An Effective Model of Facets Formation Dima Ioffe 1 Technion April - - PowerPoint PPT Presentation

An Effective Model of Facets Formation

Dima Ioffe1

Technion

April 2015

1Based on joint works with Senya Shlosman, Fabio Toninelli, Yvan Velenik

and Vitali Wachtel

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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.

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Macroscopic Variational Problem

Surface tension τβ and bulk susceptibility Dβ are coming from an effective SOS-type model at inverse temperature β.

γ7

B = [−1, 1]2

Nested family of loops L = (γ1, . . . , γ7) γ6 γ2 γ1 γ3 γ5 γ4

τβ(γ) =

• γ τβ(ns)ds.

τβ(L) =

ℓ τβ(γℓ).

a(γ) - area inside γ. a(L) =

ℓ a(γℓ).

min

L

• (δ − a(L))2

2Dβ + τβ(L)

• (VPδ)

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Macroscopic Variational Problem: Rescaling

Let e be a lattice direction. Set v = δ τβ(e)Dβ , σβ = Dβτβ(e) and τ(·) = τβ(·) τβ(e). (1) Since, (δ − a(L))2 2Dβ + τβ(L) = δ2 2Dβ + τβ(e)

• −va(L) + τ(L) + a(L)2

2σβ

• ,

we can reformulate the family of variational problems (VPδ) as follows: min

L

• −va(L) + τ(L) + a(L)2

2σβ

• .

(VPv)

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Geometric Interpretation (Legendre-Fenchel Transform)

τ(a) = min {τ(L) : a(L) = a} Given v ≥ 0, find min

L

• −va + τ(a) + a2

2σβ

• .

(VPv)

τ(a) + a2 2σβ v∗ 2 v∗ 1 v∗ 3 a− 1 a+ 1 a− 2 a+ 2 a− 3 a

If the graph of a → τ(a) +

a2 2σβ

is not convex, then an (infinite) sequence of first

• rder transitions with:

Transition slopes v ∗

1, v ∗ 2, . . . .

Transition areas a±

ℓ .

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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 ∈ S1

has radius 1. Consider: min

γ⊂B; a(γ)=b τ(γ)

(STb)

Wb Pb r r

B B

Wulff Plaquette of area b Wulff Shape of area b

Define w = a(W). Wb solves (STb) for b = r 2w ∈ [0, w] Pb solves (STb) for b = 4−r 2(4−w) ∈ [w, 4].

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Isoperimetric Stacks

min

L

• −va(L) + τ(L) + a(L)2

2σβ

• .

(VPv)

Define: Sb = Wb1 Ib∈[0,w) + Pb1 Ib∈[w,4] Each nested family L of loops could be recorded as an integer valued function u : B1 → N ∪ 0. Set bℓ = |x : u(x) ≥ ℓ|. Rearrangement: L∗ = {Sb1, Sb2, . . .} - nested family of loops. a(L∗) = a(L) but τ(L∗) ≤ τ(L).

−1 −1 1 1 u u∗

Hence only stacks of Wulff plaquettes and shapes matter.

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Regular Isoperimetric Stacks of Type 1 and 2 Recall w = a(W). For any b ∈ (0, w), respectively, b ∈ (w, 4),

d dbτ (Wb) = 1 r(b) and d dbτ (Pb) = 1 r(b).

Which means that optimal stacks of area a could be one of two types: Stacks L1

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 L2

ℓ(a) of type 2. These contain ℓ identical Wulff plaquettes of

the same radius r ∈ [0, 1].

Set ℓ∗ ∆ =

4 4−w (and assume ℓ∗ ∈ N). Then, relevant area ranges are:

Range(L1

ℓ) =

• [4(ℓ − 1), ℓw],

if ℓ < ℓ∗ [ℓw, 4(ℓ − 1)], if ℓ > ℓ∗ and Range(L2

ℓ) = [ℓw, 4ℓ]

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Structure of Solutions to (VPv )

min

L

• −va(L) + τ(L) + a(L)2

2σβ

• .

