (φ) = x Λ,y Λ, x y =1 V (φ x φ y )+2 x Λ,y Λ, x y =1 V (φ x φ y ) Bonds b = (x, y), x Λ, y Λ Λ, x y = 1 Gradients ηb = φ b = φ x φ y for b = (x, y)

10 p. 3 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 Hamiltonian H ψ Λ (φ) = x Λ,y Λ, x y =1 V (φ x φ y )+2 x Λ,y Λ, x y =1 V (φ x φ y ) Bonds b = (x, y), x Λ, y Λ Λ, x y = 1 Gradients ηb = φ b = φ x φ y for b = (x, y) β = 1 T > 0, where T is the temperature

11 p. 3 Potentials V : R R, even, diverges at ± Example: Gaussian case: V (s) = s 2 Hamiltonian H ψ Λ (φ) = x Λ,y Λ, x y =1 V (φ x φ y )+2 x Λ,y Λ, x y =1 V (φ x φ y ) Bonds b = (x, y), x Λ, y Λ Λ, x y = 1 Gradients ηb = φ b = φ x φ y for b = (x, y) β = 1 T > 0, where T is the temperature Finite volume surface tension σλ (u): macroscopic energy of a surface with tilt u = (u 1,..., u d ) R d.

12 p. 4 φ-gibbs Measure The finite volume Gibbs measure in Λ ν ψ Λ (dφ) = 1 Z ψ Λ exp { βh ψλ (φ) } x Λ dφ x.

13 p. 4 φ-gibbs Measure The finite volume Gibbs measure in Λ ν ψ Λ (dφ) = 1 Z ψ Λ exp { βh ψλ (φ) } x Λ dφ x. The probability measure ν P(R Z d ) is called a Gibbs measure for the φ-field if ν( F Λ c)(ψ) = ν ψ Λ ( ), ν a.e. ψ, for every Λ Z d, where F Λ c is the σ-field of R Zd generated by {φ(x) : x / Λ}.

14 p. 5 φ-gibbs Measures Gibbs measures do not exist for d = 1, 2.

15 p. 5 φ-gibbs Measures Gibbs measures do not exist for d = 1, 2. The Hamiltonian is a functional of the gradient fields φ, so we can consider the distribution of the φ-field under the Gibbs measure ν. We call this measure the φ-gibbs measure µ.

16 p. 5 φ-gibbs Measures Gibbs measures do not exist for d = 1, 2. The Hamiltonian is a functional of the gradient fields φ, so we can consider the distribution of the φ-field under the Gibbs measure ν. We call this measure the φ-gibbs measure µ. φ-gibbs measures exist for d 1.

17 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u).

18 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u). 2. Existence of shift-invariant φ- Gibbs measure µ.

19 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u). 2. Existence of shift-invariant φ- Gibbs measure µ. 3. Uniqueness of shift-invariant φ- Gibbs measure µ under additional assumptions on µ.

20 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u). 2. Existence of shift-invariant φ- Gibbs measure µ. 3. Uniqueness of shift-invariant φ- Gibbs measure µ under additional assumptions on µ. 4. Decay of covariances with respect to µ.

21 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u). 2. Existence of shift-invariant φ- Gibbs measure µ. 3. Uniqueness of shift-invariant φ- Gibbs measure µ under additional assumptions on µ. 4. Decay of covariances with respect to µ. 5. Central limit theorem (CLT) results.

22 p. 6 Questions 1. Existence and (strict) convexity of the surface tension σ(u) = lim Λ σ Λ(u). 2. Existence of shift-invariant φ- Gibbs measure µ. 3. Uniqueness of shift-invariant φ- Gibbs measure µ under additional assumptions on µ. 4. Decay of covariances with respect to µ. 5. Central limit theorem (CLT) results. 6. Large deviations (LDP) results.

23 Results: Strictly Convex Potentials p Funaki-Spohn (CMP, 1997): Existence and convexity of the surface tension for d 1.

24 Results: Strictly Convex Potentials p Funaki-Spohn (CMP, 1997): Existence and convexity of the surface tension for d Deuschel-Giacomin-Ioffe (PTRF, 2000)/Giacomin-Olla-Spohn (AOP, 2001): Strict convexity of the surface tension for d 1.

25 Results: Strictly Convex Potentials p Funaki-Spohn (CMP, 1997): Existence and convexity of the surface tension for d Deuschel-Giacomin-Ioffe (PTRF, 2000)/Giacomin-Olla-Spohn (AOP, 2001): Strict convexity of the surface tension for d Funaki-Spohn (CMP, 1997): Assume C 1 V C 2. For every u = (u 1,..., u d ) R d there exists a unique shift-invariant φ- Gibbs measure µ with E µ [φ(e i ) φ(0)] = u i, for all i = 1,..., d.

26 p. 8 Results: Strictly Convex Potentials 4. Naddaf-Spencer (CMP, 1997)/Giacomin-Olla-Spohn (AOP, 2001): CLT ǫ d ǫ 0 2 f(ǫx)[φ x+ei φ x u i ] N(0, σu(f)). 2 x Z d

27 p. 8 Results: Strictly Convex Potentials 4. Naddaf-Spencer (CMP, 1997)/Giacomin-Olla-Spohn (AOP, 2001): CLT ǫ d ǫ 0 2 f(ǫx)[φ x+ei φ x u i ] N(0, σu(f)). 2 x Z d 5. Deuschel-Giacomin-Ioffe (PTRF, 2000): LDP

28 p. 9 Techniques Brascamp-Lieb Inequality: for all x Λ ν ψ Λ var ν ψ(φ x ) var Λ ν ψ(φ x ), Λ is the Gibbs measure with potential Ṽ (s) = C 1 s 2.

