An Effective Model of Facets Formation

Transcription

1 An Effective Model of Facets Formation Dima Ioffe 1 Technion April Based on joint works with Senya Shlosman, Fabio Toninelli, Yvan Velenik and Vitali Wachtel Dima Ioffe (Technion ) Microscopic Facets April / 33

2 Plan of the Talk A macroscopic variational problem. Low temperature 3D Ising model, Wulff shapes and (unknown) structure of microscopic facets. Facets on SOS surfaces with bulk Bernoulli fields. Fluctuations of level lines. Dima Ioffe (Technion ) Microscopic Facets April / 33

3 Macroscopic Variational Problem Surface tension τ β and bulk susceptibility D β are coming from an effective SOS-type model at inverse temperature β. γ 3 γ 2 γ 5 γ 1 γ 4 γ 6 τ β (γ) = τ γ β(n s )ds. τ β (L) = l τ β(γ l ). a(γ) - area inside γ. a(l) = l a(γ l). B = [ 1, 1] 2 γ 7 Nested family of loops L = (γ 1,..., γ 7 ) min L { (δ a(l)) 2 2D β + τ β (L) } (VP δ ) Dima Ioffe (Technion ) Microscopic Facets April / 33

4 Macroscopic Variational Problem: Rescaling Let e be a lattice direction. Set Since, (δ a(l)) 2 v = δ, σ β = D β τ β (e) and τ( ) = τ β( ) τ β (e)d β τ β (e). (1) 2D β + τ β (L) = δ2 2D β + τ β (e) { va(l) + τ(l) + a(l)2 we can reformulate the family of variational problems (VP δ ) as follows: } min { va(l) + τ(l) + a(l)2. (VP v ) L 2σ β 2σ β }, Dima Ioffe (Technion ) Microscopic Facets April / 33

5 Geometric Interpretation (Legendre-Fenchel Transform) τ(a) = min {τ(l) : a(l) = a} Given v 0, find min { va + τ(a) + a2 L 2σ β }. (VP v ) τ(a) + a2 2σ β a 1 a+ 1 v 1 a 2 a+ 2 v 2 a a 3 v 3 If the graph of a τ(a) + a2 2σ β is not convex, then an (infinite) sequence of first order transitions with: Transition slopes v 1, v 2,.... Transition areas a ± l. Dima Ioffe (Technion ) Microscopic Facets April / 33

6 Wulff Shapes and Wulff Plaquettes Recall B = [ 1, 1] 2. The rescaled surface tension τ(e) = 1. The Wulff shape W { x : x n τ(n) n S 1} has radius 1. Consider: min τ(γ) (ST b) γ B; a(γ)=b B r W b Wulff Shape of area b r Pb B Wulff Plaquette of area b Define w = a(w). W b solves (ST b ) for b = r 2 w [0, w] P b solves (ST b ) for b = 4 r 2 (4 w) [w, 4]. Dima Ioffe (Technion ) Microscopic Facets April / 33

7 Define: min L Isoperimetric Stacks { va(l) + τ(l) + a(l)2 2σ β S b = W b 1I b [0,w) + P b 1I b [w,4] }. (VP v ) Each nested family L of loops could be recorded as an integer valued function u : B 1 N 0. Set b l = x : u(x) l. Rearrangement: L = {S b1, S b2,...} - nested family of loops. a(l ) = a(l) but τ(l ) τ(l). u u Hence only stacks of Wulff plaquettes and shapes matter. Dima Ioffe (Technion ) Microscopic Facets April / 33

8 Regular Isoperimetric Stacks of Type 1 and 2 Recall w = a(w). For any b (0, w), respectively, b (w, 4), d db τ (W b) = 1 r(b) and d db τ (P b) = 1 r(b). Which means that optimal stacks of area a could be one of two types: Stacks L 1 l (a) of type 1. These contain l 1 identical Wulff plaquettes and a Wulff shape, all of the same radius r [0, 1]. Stacks L 2 l (a) of type 2. These contain l identical Wulff plaquettes of the same radius r [0, 1]. Set l = 4 (and assume 4 w l N). Then, relevant area ranges are: { Range(L 1 [4(l 1), lw], if l < l l) = and Range(L 2 l) = [lw, 4l] [lw, 4(l 1)], if l > l Dima Ioffe (Technion ) Microscopic Facets April / 33