(VPv)

• If w ≤ 2σβ, then stacks of type 1 are never optimal, and (recall

Range(L2

ℓ) = [ℓw, 4ℓ]):

3w 12 etc v∗ 1 Transition slopes 0 < v∗ 1 < v∗ 2 < . . . L2 1 L2 3 v∗ 4 L2 2 v∗ 3 v∗ 2 w a 4 2w 8

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Structure of solutions to (VPv ) Recall ℓ∗ ∆ =

4 4−w ∈ N. For ℓ < ℓ∗ the area ranges are:

Range(L1

ℓ) = [4(ℓ − 1), ℓw] and Range(L2 ℓ) = [ℓw, 4ℓ]

• If w > 2σβ, then then there exists a number 1 ≤ k < ℓ∗

1 − σβ

8

• such

that stacks L1

ℓ show up for any ℓ = 1, . . . , k:

2w a+ 2 a− k Type 2 L2 2 L2 1 L1 1 L1 2 L2 k a+ k kw L1 k a− 1 a w a+ 1 a− 2

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3D Ising model

∂ΛN ΛN ⊂ Z3 |ΛN| = N3

The Gibbs State − H−

N = 1

2

• x∼y

σxσy−

• x∈∂ΛN

σx P−

N,β(σ) ∼ e−βH−

N

Low Temperature β ≫ 1 ⇒ m∗(β) > 0. Phase Segregation: Fix m > −m∗ and consider Pm,−

N,β (·) = P− N,β

• ·
• σx = mN3

.

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Microscopic 3D Wulff shape

Typical Picture under Pm,−

N,β

ΓN

Volume of the microscopic Wulff droplet |ΓN| ≈ m + m∗ 2m∗ N3 Theorem (Bodineau, Cerf-Pisztora): As N → ∞ the scaled shape

1 N ΓN converges to the macroscopic Wulff shape.

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3D Surface Tension and Macroscopic Wulff Shape

n + − M + − n Kβ h

ξβ(n) = − lim

M→∞

| cos n| M2 log Z ±

M

Z −

M

. ξβ = max

h∈∂Kβ h · n

Dilated Wulff Shape Km

β =

m + m∗ 2m∗|Kβ| 1/3 Kβ

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Bodineau, Cerf-Pisztora Result

N(Km β + u) ΓN Nu

Define (on unit box Λ ⊂ R3) φN(t) = 1 I{Nt∈ΓN} − 1 I{Nt∈ΓN}. Define χm(t) = 1 I{t∈Km

β} − 1

I{t∈Km

β}

Then, under

• Pm,−

N,β

• ,

lim

N→∞ min u φN(·) − χm(u + ·)L1(Λ) = 0

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Macroscopic Facets

Kβ n Fn ξβ - support function of Kβ. Then Fn = ∂ξβ(n). Set ei - lattice direction. Dobrushin ’72, Miracle-Sole ’94: For β ≫ 1 Fei is a proper 2D facet

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Microscopic Facets

Zooming Bodineau, Cerf-Pisztora picture, what happens?

NFe1 NFe1 NFe1 ΓN OR ΓN OR ΓN

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SOS Model BN = {−N, . . . , N}2

ΓN VN

BN = {−N, . . . , N}2

Ak k ℓN

Bodineau, Schonmann, Shlosman ’05 PN (ΓN = γ) ∼ e−β|γ| Pm

N (·) = PN

• ·
• VN ≥ mN3

Result: There exists a(β) ց 0 such that ℓN = max

• k : Ak ≥ a(β)N2

satisfies AℓN−1 ≥ (1 − a(β))N2.

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Effective Model of Microscopic Facets

ΓN VN SN

BN = {−N, . . . , N}2

ps β pv β

Configuration:

• ΓN, {ξv

i }i∈VN ,

• ξs

j

• j∈SN
• .

