Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

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DOI:10.1214/09-AOP492

©Institute of Mathematical Statistics, 2010

HOMOGENEOUS NUCLEATION FOR GLAUBER AND KAWASAKI DYNAMICS IN LARGE VOLUMES AT LOW TEMPERATURES

BY ANTONBOVIER1, FRANK DENHOLLANDER2 ANDCRISTIANSPITONI

Rheinische Friedrich–Wilhelms-Universitaet Bonn, Leiden University and EURANDOM and Leiden University and EURANDOM

In this paper, we study metastability in large volumes at low tempera-tures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Letβdenote the inverse temperature and letβ ⊂Z2be a square box with periodic

bound-ary conditions such that limβ→∞|β| = ∞. We run the dynamics onβ,

starting from a random initial configuration where all of the droplets (clus-ters of plus-spins and clus(clus-ters of particles, respectively) are small. For large βand for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place un-der the dynamics. This transition is triggered by the appearance of a sin-glecritical dropletsomewhere inβ. Using potential-theoretic methods, we

compute theaverage nucleation time(the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to 1 asβ→ ∞. It turns out that this time grows asKeβ/|β|for Glauber dynamics and as

Kβeβ/|β|for Kawasaki dynamics, whereis the local canonical (resp.

grand-canonical) energy, to create a critical droplet andKis a constant re-flecting the geometry of the critical droplet, provided these times tend to in-finity (which puts a growth restriction on|β|). The fact that the average

nucleation time is inversely proportional to|β|is referred to as

homoge-neous nucleationbecause it says that the critical droplet for the transition appears essentially independently in small boxes that partitionβ.

1. Introduction and main results.

1.1. Background. In a recent series of papers, Gaudillière et al. [12–14] study a system of lattice gas particles subject to Kawasaki hopping dynamics in a large box at low temperature and low density. Using the so-calledpathwise approachto

Received June 2008; revised May 2009.

1Supported by DFG and NWO through the Dutch–German Bilateral Research Group on “Random

Spatial Models from Physics and Biology” (2003–2009).

2Supported by NWO through Grant no. 613.000.556.

AMS 2000 subject classifications.60K35, 82C26.

Key words and phrases.Glauber dynamics, Kawasaki dynamics, critical droplet, metastable tran-sition time, last-exit biased distribution, Dirichlet principle, Berman–Konsowa principle, capacity, flow, cluster expansion.

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metastability (see Olivieri and Vares [23]), they show that the transition time be-tween the metastable state (the gas phase with only small droplets) and the stable state (the liquid phase with one or more large droplets) is inversely proportional to the volume of the large box, provided that the latter does not grow too fast with the inverse temperature. This type of behavior is calledhomogeneous nucleation be-cause it corresponds to the situation where the critical droplet triggering the nucle-ation appears essentially independently in small boxes that partition the large box. Thenucleation time(the first time a critical droplet appears and starts growing) is computed up to a multiplicative error that is small on the scale of the exponential of the inverse temperature. The techniques developed in [12–14] center around the idea of approximating the low temperature and low density Kawasaki lattice gas by an ideal gas without interaction and showing that this ideal gas stays close to equilibrium while exchanging particles with droplets that are growing and shrink-ing. In this way, the large system is shown to behave essentially like the union of many small independent systems, leading to homogeneous nucleation. The proofs are long and complicated, but they provide considerable detail about the typical trajectory of the system prior to and shortly after the onset of nucleation.

In the present paper, we consider the same problem, both for Ising spins subject to Glauber spin-flip dynamics and for lattice gas particles subject to Kawasaki hopping dynamics. Using the potential-theoretic approach to metastability (see Bovier [5]), we improve upon an aspect of the results in [12–14], namely, we compute the average nucleation time up to a multiplicative error that tends to 1 as the temperature tends to 0, thereby providing a very sharp estimate of the time at which the gas starts to condensate.

