An Effective Model of Facets Formation Dima Ioffe 1 Technion April 2015 1 Based on joint works with Senya Shlosman, Fabio Toninelli, Yvan Velenik and Vitali Wachtel Dima Ioffe (Technion ) Microscopic Facets April 2015 1 / 33
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 2015 2 / 33
Macroscopic Variational Problem Surface tension τ β and bulk susceptibility D β are coming from an effective SOS-type model at inverse temperature β . � τ β ( γ ) = γ τ β ( n s ) d s . τ β ( L ) = � ℓ τ β ( γ ℓ ). γ 2 γ 3 a( γ ) - area inside γ . γ 5 a( L ) = � γ 6 ℓ a( γ ℓ ). γ 4 γ 1 γ 7 � � ( δ − a( L )) 2 B = [ − 1 , 1]2 min + τ β ( L ) 2D β L Nested family of loops L = ( γ 1 , . . . , γ 7) ( VP δ ) Dima Ioffe (Technion ) Microscopic Facets April 2015 3 / 33
Macroscopic Variational Problem: Rescaling Let e be a lattice direction. Set δ , σ β = D β τ β (e) and τ ( · ) = τ β ( · ) v = τ β (e) . (1) τ β (e)D β Since, ( δ − a( L )) 2 + τ β ( L ) = δ 2 − v a( L ) + τ ( L ) + a( L ) 2 � � + τ β (e) , 2D β 2D β 2 σ β we can reformulate the family of variational problems ( VP δ ) as follows: − va ( L ) + τ ( L ) + a ( L ) 2 � � min ( VP v ) . 2 σ β L Dima Ioffe (Technion ) Microscopic Facets April 2015 4 / 33
Geometric Interpretation (Legendre-Fenchel Transform) τ ( a ) = min { τ ( L ) : a( L ) = a } − va + τ ( a ) + a 2 � � Given v ≥ 0, find min ( VP v ) . 2 σ β L a 2 If the graph of a �→ τ ( a ) + 2 σ β τ ( a ) + a 2 v ∗ is not convex, then an 2 σ β 3 (infinite) sequence of first v ∗ order transitions with: 2 Transition slopes v ∗ 1 , v ∗ 2 , . . . . v ∗ 1 a Transition areas a ± ℓ . a − a + a − a − a + 2 2 1 1 3 Dima Ioffe (Technion ) Microscopic Facets April 2015 5 / 33
Wulff Shapes and Wulff Plaquettes Recall B = [ − 1 , 1] 2 . The rescaled surface tension τ (e) = 1. The Wulff shape x : x · n ≤ τ ( n ) ∀ n ∈ S 1 � W � ∂ � has radius 1. Consider: γ ⊂ B ; a ( γ )= b τ ( γ ) min ( ST b ) Define w = a (W). B B r W b solves ( ST b ) r for b = r 2 w ∈ [0 , w ] W b P b P b solves ( ST b ) for Wulff Shape of area b Wulff Plaquette of area b b = 4 − r 2 (4 − w ) ∈ [ w , 4]. Dima Ioffe (Technion ) Microscopic Facets April 2015 6 / 33
Isoperimetric Stacks − va ( L ) + τ ( L ) + a ( L ) 2 � � min . ( VP v ) 2 σ β L Define: S b = W b 1 I b ∈ [0 , w ) + P b 1 I b ∈ [ w , 4] 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 b 1 , S b 2 , . . . } - nested family of loops. a( L ∗ ) = a( L ) but τ ( L ∗ ) ≤ τ ( L ) . u ∗ u − 1 1 − 1 1 Hence only stacks of Wulff plaquettes and shapes matter. Dima Ioffe (Technion ) Microscopic Facets April 2015 7 / 33
Regular Isoperimetric Stacks of Type 1 and 2 Recall w = a (W). For any b ∈ (0 , w ), respectively, b ∈ ( w , 4), d 1 d 1 d b τ (W b ) = and d b τ (P b ) = r ( b ) . 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 − w (and assume ℓ ∗ �∈ N ). Then, relevant area ranges are: 4 = � if ℓ < ℓ ∗ [4( ℓ − 1) , ℓ w ] , R ange ( L 1 if ℓ > ℓ ∗ and R ange ( L 2 ℓ ) = ℓ ) = [ ℓ w , 4 ℓ ] [ ℓ w , 4( ℓ − 1)] , Dima Ioffe (Technion ) Microscopic Facets April 2015 8 / 33
Structure of Solutions to ( VP v ) − va ( L ) + τ ( L ) + a ( L ) 2 � � min ( VP v ) . 2 σ β L • If w ≤ 2 σ β , then stacks of type 1 are never optimal, and (recall R ange ( L 2 ℓ ) = [ ℓ w , 4 ℓ ]): v ∗ etc L 2 4 3 v ∗ L 2 3 2 v ∗ L 2 2 1 v ∗ 1 a 0 w 4 2 w 8 3 w 12 Transition slopes 0 < v ∗ 1 < v ∗ 2 < . . . Dima Ioffe (Technion ) Microscopic Facets April 2015 9 / 33