29 p. 9 Techniques Brascamp-Lieb Inequality: for all x Λ ν ψ Λ var ν ψ(φ x ) var Λ ν ψ(φ x ), Λ is the Gibbs measure with potential Ṽ (s) = C 1 s 2. Random Walk Representation: Representation of the Covariance Matrix in terms of the Green function of a particular random walk.

30 p. 10 Results: Non-convex potentials Funaki-Spohn (CMP, 1997): Convexity of the surface tension for all potentials V.

31 p. 10 Results: Non-convex potentials Funaki-Spohn (CMP, 1997): Convexity of the surface tension for all potentials V. Cotar-Deuschel-Müller (CMP, 2009): Surface tension is strictly convex if V = V 0 + g, C 1 V 0 C 2, C 0 g < 0, β g L1 (R) small (C 0, C 1, C 2 ).

32 p. 10 Results: Non-convex potentials Funaki-Spohn (CMP, 1997): Convexity of the surface tension for all potentials V. Cotar-Deuschel-Müller (CMP, 2009): Surface tension is strictly convex if V = V 0 + g, C 1 V 0 C 2, C 0 g < 0, β g L1 (R) small (C 0, C 1, C 2 ). Adams-Kotecký-Müller (preprint, 2011): Strict convexity of the surface tension for small tilts u and large β.

33 p. 11 For the potential Non-Convex potentials e V (s) = pe k 1 s2 2 + (1 p)e k 2 s 2 2 V(s) 0 s

34 Biskup-Kotecký (PTRF, 2007): Non-uniqueness of several φ-gibbs measures with E µ [φ(e i ) φ(0)] = 0, but with different variances. p. 12

35 p. 12 Biskup-Kotecký (PTRF, 2007): Non-uniqueness of several φ-gibbs measures with E µ [φ(e i ) φ(0)] = 0, but with different variances. Biskup-Spohn (AOP, 2011): CLT for every ergodic φ-gibbs measure µ with E µ [φ(e i ) φ(0)] = 0.

36 p. 13 Non-Convex potentials Cotar-Deuschel (AIHP, to appear): Let u R d. Assume V = V 0 + g, with C 1 V 0 C 2 and < g < 0. If β g L1 (R) small (C 1, C 2 ), there exists exactly one shift-invariant φ-gibbs measure µ such that E µ [φ(e i ) φ(0)] = u i for all i = 1, 2,..., d.

37 p. 13 Non-Convex potentials Cotar-Deuschel (AIHP, to appear): Let u R d. Assume V = V 0 + g, with C 1 V 0 C 2 and < g < 0. If β g L1 (R) small (C 1, C 2 ), there exists exactly one shift-invariant φ-gibbs measure µ such that E µ [φ(e i ) φ(0)] = u i for all i = 1, 2,..., d.

38 p. 14 Interfaces with Disorder Two ways to add randomness (a) Add to the Hamiltonian the random part ξ x φ x x Λ (ξ x ) x Z d i.i.d with E P (ξ 2 x) <. Or

39 p. 14 Interfaces with Disorder Two ways to add randomness (a) Add to the Hamiltonian the random part ξ x φ x x Λ (ξ x ) x Z d i.i.d with E P (ξ 2 x) <. Or (b) Make the potentials random: V x,y i.i.d with As 2 B (x,y) V (x,y) (s) C 2 s 2 and E B x,y <.

40 p. 15 Results Translation-covariance: µ[τ v (ξ)](dη)f(η) = µ[ξ](dη)f(τ v η), for (τ v η)(b) := η(b v), for all bonds b and all v Z d.

41 p. 15 Results Translation-covariance: µ[τ v (ξ)](dη)f(η) = µ[ξ](dη)f(τ v η), for (τ v η)(b) := η(b v), for all bonds b and all v Z d. For model (a), van Enter-Külske (2007): For d = 2, non-existence of shift-covariant gradient Gibbs measures.

42 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures p. 16

43 p. 16 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures 1. for model (a) when E P (ξ(x)) = 0 and d 3

44 p. 16 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures 1. for model (a) when E P (ξ(x)) = 0 and d 3 2. for model (b), when d 2.

45 p. 16 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures 1. for model (a) when E P (ξ(x)) = 0 and d 3 2. for model (b), when d 2. Cotar-Külske (in preparation): Uniqueness of shift-covariant gradient Gibbs measures with tilt u when V uniformly strictly convex

46 p. 16 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures 1. for model (a) when E P (ξ(x)) = 0 and d 3 2. for model (b), when d 2. Cotar-Külske (in preparation): Uniqueness of shift-covariant gradient Gibbs measures with tilt u when V uniformly strictly convex 1. for model (a) when E P (ξ(x)) = 0 and d 3

47 p. 16 Cotar-Külske (2010): Existence of shift-covariant gradient Gibbs measures 1. for model (a) when E P (ξ(x)) = 0 and d 3 2. for model (b), when d 2. Cotar-Külske (in preparation): Uniqueness of shift-covariant gradient Gibbs measures with tilt u when V uniformly strictly convex 1. for model (a) when E P (ξ(x)) = 0 and d 3 2. for model (b), when d 2.