We have no results about the typical time it takes for the system to grow a large droplet after the onset of nucleation. This is a difficult problem that will be addressed in future work. All that we can prove is that the dynamics has a negligible probability to shrink down a supercritical droplet once it has managed to create one. At least this shows that the appearance of a single critical droplet indeed represents the threshold for nucleation, as was shown in [12–14]. A further restriction is that we need to draw the initial configuration according to a class

of initial distributions on the set of subcritical configurations, called thelast-exit biased distributions since these are particularly suitable for the use of potential theory. It remains a challenge to investigate to what extent this restriction can be relaxed. This problem is addressed with some success in [12–14] and will also be tackled in future work.

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In the same direction, Bianchi, Bovier, and Ioffe [3] study the dynamics of the random field Curie–Weiss model on a finite box at a fixed positive temperature.

As we will see, the basic difficulty in estimating the nucleation time is to obtain sharp upper and lower bounds oncapacities. Upper bounds follow from the Dirich-let variational principle, which represents a capacity as an infimum over a class of test functions. In [3], a new technique is developed, based on a variational princi-ple due to Berman and Konsowa [2], which represent a capacity as a supremum over a class of unit flows. This technique allows for lower bounds to be obtained and it will also be exploited here.

1.2. Ising spins subject to Glauber dynamics. We will study models in finite boxes,β, in the limit as both the inverse temperature,β, and the volume of the

box, |β|, tend to infinity. Specifically, we let β ⊂Z2 be a square box with

odd side length, centered at the origin with periodic boundary conditions. A spin configuration is denoted byσ = {σ (x):xβ}, withσ (x)representing the spin

at sitex, and is an element of= {−1,+1}β. It will frequently be convenient to

identify a configurationσ with itssupport, defined as supp[σ] = {xβ:σ (x)= +1}.

The interaction is defined by the the usual Ising Hamiltonian

Hβ(σ )= − J

2

(x,y)β

xy

σ (x)σ (y)h

2

xβ

σ (x), σXβ,

(1.1)

whereJ >0 is the pair potential,h >0 is the magnetic field andxymeans that

x andy are nearest neighbors. TheGibbs measureassociated with is

μβ(σ )=

1

eβHβ(σ ), σX

β,

(1.2)

where is the normalizing partition function.

The dynamics of the model will the continuous-time Markov chain,(σ (t))t≥0,

with state space, whose transition rates are given by

cβ(σ, σ)=

eβ[Hβ(σ)Hβ(σ )]+, forσ=σx for somex β,

0, otherwise,

(1.3)

whereσx is the configuration obtained fromσby flipping the spin at sitexand[·]+ denotes the positive part. We refer to this Markov process asGlauber dynamics. It is ergodic and reversible with respect to its unique invariant measure,μβ, that is,

μβ(σ )cβ(σ, σ)=μβ(σ)cβ(σ, σ )σ, σXβ.

(1.4)

Glauber dynamics exhibits metastable behavior in the regime 0< h <2J, β→ ∞.

(1.5)

To understand this, let us briefly recall what happens in a finiteβ-independent box

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FIG. 1. A critical droplet for Glauber dynamics on.The shaded area represents the(+1)-spins; the unshaded area represents the(−1)-spins[see(1.6)].

(resp.+1). As was shown by Neves and Schonmann [22], for Glauber dynamics restricted towith periodic boundary conditions and subject to (1.5), thecritical droplets for the crossover from to are elements of the set of all those

configurations where the(+1)-spins form anc×(c−1)quasi-square (in either

of the both orientations) with a protuberance attached to one of its longest sides, where

c=

2J

h

(1.6)

(see Figures 1 and2; for nondegeneracy reasons, it is assumed that 2J / h /∈N). The quasi-squares without the protuberance are calledproto-critical droplets.