9 min L Structure of Solutions to (VP v ) { va(l) + τ(l) + a(l)2 2σ β }. (VP v ) If w 2σ β, then stacks of type 1 are never optimal, and (recall Range(L 2 l ) = [lw, 4l]): L 2 3 v 4 etc L 2 2 v 3 L 2 1 v 2 v 1 a 0 w 4 2w 8 3w 12 Transition slopes 0 < v 1 < v 2 <... Dima Ioffe (Technion ) Microscopic Facets April / 33

10 Structure of solutions to (VP v ) Recall l = 4 4 w N. For l < l the area ranges are: Range(L 1 l) = [4(l 1), lw] and Range(L 2 l) = [lw, 4l] If w > 2σ β, then then there exists a number 1 k < l ( 1 σ ) β 8 such that stacks L 1 l show up for any l = 1,..., k: L 1 k L 2 k Type 2 L 1 2 L 2 2 L 1 1 L 2 1 a 0 a w a 2w a a a k kw a + k Dima Ioffe (Technion ) Microscopic Facets April / 33

11 3D Ising model Λ N Λ N Z 3 Λ N = N 3 The Gibbs State H N = 1 σ x σ y 2 x y P N,β (σ) e βh N x Λ N σ x Low Temperature β 1 m (β) > 0. Phase Segregation: Fix m > m and consider P m, N,β ( ) = P N,β ( σ x = mn 3 ). Dima Ioffe (Technion ) Microscopic Facets April / 33

12 Microscopic 3D Wulff shape Typical Picture under P m, N,β Γ N Volume of the microscopic Wulff droplet Γ N m + m 2m N 3 Theorem (Bodineau, Cerf-Pisztora): As N the scaled shape 1 N Γ N converges to the macroscopic Wulff shape. Dima Ioffe (Technion ) Microscopic Facets April / 33

13 3D Surface Tension and Macroscopic Wulff Shape + n + cos n ξ β (n) = lim M M 2 log Z ± M Z. M n h M K β ξ β = max h K β h n Dilated Wulff Shape ( ) m + m K m 1/3 β = K 2m β K β Dima Ioffe (Technion ) Microscopic Facets April / 33

14 Bodineau, Cerf-Pisztora Result Γ N Define (on unit box Λ R 3 ) Nu φ N (t) = 1I {Nt ΓN } 1I {Nt ΓN }. Define χ m (t) = 1I {t K m β } 1I {t K m β} N(K m β + u) Then, under { P m, N,β }, lim min φ N ( ) χ m (u + ) L1 (Λ) = 0 N u Dima Ioffe (Technion ) Microscopic Facets April / 33

15 Macroscopic Facets n K β Fn ξ β - support function of K β. Then F n = ξ β (n). Set e i - lattice direction. Dobrushin 72, Miracle-Sole 94: For β 1 F ei is a proper 2D facet Dima Ioffe (Technion ) Microscopic Facets April / 33

16 Microscopic Facets Zooming Bodineau, Cerf-Pisztora picture, what happens? Γ N OR Γ N NFe 1 OR Γ N NFe 1 NFe 1 Dima Ioffe (Technion ) Microscopic Facets April / 33

17 SOS Model l N k 0 Γ N V N B N = { N,..., N} 2 A k B N = { N,..., N} 2 Bodineau, Schonmann, Shlosman 05 P N (Γ N = γ) e β γ P m N ( ) = P N ( VN mn 3) Result: There exists a(β) 0 such that l N = max { k : A k a(β)n 2} satisfies A ln 1 (1 a(β))n 2. Dima Ioffe (Technion ) Microscopic Facets April / 33

18 Effective Model of Microscopic Facets p v β p s β V N Γ N S N B N = { N,..., N} 2 Configuration: ( Γ N, {ξi v} i V N, { ) ξj s. }j SN Total number of particles: Ξ N = ξi v + ξj s i V N j S N Γ - area of Γ B p (ξ) = p ξ (1 p) 1 ξ β large Dima Ioffe (Technion ) Microscopic Facets April / 33

19 Contour Representation of Γ Orientation of contours: Positive and negative (holes) α(γ) - signed area. γ - length. Compatibility γ γ For Γ = {γ i } Γ γ i, α(γ) = α(γ i ) Dima Ioffe (Technion ) Microscopic Facets April / 33