Total number of particles: ΞN =

• i∈VN

ξv

i +

• j∈SN

ξs

j

|Γ| - area of Γ Bp(ξ) = pξ(1 − p)1−ξ β large

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Contour Representation of Γ

Orientation of contours: Positive and negative (holes) α(γ) - signed area. |γ| - length. Compatibility γ ∼ γ′ For Γ = {γi} |Γ| ∼

• |γi|, α(Γ)

∆

=

• α(γi)

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Creation of Facets

ps pv α(ΓN)

ΞN - total number of particles EN (ΞN) = ps + pv 2 N3 ∆ = pN3 Consider PA

N (·) = PN

• ·
• ΞN = pN3 + AN2

2D Surface Tension: log P

• α(ΓN) = aN2

≈ −Nτβ(a). Bulk Fluctuations: ∆ = 2(ps − pv), EN

• ΞN
• α(ΓN)
• = pN3 + ∆α(ΓN).

log PN

• ΞN = pN3 + AN2

α(ΓN) = aN2 ≈ −(AN2 − ∆aN2)2 N3R = −N (δ − a)2 2Dβ , where R = ps(1 − ps) + pv(1 − pv), Dβ = R/(2∆2) and δ = A/∆. Hence mina

• (δ−a)2

2Dβ

+ τβ(a)

• .

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Surface Tension and Macroscopic Variational Problem

C Nx γ

wβ(γ) = e−β|γ|−

C∼γ Φβ(γ)

Gβ(Nx) =

• γ:0→Nx

wβ(γ). τβ(x) = − lim

N→∞

1 N log Gβ(Nx). τβ(γ) =

• γ

τβ(ns)ds.

Macroscopic Variational Problem

Recall ∆ = 2(ps − pv), R = ps(1 − ps) + pv(1 − pv) , Dβ = R/(2∆2) and δ = A/∆. (VP)δ min

a

(δ − a)2 2Dβ + min

a(L)=a τβ(L)

• .

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Reduction to Large Contours Fix β ≫ 1. Bulk fluctuations simplify analysis of PA

• N. Recall the contour

representation Γ = {γi}. Lemma 1 (No intermediate contours). ∀A > 0 there exists ǫ = ǫ(A) > 0 such that PA

N

• ∃γi : 1

ǫ log N ≤ |γi| ≤ ǫN

• = o(1).

Lemma 2 (Irrelevance of small contours) PA

N

• α(γi)1

I{|γi|≤ǫ−1 log N}

• ≫ N
• = o(1).

Definition: γ is large if |γ| ≥ ǫN.

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Reduced Model of Large Contours

• A. Fix A > 0 and forget about intermediate contours 1

ǫ log N ≤ |γ| ≤ ǫN.

• B. Expand with respect to small contours |γ| ≤ 1

ǫ log N.

For Γ = {γi} collection of large contours the effective weight is ˆ PN(Γ) ∝ exp

• −β |γi| −

C∼Γ Φβ(C)

• .

The family of clusters C depends on N and A. However cluster weights Φβ(C) remain the same, and they are small: For all β sufficiently large |Φβ(γ; C)| ≤ ce−β(diam(C)+1) Reduced Model of Large Contours and Bulk Particles: ˆ PN (Γ, ξv, ξs) = ˆ PN(Γ)

• i∈ ˆ

VN

Bpv (ξv

i )

• j∈ ˆ

SN

Bps(ξs

j )

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Limit Shapes Result (I., Shlosman 2015) Recall: ∆ = 2(ps − pv), R = ps(1 − ps) + pv(1 − pv) , Dβ = R/(2∆2) and δ = A/∆. (VP)δ mina

• (δ−a)2

D

+ mina(L)=a τβ(L)

• .

a+ k∗ A1 A2 A3 Type 2 k∗ Layers Ak∗ a+ 1 a− 1 a− 2 a+ 2 a− k∗

Set: ˆ PA

N (·) = ˆ

PN

• ·
• ΞN = pN3 + AN2

. Then for any ν > 0 and any A ≥ 0, the (random) collection of large contours Γ satisfies: limN→∞ ˆ PA

N

• minL∗−solutions to (VP)δ dH

1

N Γ, L∗

< ν

• = 1

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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 ց 0.

2

Spontaneous appearance of a droplet of linear size N2/3 in the context of the 2D Ising model (any β > βc) 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-Sol´ e (JSP 2011) proved a sequence of layering transitions as h ց 0.