Let us now return to our setting with finiteβ-dependent volumesβ⊂Z2. We

will start our dynamics on β from initial configurations in which all droplets

are “sufficiently small.” To make this notion precise, letCB(σ ), σ, be the

FIG. 2. A nucleation path fromtofor Glauber dynamics.in(1.10)is the minimal energy

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configuration that is obtained from σ by a “bootstrap percolation map,” that is, by circumscribing all of the droplets inσ with rectangles and continuing to do so in an iterative manner until a union of disjoint rectangles is obtained (see Kotecký and Olivieri [19]). We callCB(σ )subcriticalif all of its rectangles fit inside

proto-critical droplets and are at distance≥2 from each other (i.e., are noninteracting).

DEFINITION1.1. (a)S= {σ:CB(σ )is subcritical};

(b)P = {σS:cβ(σ, σ) >0 for someσSc};

(c)C= {σSc:cβ(σ, σ) >0 for someσS}.

We refer toS,P andCas the set ofsubcritical,proto-criticalandcritical con-figurations, respectively. Note that, for everyσ, each step in the bootstrap

percolation mapσCB(σ )deceases the energy and therefore the Glauber

dy-namics moves from σ toCB(σ ) in a time of order 1. This is whyCB(σ ), rather

thanσ, appears in the definition ofS.

For1, 2∈N, let R1,2(x)β be the 2 rectangle whose lower-left

corner isx. We always take1≤2and allow for both orientations of the

rectan-gle. For L=1, . . . ,2c−3, let QL(x) denote theLth element in thecanonical

sequence of growing squares and quasi-squares

R1,2(x), R2,2(x), R2,3(x), R3,3(x), . . . , Rc−1,c−1(x), Rc−1,c(x). (1.7)

In what follows, we will choose to start the dynamics in a way that is suitable for the use of potential theory, as follows. First, we take the initial law to be con-centrated on one of the setsSLSdefined by

SL= {σS: each rectangle inCB(σ )

(1.8)

fits insideQL(x)for somexβ},

whereLis any integer satisfying

L∗≤L≤2c−3

(1.9)

withL∗=min

1≤L≤2c−3 : lim β→∞

μβ(SL) μβ(S) =

1

.

In words,SL is the subset of those subcritical configurations whose droplets fit

inside a square or quasi-square labeledL, withLchosen large enough so thatSL

is typical withinS under the Gibbs measureμβ asβ→ ∞(our results will not

depend on the choice ofLsubject to these restrictions). Second, we take the initial law to be biased according to the last exit ofSL for the transition from SL to a

target set inSc. (Different choices will be made for the target set and the precise definition of the biased law will be given in Section2.2.) This is ahighly specific

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REMARKS. (1) Note that S2c−3 =S, which implies that the range of L -values in (1.9) is nonempty. The value of L∗ depends on how fast β grows

with β. In Appendix C.1, we will show that, for every 1 ≤ L ≤ 2c − 4,

limβ→∞μβ(SL)/μβ(S)=1 if and only if limβ→∞|β|eβL+1=0, withL+1

the energy needed to create a dropletQL+1(0)at the origin. Thus, if|β| =eθβ,

thenL∗=L(θ )=(2c−3)∧min{L∈N:L+1> θ}, which increases stepwise

from 1 to 2c−3 asθ increases from 0 todefined in (1.10).

(2) If we draw the initial configuration σ0 from some subset of S that has a

strong recurrence property under the dynamics, then the choice of initial distribu-tion on this subset should not matter. This issue will be addressed in future work.

To state our main theorem for Glauber dynamics, we need some further nota-tion. The key quantity for the nucleation process is

=J[4c] −h[c(c−1)+1],

(1.10)

which is the energy needed to create a critical droplet of (+1)-spins at a given location in a sea of(−1)-spins (see Figures1and2). Forσ, letPσ denote

the law of the dynamics starting fromσ and, forνa probability distribution onX, put

Pν(·)=

σXβ

Pσ(·)ν(σ ).

(1.11)

For a nonempty setA, let

τA=inf{t >0 :σtA, σt−∈/A}

(1.12)

denote the first time that the dynamics entersA. For nonempty and disjoint sets

A,B, letνAB denote thelast-exit biased distribution onAfor the crossover toBdefined in (2.9) in Section2.2. Put

N1=4c, N2=43(2c−1).