20 Creation of Facets Ξ N - total number of particles α(γ N ) p v p s E N (Ξ N ) = ps + p v N 3 = pn 3 2 Consider P A N ( ) = P ( ΞN N = pn 3 + AN 2) 2D Surface Tension: log P ( α(γ N ) = an 2) Nτ β (a). Bulk Fluctuations: = 2(p s p v ( ), E N ΞN α(γ N ) ) = pn 3 + α(γ N ). ( log P N ΞN = pn 3 + AN 2 α(γn ) = an 2) (AN2 an 2 ) 2 N 3 R (δ a)2 = N, where R = p s (1 p s ) + p v (1 p v ), 2D β D β = R/(2 2 ) and δ = A/. Hence min a { (δ a) 2 2D β } + τ β (a). Dima Ioffe (Technion ) Microscopic Facets April / 33

21 Surface Tension and Macroscopic Variational Problem C Nx 0 γ w β (γ) = e β γ C γ Φ β(γ) G β (Nx) = γ:0 Nx w β (γ). 1 τ β (x) = lim N N log G β(nx). τ β (γ) = τ β (n s )ds. γ Macroscopic Variational Problem Recall = 2(p s p v ), R = p s (1 p s ) + p v (1 p v ), D β = R/(2 2 ) and δ = A/. (VP) δ { } (δ a) 2 min + min a 2D τ β(l). β a(l)=a Dima Ioffe (Technion ) Microscopic Facets April / 33

22 Reduction to Large Contours Fix β 1. Bulk fluctuations simplify analysis of P A N. Recall the contour representation Γ = {γ i }. Lemma 1 (No intermediate contours). A > 0 there exists ɛ = ɛ(a) > 0 such that ( γ i : 1 ) ɛ log N γ i ɛn = o(1). P A N Lemma 2 (Irrelevance of small contours) ( P A ) N α(γi )1I { γi ɛ 1 log N} N Definition: γ is large if γ ɛn. = o(1). Dima Ioffe (Technion ) Microscopic Facets April / 33

23 Reduced Model of Large Contours A. Fix A > 0 and forget about intermediate contours 1 ɛ log N γ ɛn. B. Expand with respect to small contours γ 1 ɛ log N. For Γ = {γ i } collection of large { contours the effective weight is ˆP N (Γ) exp β γ i } C Γ Φ β(c). The family of clusters C depends on N and A. However cluster weights Φ β (C) remain the same, and they are small: For all β sufficiently large Φ β (γ; C) ce β(diam(c)+1) Reduced Model of Large Contours and Bulk Particles: ˆP N (Γ, ξ v, ξ s ) = ˆP N (Γ) B pv (ξi v ) B ps (ξj s ) i ˆV N j ŜN Dima Ioffe (Technion ) Microscopic Facets April / 33

24 Limit Shapes Result (I., Shlosman 2015) Recall: = 2(p s p v ), R = { p s (1 p s ) + p v (1 p v ), } D β = R/(2 2 ) and δ = A/. (VP) δ min (δ a) 2 a D + min a(l)=a τ β (L). Layers k A k Type 2 A 1 A 2 A 3 a 1 a + a 1 2 a + 2 a k a + k Set: ˆP A N ( ) = ( ˆP N Ξ N = pn 3 + AN 2). Then for any ν > 0 and any A 0, the (random) ( collection of large contours Γ satisfies: lim N ˆP A ( ) N min L solutions to (VP) δ d 1 H N Γ, L ) < ν = 1 Dima Ioffe (Technion ) Microscopic Facets April / 33

25 1st Order (Shape) Transitions in Microscopic Models 1 For β 1 pure 2+1 SOS, conditioned to stay positive and with an additional bulk field h > 0, Chesi-Martinelli (JSP 1996) and Dinaburg-Mazel (JSP 1996) proved a sequence of layering transitions as h 0. 2 Spontaneous appearance of a droplet of linear size N 2/3 in the context of the 2D Ising model (any β > β c ) was established by Biskup, Chayes and Kotecky (CMP 03). 3 For β 1 pure 2+1 SOS, conditioned to stay positive and with an additional attractive 0-layer boundary field h, Alexander, Dunlop and Miracle-Solé (JSP 2011) proved a sequence of layering transitions as h 0. 4 For β 1 pure 2+1 SOS (without bulk Bernoulli fields) models of interfaces with zero b.c. on B N, and conditioned to stay positive on B N, Caputo, Lubetzky, Martinelli, Sly and Toninelli proved in a series of works that there are 1 4β log N macroscopic facets with asymptotically different Wulff Plaquette shapes. Dima Ioffe (Technion ) Microscopic Facets April / 33