Structure of solutions to ( VP v ) Recall ℓ ∗ ∆ 4 − w �∈ N . For ℓ < ℓ ∗ the area ranges are: 4 = R ange ( L 1 ℓ ) = [4( ℓ − 1) , ℓ w ] and R ange ( L 2 ℓ ) = [ ℓ w , 4 ℓ ] 1 − σ β • If w > 2 σ β , then then there exists a number 1 ≤ k < ℓ ∗ � � such 8 that stacks L 1 ℓ show up for any ℓ = 1 , . . . , k : L 1 L 2 Type 2 k k L 1 L 2 2 2 L 1 L 2 1 1 a a − w a + 1 a − 2 w a + a − a + 0 kw 1 2 2 k k Dima Ioffe (Technion ) Microscopic Facets April 2015 10 / 33
3D Ising model The Gibbs State ∂ Λ N N = 1 � � − H − σ x σ y − σ x Λ N ⊂ Z 3 2 x ∼ y x ∈ ∂ Λ N N ,β ( σ ) ∼ e − β H − P − N | Λ N | = N 3 Low Temperature β ≫ 1 ⇒ m ∗ ( β ) > 0. Phase Segregation: Fix m > − m ∗ and consider � σ x = mN 3 � P m , − N ,β ( · ) = P − � � � · . N ,β Dima Ioffe (Technion ) Microscopic Facets April 2015 11 / 33
Microscopic 3D Wulff shape Typical Picture under P m , − N ,β Volume of the microscopic Γ N Wulff droplet | Γ N | ≈ m + m ∗ N 3 2 m ∗ 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 2015 12 / 33
3D Surface Tension and Macroscopic Wulff Shape n + log Z ± | cos n | M ξ β ( n ) = − lim . + Z − M 2 M →∞ M − − ξ β = max h ∈ ∂ K β h · n M n Dilated Wulff Shape h � m + m ∗ � 1 / 3 K m β = K β K β 2 m ∗ | K β | Dima Ioffe (Technion ) Microscopic Facets April 2015 13 / 33
Bodineau, Cerf-Pisztora Result Define (on unit box Λ ⊂ R 3 ) Γ N φ N ( t ) = 1 I { Nt ∈ Γ N } − 1 I { Nt �∈ Γ N } . Nu Define χ m ( t ) = 1 β } − 1 I { t ∈ K m I { t �∈ K m β } N ( K m β + u ) P m , − � � Then, under , N ,β u � φ N ( · ) − χ m ( u + · ) � L 1 (Λ) = 0 N →∞ min lim Dima Ioffe (Technion ) Microscopic Facets April 2015 14 / 33
Macroscopic Facets n Fn ξ β - support function of K β . Then F n = ∂ξ β ( n ) . K β Set e i - lattice direction. Dobrushin ’72, Miracle-Sole ’94: For β ≫ 1 F e i is a proper 2D facet Dima Ioffe (Technion ) Microscopic Facets April 2015 15 / 33
Microscopic Facets Zooming Bodineau, Cerf-Pisztora picture, what happens? Γ N NF e 1 OR Γ N NF e 1 OR Γ N NF e 1 Dima Ioffe (Technion ) Microscopic Facets April 2015 16 / 33
SOS Model Bodineau, Schonmann, Γ N Shlosman ’05 P N (Γ N = γ ) ∼ e − β | γ | VN P m � V N ≥ mN 3 � � � N ( · ) = P N · B N = {− N, . . . , N } 2 Result: There exists a ( β ) ց 0 such that ℓN k k : A k ≥ a ( β ) N 2 � � ℓ N = max Ak 0 satisfies A ℓ N − 1 ≥ (1 − a ( β )) N 2 . B N = {− N, . . . , N } 2 Dima Ioffe (Technion ) Microscopic Facets April 2015 17 / 33
Effective Model of Microscopic Facets pv Configuration: β VN � � Γ N , { ξ v � ξ s � i } i ∈ V N , . j j ∈ S N Total number of particles: Γ N � � ξ v ξ s Ξ N = i + j i ∈ V N j ∈ S N ps β SN | Γ | - area of Γ B p ( ξ ) = p ξ (1 − p ) 1 − ξ B N = {− N, . . . , N } 2 β large Dima Ioffe (Technion ) Microscopic Facets April 2015 18 / 33
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 2015 19 / 33
Creation of Facets Ξ N - total number of particles E N (Ξ N ) = p s + p v pv N 3 ∆ = pN 3 2 Consider α (Γ N ) ps � Ξ N = pN 3 + AN 2 � P A � � N ( · ) = P N · � α (Γ N ) = aN 2 � 2D Surface Tension: log P ≈ − N τ β ( a ). Bulk Fluctuations: ∆ = 2( p s − p v ), E N = pN 3 + ∆ α (Γ N ). � � � Ξ N � α (Γ N ) ≈ − ( AN 2 − ∆ aN 2 ) 2 Ξ N = pN 3 + AN 2 � � α (Γ N ) = aN 2 � � log P N N 3 R = − N ( δ − a ) 2 where R = p s (1 − p s ) + p v (1 − p v ) , , 2 D β � ( δ − a ) 2 � D β = R / (2∆ 2 ) and δ = A / ∆. Hence min a + τ β ( a ) . 2 D β Dima Ioffe (Technion ) Microscopic Facets April 2015 20 / 33
Surface Tension and Macroscopic Variational Problem � G β ( Nx ) = w β ( γ ) . C γ :0 → Nx Nx 1 τ β ( x ) = − lim N log G β ( Nx ) . N →∞ γ 0 � τ β ( γ ) = τ β ( n s ) d s . w β ( γ ) = e − β | γ |− � C�∼ γ Φ β ( γ ) 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 / ∆. � ( δ − a ) 2 � (VP) δ min + min a ( L )= a τ β ( L ) . 2 D β a Dima Ioffe (Technion ) Microscopic Facets April 2015 21 / 33
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