(1.13)

ForM∈NwithMc, define

DM= {σ:∃xβ such that supp[CB(σ )] ⊃RM,M(x)},

(1.14)

that is, the set of configurations containing a supercritical droplet of sizeM. For our results below to be valid, we need to assume that

lim

β→∞|β| = ∞, βlim→∞|β|e

β=0.

(1.15)

THEOREM1.2. In the regime(1.5),subject to(1.9)and(1.15),the following hold:

(a)

lim

β→∞|β|e

βE

νSSc

L

Sc)= 1

N1;

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(b)

lim

β→∞|β|e

βE

νSSc\C

L

Sc\C)= 1

N2;

(1.17)

(c)

lim

β→∞|β|e

βE

νDM SL

DM)= 1

N2 ∀

cM≤2c−1.

(1.18)

The proof of Theorem1.2will be given in Section3. Part (a) says that the av-erage time to create a critical droplet is[1+o(1)]eβ/N1|β|. Parts (b) and (c)

say that the average time to go beyond this critical droplet and to grow a droplet that is twice as large is [1+o(1)]eβ/N2|β|. The factor N1 counts the

num-ber ofshapesof the critical droplet, while |β|counts the number of locations. The average times to create a critical and a supercritical droplet differ by a fac-tor N2/N1<1. This is because once the dynamics is “on top of the hill” C, it

has a positive probability to “fall back” to S. On average, the dynamics makes

N1/N2>1 attempts to reach the topCbefore it finally “falls over” toSc\C. After

that, it rapidly grows a large droplet (see Figure2).

REMARKS. (1) The second condition in (1.15) will not actually be used in the proof of Theorem1.2(a). If this condition fails, then there is a positive probability to see a proto-critical droplet inβ under the starting measureνS

c

SL and nucleation sets in immediately. Theorem1.2(a) continues to be true, but it no longer describes metastable behavior.

(2) In AppendixD, we will show that theaverageprobability under the Gibbs measureμβof destroying a supercritical droplet and returning to a configuration in SLis exponentially small inβ. Hence, the crossover fromSLtoSc\Crepresents

the true threshold for nucleation and Theorem1.2(b) represents the true nucleation time.

(3) We expect Theorem 1.2(c) to hold for values of M that grow with β as

M=eo(β). As we will see in Section3.3, the necessary capacity estimates carry over, but the necessary equilibrium potential estimates are not yet available. This problem will be addressed in future work.

(4) Theorem1.2should be compared with the results in Bovier and Manzo [8] for the case of a finite β-independent box (large enough to accommodate a critical droplet). In that case, if the dynamics starts from , then the average

time it needs to hit C (the set of configurations in with a critical droplet),

respectively,equals

Keβ[1+o(1)] withK=K(, c)=

1

N

1

||forN=N1(resp.N2).

(1.19)

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time when the plus-phase appears near the origin. This time, which depends on how a supercritical droplet grows and eventually invades the origin, was studied by Dehghanpour and Schonmann [10, 11], Shlosman and Schonmann [24] and, more recently, by Cerf and Manzo [9].

1.3. Lattice gas subject to Kawasaki dynamics. We next consider the lattice gas subject to Kawasaki dynamics and state a similar result for homogeneous nu-cleation. Some aspects are similar to what we have seen for Glauber dynamics, but there are notable differences.

A lattice gas configuration is denoted byσ= {σ (x):x}, withσ (x)

repre-senting the number of particles at sitex, and is an element of= {0,1}β. The

Hamiltonian is given by

Hβ(σ )= −U

(x,y)β

xy

σ (x)σ (y), σXβ,

(1.20)

where−U <0 is the binding energy andxymeans thatxandyare neighboring sites. Thus, we are working in thecanonical ensemble, that is, there is no term analogous to the second term in (1.1). The number of particles inβ is

= ρβ|β|,

(1.21)

whereρβ is the particle density, which is chosen to be

ρβ=eβ, >0.

