an effective model of facets formation

An Effective Model of Facets Formation

Dima Ioffe1

Technion

April 2015

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


Macroscopic Variational Problem

Surface tension τβ and bulk susceptibility Dβ are coming from aneffective 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) =∫γτβ(ns)ds.

τβ(L) =∑

` τβ(γ`).

a(γ) - area inside γ.

a(L) =∑

` a(γ`).

minL

{(δ − a(L))2

2Dβ

+ τβ(L)

}(VPδ)


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) asfollows:

minL

{−va(L) + τ(L) +

a(L)2

2σβ

}. (VPv )


Geometric Interpretation (Legendre-Fenchel Transform)

τ(a) = min {τ(L) : a(L) = a}

Given v ≥ 0, find minL

{−va + τ(a) +

a2

2σβ

}. (VPv )

τ (a) + a2

2σβ

v∗2

v∗1

v∗3

a−1 a+

1a−2

a+2 a−

3

a

If the graph of a 7→ τ(a) + a2

2σβ

is not convex, then an(infinite) sequence of firstorder transitions with:

Transition slopesv ∗1 , v

∗2 , . . . .

Transition areas a±` .


Wulff Shapes and Wulff Plaquettes

Recall B = [−1, 1]2. The rescaled surface tension τ(e) = 1. The Wulffshape

W , ∂{x : x · n ≤ τ(n) ∀ n ∈ S1

}has radius 1. Consider:

minγ⊂B; a(γ)=b

τ(γ) (STb)

Wb Pb

r

rB B

Wulff Plaquette of area bWulff Shape of area b

Define w = a(W).

Wb solves (STb)forb = r 2w ∈ [0,w ]

Pb solves (STb) forb = 4−r 2(4−w) ∈[w , 4].


Isoperimetric Stacks

minL

{−va(L) + τ(L) +

a(L)2

2σβ

}. (VPv )

Define:Sb = Wb1Ib∈[0,w) + Pb1Ib∈[w ,4]

Each nested family L of loops could be recorded as an integer valuedfunction u : B1 7→ N ∪ 0. Set b` = |x : u(x) ≥ `|.Rearrangement: L∗ = {Sb1 , Sb2 , . . .} - nested family of loops.

a(L∗) = a(L) but τ(L∗) ≤ τ(L).

−1 −1 11

u u∗

Hence only stacks of Wulff plaquettes and shapes matter.


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` (a) of type 1. These contain `− 1 identical Wulff

plaquettes and a Wulff shape, all of the same radius r ∈ [0, 1].

Stacks L2` (a) of type 2. These contain ` identical Wulff plaquettes of

the same radius r ∈ [0, 1].

Set `∗∆= 4

4−w (and assume `∗ 6∈ N). Then, relevant area ranges are:

Range(L1`) =

{[4(`− 1), `w ], if ` < `∗

[`w , 4(`− 1)], if ` > `∗and Range(L2

`) = [`w , 4`]


Structure of Solutions to (VPv )

minL

{−va(L) + τ(L) +

a(L)2

2σβ

}. (VPv )

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

Range(L2` ) = [`w , 4`]):

3w 120

etc

v∗1

Transition slopes 0 < v∗1< v∗

2< . . .

L21

L23

v∗4

L22

v∗3

v∗2

w

a

4 2w 8


Structure of solutions to (VPv )

Recall `∗∆= 4

4−w 6∈ N. For ` < `∗ the area ranges are:

Range(L1`) = [4(`− 1), `w ] and Range(L2

`) = [`w , 4`]

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

8

)such

that stacks L1` show up for any ` = 1, . . . , k :

0 2w a+2

a−k

Type 2

L22

L21

L11

L12

L2k

a+k

kw

L1k

a−1

a

w a+1

a−2


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

).


Microscopic 3D Wulff shape

Typical Picture under Pm,−N,β

ΓNVolume of the microscopicWulff droplet

|ΓN | ≈m + m∗

2m∗N3

Theorem (Bodineau, Cerf-Pisztora): As N →∞ the scaled shape1N

ΓN converges to the macroscopic Wulff shape.


3D Surface Tension and Macroscopic Wulff Shape

n +

−

M

+

−

n

Kβ

h

ξβ(n) = − limM→∞

| cos n|M2

logZ±MZ−M

.