4

For β ≫ 1 pure 2+1 SOS (without bulk Bernoulli fields) models of interfaces with zero b.c. on ∂BN, and conditioned to stay positive on BN, Caputo, Lubetzky, Martinelli, Sly and Toninelli proved in a series of works 2012-14 that there are ⌊ 1

4β log N⌋ macroscopic facets with asymptotically

different Wulff Plaquette shapes.

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Difficulty: Control of Interactions of Contours in Macroscopic Stacks

C γ2 γ1

• C∼γ1∪γ2

Φβ(C) =

• C∼γ1

Φβ(C) +

• C∼γ2

Φβ(C) −

• C∼γ1∩γ2

Φβ(C) As β ր ∞ 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.

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Interaction Between ℓ Contours

γ1 γ2 γ3 γℓ

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A general Result for Ising Polymers

C3 C4 x y γ C2 C1

wβ(γ) = e−β|γ|+

C∼γ Φβ(γ;C).

Assumption: |Φβ(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.

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Effective RW Representation of Ising Polymers Portion of a Contour Between x and y

x y ˆ γ1 ˆ γ2 ˆ γm ξ1 ξ2 x y ξm

eτβ(y−x)Gβ(y − x) ∼ =

m

• ˆ

γ1,...ˆ γm

ρβ(ˆ γi) {ρβ(·)} is a probability distribution on the set of irreducible animals. ξ1 = (T1, X1), ξ2 = (T2, X2), . . . steps of the effective random walk.

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Fluctuations of Facets near Flat Boundaries

ΓN VN N Nα

• Bulk fluctuation price for VN is ∼ VNN2

N3

∼ VN

N .

• Repulsion price for staying Nα away from the boundary is N1−2α.

Therefore, N1−2α ∼ VN

N ∼ N1+α N

= Nα gives α = 1/3.

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Random Walks under Area Tilts

VN −N N a b X-trajectory of RW

• Partition Function:

Z a,b

N,+,λN =

• X∈T a,b

N,+

e−λNVNp(X)

• Scaling: xN(t) = λ1/3

N X(λ−2/3 N

t).

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Random Walks under Area Tilts

VN −N N a b X-trajectory of RW

• Partition Function:

Z a,b

N,+,λN =

• X∈T a,b

N,+

e−λNVNp(X)

• Scaling: xN(t) = λ1/3

N X(λ−2/3 N

t). In general: {Φλ} - family of self-potentials, define Φλ(X) = N

−N Φλ(Xi).

Z a,b

N,+,λN =

• X∈T a,b

N,+

e−ΦλN (X)p(X)

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Random Walks under Area Tilts: Ferrari-Spohn Diffusion

VN −N N a b X-trajectory of RW

Φλ(X) =

N

• −N

Φλ(Xi) and Z a,b

N,+,λN =

• X∈T a,b

N,+

e−ΦλN (X)p(X). Scale: H2

λΦλ(Hλ) = 1.

Assumption: limλ→0 H2

λΦλ(Hλr) = q(r) and limr→∞ q(r) = ∞.

Sturm-Liouville operator on R+: Set σ2 =

x x2p(x) and

L = σ2

2 d2 dr2 − q(r). Let ϕ1 - the leading eigenfunction of L.

Theorem (I, Shlosman, Velenik (CMP 2015)) The rescaled walk xN(t) = H−1

λN X(H2 λNt) converges to ergodic diffusion with generator

σ2 2ϕ2

1(r)

d dr

• ϕ2

1(r) d

dr

• .

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ℓ Ordered Random Walks under Area Tilts

b2 a2 a1 b1 X2 X1 Xℓ −N N aℓ bℓ

Z a,b

N,+,λN =

• X∈T a,b

N,+

e− ℓ

m=1 ΦλN (Xm)

ℓ

• m=1

p(Xm). Work in Progress: Let ϕ1, . . . , ϕℓ be first ℓ eigenfunctions of L. Define ∆ℓ(r) = det (ϕi(rj)). Then, the rescaled process H−1

λN X(H2 λNt) converges

to ergodic diffusion on {r : 0 < rℓ < · · · < r2 < r1} with generator

σ2 2∆2

ℓ(r)div

• ∆2

ℓ(r)∇

• .

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