26 Difficulty: Control of Interactions of Contours in Macroscopic Stacks C γ 1 γ 2 Φ β (C) = Φ β (C) + Φ β (C) C γ 1 γ 2 C γ 1 C γ 2 C γ 1 γ 2 Φ β (C) As β the interaction becomes small, but fluctuations of contours (level lines) also become small, at least along axis directions. Hence we are dealing with small attraction vrs small entropic repulsion. Dima Ioffe (Technion ) Microscopic Facets April / 33

27 Interaction Between l Contours γ 1 γ 2 γ3 γ l Dima Ioffe (Technion ) Microscopic Facets April / 33

28 A general Result for Ising Polymers x C 4 C 1 y w β (γ) = e β γ + C γ Φβ(γ;C). C 3 Assumption: γ C 2 Φ β (γ; C) ce χβ(diam(c)+1). Theorem (I, Shlosman, Toninelli, JSP 2015) If χ > 1 2, then repulsion wins over attraction for all β sufficiently large, in the sense that half-space surface tension equals to the full space surface tension. Remarks: a. In the case of SOS interfaces χ = 1. b. The theorem takes care of an interaction between one contour and a hard wall. Interactions between two, and more generally l, ordered contours still has to be worked out. Dima Ioffe (Technion ) Microscopic Facets April / 33

29 Effective RW Representation of Ising Polymers Portion of a Contour Between x and y x y ˆγ 1 ˆγ 2 ˆγm x ξ 1 ξ 2 ξm y e τβ(y x) G β (y x) = m ˆγ 1,...ˆγ ρβ m (ˆγ i ) {ρ β ( )} is a probability distribution on the set of irreducible animals. ξ 1 = (T 1, X 1 ), ξ 2 = (T 2, X 2 ),... steps of the effective random walk. Dima Ioffe (Technion ) Microscopic Facets April / 33

30 Fluctuations of Facets near Flat Boundaries N N α Γ N V N Bulk fluctuation price for V N is V NN 2 V N 3 N N. Repulsion price for staying N α away from the boundary is N 1 2α. Therefore, N 1 2α V N N N1+α N = Nα gives α = 1/3. Dima Ioffe (Technion ) Microscopic Facets April / 33

31 Random Walks under Area Tilts a X-trajectory of RW b V N N N Partition Function: Z a,b N,+,λ N = e λ NV N p(x) X T a,b N,+ Scaling: x N (t) = λ 1/3 N X(λ 2/3 N t). Dima Ioffe (Technion ) Microscopic Facets April / 33

32 Random Walks under Area Tilts a X-trajectory of RW b V N N N Partition Function: Z a,b N,+,λ N = e λ NV N p(x) X T a,b N,+ Scaling: x N (t) = λ 1/3 N X(λ 2/3 N t). In general: {Φ λ } - family of self-potentials, define Φ λ (X) = N N Φ λ(x i ). Z a,b N,+,λ N = X T a,b N,+ e Φ λ N (X) p(x) Dima Ioffe (Technion ) Microscopic Facets April / 33

33 Random Walks under Area Tilts: Ferrari-Spohn Diffusion a X-trajectory of RW b V N N Φ λ (X) = N N Φ λ (X i ) and Z a,b N,+,λ N = X T a,b N,+ N e Φ λ N (X) p(x). Scale: Hλ 2Φ λ(h λ ) = 1. Assumption: lim λ 0 Hλ 2Φ λ(h λ r) = q(r) and lim r q(r) =. Sturm-Liouville operator on R + : Set σ 2 = x x 2 p(x) and L = σ2 2 d 2 dr 2 q(r). Let ϕ 1 - the leading eigenfunction of L. Theorem (I, Shlosman, Velenik (CMP 2015)) The rescaled walk x N (t) = H 1 λ N X(Hλ 2 N t) converges to ergodic diffusion with generator σ 2 ( d 2ϕ 2 1 (r) ϕ 2 dr 1(r) d ). dr Dima Ioffe (Technion ) Microscopic Facets April / 33

34 l Ordered Random Walks under Area Tilts a 1 X 1 a 2 a l X 2 b 1 b 2 X l b l N Z a,b N,+,λ N = X T a,b N,+ e l m=1 Φ λ N (X m ) N l p(x m ). m=1 Work in Progress: Let ϕ 1,..., ϕ l be first l eigenfunctions of L. Define l (r) = det (ϕ i (r j )). Then, the rescaled process H 1 λ N X(Hλ 2 N t) converges to ergodic diffusion on {r : 0 < r l < < r 2 < r 1 } with generator σ ( 2 2 2(r)div 2 l (r) ). l Dima Ioffe (Technion ) Microscopic Facets April / 33