ξβ = maxh∈∂Kβ

h · n

Dilated Wulff Shape

Kmβ =

(m + m∗

2m∗|Kβ|

)1/3

Kβ


Bodineau, Cerf-Pisztora Result

N (Kmβ + u)

ΓN

Nu

Define (on unit box Λ ⊂ R3)

φN(t) = 1I{Nt∈ΓN} − 1I{Nt 6∈ΓN}.

Defineχm(t) = 1I{t∈Km

β} − 1I{t 6∈Kmβ}

Then, under{Pm,−N,β

},

limN→∞

minu‖φN(·)− χm(u + ·)‖L1(Λ) = 0


Macroscopic Facets

Kβ

nFn 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


Microscopic Facets

Zooming Bodineau, Cerf-Pisztora picture, what happens?

NFe1

NFe1

NFe1

ΓN

OR

ΓN

OR

ΓN


SOS Model

BN = {−N, . . . , N}2

ΓN

VN

BN = {−N, . . . , N}2

Ak0

k

`N

Bodineau, Schonmann,Shlosman ’05

PN (ΓN = γ) ∼ e−β|γ|

PmN (·) = PN

(·∣∣VN ≥ mN3

)Result: There exists a(β)↘ 0such that

`N = max{k : Ak ≥ a(β)N2

}satisfies A`N−1 ≥ (1− a(β))N2.


Effective Model of Microscopic Facets

ΓN

VN

SN

BN = {−N, . . . , N}2

psβ

pvβ Configuration:(

ΓN , {ξvi }i∈VN,{ξsj}j∈SN

).

Total number of particles:

ΞN =∑i∈VN

ξvi +∑j∈SN

ξsj

|Γ| - area of Γ

Bp(ξ) = pξ(1− p)1−ξ

β large

Probability Distribution:

PN (Γ, ξv , ξs) ∝ e−β|Γ|∏i∈VN

Bpv (ξvi )∏j∈SN

Bps (ξsj ).


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)


Creation of Facets

ps

pv

α(ΓN )

ΞN - total number of particles

EN (ΞN) =ps + pv

2N3 ∆

= pN3

Consider

PAN (·) = PN

(·∣∣ΞN = pN3 + AN2

)2D Surface Tension: logP

(α(ΓN) = aN2

)≈ −Nτβ(a).

Bulk Fluctuations: ∆ = 2(ps − pv ), EN

(ΞN

∣∣α(ΓN))

= pN3 + ∆α(ΓN).

logPN

(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)

}.


Surface Tension and Macroscopic Variational Problem

CNx

0 γ

wβ(γ) = e−β|γ|−∑C6∼γ Φβ(γ)

Gβ(Nx) =∑

γ:0→Nx

wβ(γ).

τβ(x) = − limN→∞

1

NlogGβ(Nx).

τβ(γ) =

∫γτβ(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 mina

{(δ − a)2

2Dβ+ min

a(L)=aτβ(L)

}.


Reduction to Large Contours

Fix β � 1. Bulk fluctuations simplify analysis of PAN . Recall the contour

representation Γ = {γi}.

Lemma 1 (No intermediate contours). ∀A > 0 there exists ε = ε(A) > 0such that

PAN

(∃γi :

1

εlogN ≤ |γi | ≤ εN

)= o(1).

Lemma 2 (Irrelevance of small contours)

PAN

(∣∣∑α(γi )1I{|γi |≤ε−1 log N}∣∣� N

)= o(1).

Definition: γ is large if |γ| ≥ εN.


Reduced Model of Large Contours

A. Fix A > 0 and forget about intermediate contours 1ε logN ≤ |γ| ≤ εN.

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

For Γ = {γi} collection of large contours the effective weight is

PN(Γ) ∝ exp{−β∑ |γi | −∑C6∼Γ Φβ(C)

}.