(1.22) Put

X(nβ)

β = {σ:|supp[σ]| =},

(1.23)

where supp[σ] = {xβ:σ (x)=1}.

REMARK. If we were to work in the grand-canonical ensemble, then we would have to consider the Hamiltonian

Hgc(σ )= −U (x,y)β

xy

σ (x)σ (y)+ xβ

σ (x), σXβ,

(1.24)

where >0 is an activity parameter taking over the role ofhin (1.1). The second term would mimic the presence of an infinite gas reservoir with densityρβ outside β. Such a Hamiltonian was used in earlier work on Kawasaki dynamics, when

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The dynamics of the model will be the continuous-time Markov chain,(σt)t≥0,

with state spaceXβ(nβ), whose transition rates are

cβ(σ, σ)=

⎧ ⎨ ⎩

eβ[Hβ(σ)Hβ(σ )]+,

forσ=σx,y for somex, yβ withxy,

0, otherwise,

(1.25)

where σx,y is the configuration obtained from σ by interchanging the values at sitesxandy. We refer to this Markov process asKawasaki dynamics. It is ergodic and reversible with respect to the canonical Gibbs measure

μβ(σ )=

1

Z(nββ)

eβHβ(σ ), σX(nβ)

β ,

(1.26)

whereZβ(nβ)is the normalizing partition function. Note that the dynamics preserves particles, that is, it isconservative.

Kawasaki dynamics exhibits metastable behavior in the regime

U < <2U, β→ ∞.

(1.27)

This is again inferred from the behavior of the model in a finiteβ-independent box

⊂Z2. Let (resp.) denote the configurations where all of the sites in

are vacant (resp. occupied). For Kawasaki dynamics onwith anopen boundary, where particles are annihilated at rate 1 and created at rate eβ, it was shown in den Hollander, Olivieri, and Scoppola [18] and in Bovier, den Hollander, and Nardi [7] that, subject to (1.27) and for the Hamiltonian in (1.24), the critical droplets for the crossover from to are the set of all those configurations

where the particles form either:

(1) an(c−2)×(c−2)square with four bars attached to the four sides with

total length 3c−3; or

(2) an(c−1)×(c−3)rectangle with four bars attached to the four sides with

total length 3c−2,

plus afree particleanywhere in the box, where

c=

U

2U

(1.28)

[see Figures3and4; for nondegeneracy reasons, it is assumed thatU/(2U) /

N].

Let us now return to our setting with finiteβ-dependent volumes. We define a reference distance,, as

L2β=e(δβ)β= 1

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FIG. 3. A critical droplet for Kawasaki dynamics on(a proto-critical droplet plus a free particle). The shaded area represents the particles;the unshaded area represents the vacancies[see(1.28)]. The proto-critical droplet for Kawasaki dynamics drawn in the figure has the same shape as the critical droplet for Glauber dynamics, but there are other shapes as well[see(1)and (2)below (1.27)]. A proto-critical droplet for Kawasaki dynamics becomes critical when a free particle is added.

with

lim

β→∞δβ=0, βlim→∞βδβ= ∞,

(1.30)

FIG. 4. A nucleation path from to for Kawasaki dynamics onwith open boundary.

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that is, is marginally below the typical inter-particle distance. We assume

to be odd and write BLβ,Lβ(x), xβ, for the square box with side length whose center isx.

DEFINITION1.3. (a)S= {σXβ(nβ):|supp[σ]∩BLβ,Lβ(x)| ≤c(c−1)+1

xβ};

(b)P = {σS:cβ(σ, σ) >0 for someσSc};

(c)C= {σSc:cβ(σ, σ) >0 for someσS};

(d)C−= {σC:∃xβ such thatBLβ,Lβ(x)contains a proto-critical droplet plus a free particle at distance};

(e)C+= the set of configurations obtained fromC−by moving the free particle to a site at distance 2 from the proto-critical droplet, that is, next to its boundary;

(f)C˜= the set of configurations “interpolating” betweenC−andC+, that is, the free particle is somewhere between the boundary of the proto-critical droplet and the boundary of the box of size around it.

As before, we refer to S,P andC as the set ofsubcritical,proto-criticaland

critical configurations, respectively. Note that, for every σS, the number of particles in a box of size does not exceed the number of particles in a

proto-critical droplet. These particles do not have to form a cluster or to be near each other because the Kawasaki dynamics brings them together in a time of orderL2β= o(1/ρβ).

The initial law will again be concentrated on setsSLS, this time defined by SL=

σXβ(nβ):|supp[σ] ∩BLβ,Lβ(x)| ≤Lxβ

,

(1.31)

whereLis any integer satisfying

L∗≤Lc(c−1)+1

(1.32)

withL∗=min

1≤Lc(c−1)+1 : lim β→∞

μβ(SL) μβ(S) =

1

.

In words,SLis the subset of those subcritical configurations for which no box of

size carries more thanLparticles, withLagain chosen such thatSLis typical

withinS under the Gibbs measureμβ asβ→ ∞.

REMARK. Note that Sc(c−1)+1 =S. As for Glauber dynamics, the value of L∗ depends on how fast β grows with β. In Appendix C.2, we will show

that for every 1≤Lc(c−1), limβ→∞μβ(SL)/μβ(S)=1 if and only if

limβ→∞|β|eβ(L+1−)=0, withL+1 the energy needed to create a droplet

of L+1 particles, closest in shape to a square or quasi-square, in BLβ,Lβ(0) under the grand-canonical Hamiltonian on this box. Thus, if |β| =eθβ, then L∗=L(θ )= [c(c−1)+1] ∧min{L∈N:L+1− > θ}, which increases

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Set

= −U[(c−1)2+c(c−1)+1] +[c(c−1)+2],

(1.33)

which is the energy of a critical droplet at a given location with respect to the

grand-canonical Hamiltonian given by (1.24) (see Figures 3 and 4). Put N = 1

3

2

c(2c−1). ForM∈NwithMc, define

DM= {σ:∃xβ such that supp[(σ )] ⊃RM,M(x)},

(1.34)

that is, the set of configurations containing a supercritical droplet of sizeM. For our results below to be valid, we need to assume that

lim

β→∞|β|ρβ= ∞, βlim→∞|β|e

β=0.

(1.35)

This first condition says that the number of particles tends to infinity and ensures that the formation of a critical droplet somewhere does not globally deplete the surrounding gas.

THEOREM1.4. In the regime(1.27),subject to(1.32)and(1.35),the follow-ing hold:

(a)

lim

β→∞|β|

4π βe

βE

νS(Sc\ ˜C)C+

L

τ(Sc\ ˜C)C+

= 1

N;

(1.36)

(b)

lim

β→∞|β|

4π βe

βE

νDM SL

DM)= 1

NcM≤2c−1.

(1.37)

The proof of Theorem 1.4, which is an analog of Theorem 1.2, will be given in Section4. Part (a) says that the average time to create a critical droplet is[1+

o(1)](β/4π )eβN|β|. The factorβ/4πcomes from the simple random walk

that is performed by the free particle “from the gas to the proto-critical droplet” (i.e., the dynamics goes fromC− toC+), which slows down the nucleation. The factorN counts the number of shapes of the proto-critical droplet (see Bovier, den Hollander and Nardi [7]). Part (b) says that once the critical droplet is created, it rapidly grows to a droplet that has twice the size.

REMARKS. (1) As for Theorem1.2(c), we expect Theorem1.4(b) to hold for values ofMthat grow withβ asM=eo(β). See Section4.2for more details.

(2) In AppendixD, we will show that theaverageprobability under the Gibbs measureμβ of destroying a supercritical droplet and returning to a configuration

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represents the true threshold for nucleation and Theorem1.4(a) represents the true nucleation time.

(3) It was shown in Bovier, den Hollander and Nardi [7] that the average crossover time in a finite boxequals

Keβ[1+o(1)] withK=K(, c)

log||

4π

1

N||, →Z 2.