The family of clusters C depends on N and A. However cluster weightsΦβ(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:

PN (Γ, ξv , ξs) = PN(Γ)∏i∈VN

Bpv (ξvi )∏j∈SN

Bps (ξsj )


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−

2a+2

a−k∗

Set: PAN (·) = PN

(·∣∣ΞN = pN3 + AN2

). Then for any ν > 0 and any

A ≥ 0, the (random) collection of large contours Γ satisfies:

limN→∞ PAN

(minL∗−solutions to (VP)δ

dH(

1N Γ,L∗

)< ν

)= 1


1st Order (Shape) Transitions in Microscopic Models

1 For β � 1 pure 2+1 SOS, conditioned to stay positive and with anadditional 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 N2/3 in the context ofthe 2D Ising model (any β > βc) was established by Biskup, Chayes andKotecky (CMP’03).

3 For β � 1 pure 2+1 SOS, conditioned to stay positive and with anadditional attractive 0-layer boundary field h, Alexander, Dunlop andMiracle-Sole (JSP 2011) proved a sequence of layering transitions as h↘ 0.

4 For β � 1 pure 2+1 SOS (without bulk Bernoulli fields) models ofinterfaces with zero b.c. on ∂BN , and conditioned to stay positive on BN ,Caputo, Lubetzky, Martinelli, Sly and Toninelli proved in a series of works2012-14 that there are b 1

4β logNc macroscopic facets with asymptoticallydifferent Wulff Plaquette shapes.


Difficulty: Control of Interactions of Contours in Macroscopic Stacks

C

γ2

γ1

∑C6∼γ1∪γ2

Φβ(C) =∑C6∼γ1

Φβ(C) +∑C6∼γ2

Φβ(C)−∑

C6∼γ1∩γ2

Φβ(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.


Interaction Between ` Contours

γ1

γ2

γ3

γℓ


A general Result for Ising Polymers

C3

C4

x

y

γ C2

C1 wβ(γ) = e−β|γ|+∑C6∼γ Φβ(γ;C).

Assumption:

|Φβ(γ; C)| ≤ ce−χβ(diam(C)+1).

Theorem (I, Shlosman, Toninelli , JSP 2015)

If χ > 12 , 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 `, ordered

contours still has to be worked out.Dima Ioffe (Technion ) Microscopic Facets April 2015 28 / 33

Effective RW Representation of Ising Polymers

Portion of a Contour Between x and y

xy

γ1 γ2γm

ξ1 ξ2xy

ξm

eτβ(y−x)Gβ(y − x) ∼=∑

m

∑γ1,...γm

∏ρβ(γi )

{ρβ(·)} is a probability distribution on the set of irreducible animals.

ξ1 = (T1,X1), ξ2 = (T2,X2), . . . steps of the effective random walk.


Fluctuations of Facets near Flat Boundaries

ΓN VN

N

Nα

• Bulk fluctuation price for VN is ∼ VNN2

N3 ∼ VNN .

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

Therefore, N1−2α ∼ VNN ∼ N1+α

N = Nα gives α = 1/3.


Random Walks under Area Tilts

VN

−N N

a

b

X-trajectory of RW

• Partition Function:

Z a,bN,+,λN

=∑

X∈T a,bN,+

e−λNVN p(X)

• Scaling: xN(t) = λ1/3N X(λ

−2/3N t).

In general: {Φλ} - family ofself-potentials, defineΦλ(X) =

∑N−N Φλ(Xi ).

Z a,bN,+,λN

=∑

X∈T a,bN,+

e−ΦλN (X)p(X)


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,bN,+,λN

=∑

X∈T a,bN,+

e−ΦλN (X)p(X).

Scale: H2λΦλ(Hλ) = 1.

Assumption: limλ→0 H2λΦλ(Hλr) = q(r) and limr→∞ q(r) =∞.

Sturm-Liouville operator on R+: Set σ2 =∑

x x2p(x) and

L = σ2

2d2

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

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

λNX(H2

λNt) converges to ergodic diffusion with generator

σ2

2ϕ21(r)

d

dr

(ϕ2

1(r)d

dr

).


` Ordered Random Walks under Area Tilts

b2

a2

a1

b1X2

X1

X`

−N N

a`

b`

Za,bN,+,λN

=∑

X∈T a,bN,+

e−∑`

m=1 ΦλN (Xm)∏m=1

p(Xm).

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

λNX(H2

λNt) converges

to ergodic diffusion on {r : 0 < r` < · · · < r2 < r1} with generatorσ2

2∆2`(r)

div(∆2` (r)∇

).


an effective model of facets formation

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