(1.38)

This matches the |β|-dependence in Theorem 1.4, with the logarithmic factor

in (1.38) accounting for the extra factor β in Theorem1.4compared to Theo-rem1.2. Note that this factor is particularly interesting since it says that the effec-tive box sizeresponsible for the formation of a critical droplet is.

1.4. Outline. The remainder of this paper is organized as follows. In Sec-tion2, we present a brief sketch of the basic ingredients of the potential-theoretic approach to metastability. In particular, we exhibit a relation between average crossover times and capacities and we state two variational representations for capacities, the first of which is suitable for deriving upper bounds and the sec-ond for deriving lower bounds. Section3contains the proof of our results for the case of Glauber dynamics. Technically, this will be relatively easy and will give a first flavor of how our method works. In Section4, we deal with Kawasaki dy-namics. Here, we will encounter several rather more difficult issues, all coming from the fact that Kawasaki dynamics isconservative. The first issue is to under-stand why the constant, representing the local energetic cost to create a critical droplet, involves the grand-canonical Hamiltonian, even though we are working in the canonical ensemble. This mystery will, of course, be resolved by the observa-tion that the formaobserva-tion of a critical droplet reduces the entropy of the system: the precise computation of this entropy loss yields via equivalence of ensembles. The second problem is to control the probability of a particle moving from the gas to the proto-critical droplet at the last stage of the nucleation. This nonlocality issue will be dealt with via upper and lower estimates. Appendices A–Dcollect some technical lemmas that are needed in Sections3and4.

The extension of our results to higher dimensions is limited only by the com-binatorial problems involved in the computation of the number of critical droplets (which is hard in the case of Kawasaki dynamics) and of the probability for sim-ple random walk to hit a critical drosim-plet of a given shape when coming from far. We will not pursue this generalization here. The relevant results for a β -independent box inZ3 can be found in Ben Arous and Cerf [1] (Glauber) and den Hollander, Nardi, Olivieri and Scoppola [17] (Kawasaki). For recent overviews on droplet growth in metastability, we refer the reader to den Hollander [15, 16] and Bovier [4, 5]. A general overview on metastability is given in the monograph by Olivieri and Vares [23].

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ob-servations. First, most quantities of physical interest can be represented in terms of Dirichlet problems associated with the generator of the dynamics. Second, the Green function of the dynamics can be expressed in terms of capacities and equi-librium potentials. Third, capacities satisfy variational principles that allow upper and lower bounds to be obtained in a flexible way. We will see that in the current setting, the implementation of these observations provides very sharp results.

2.1. Equilibrium potential and capacity. The fundamental quantity in the the-ory is theequilibrium potential,hA,B, associated with two nonempty disjoint sets

of configurations,A,BX ( orX (nβ)

β ), which, probabilistically, is given by

hA,B(σ )=

⎧ ⎨ ⎩

Pσ(τA< τB), forσ(AB)c,

1, forσA, 0, forσB, (2.1)

where

τA=inf{t >0 :σtA, σt−∈/A},

(2.2)

(σt)t≥0 is the continuous-time Markov chain with state space X and Pσ is its

law starting from σ. This function is harmonic and is the unique solution of the

Dirichlet problem

(LhA,B)(σ )=0, σ(AB)c, hA,B(σ )=1, σA,

(2.3)

hA,B(σ )=0, σB,

where the generator is the matrix with entries

L(σ, σ)=cβ(σ, σ)δσ,σcβ(σ ), σ, σX,

(2.4)

wherecβ(σ )is the total rate at which the dynamics leavesσ, cβ(σ )=

σX\{σ}

cβ(σ, σ), σX.

(2.5)

A related quantity is theequilibrium measureonA, which is defined as

eA,B(σ )= −(LhA,B)(σ ), σA.

(2.6)

The equilibrium measure also has a probabilistic meaning, namely, Pσ(τB< τA)=

eA,B(σ ) cβ(σ )

, σA.

(2.7)

The key object we will work with is thecapacity, which is defined as CAP(A,B)=

σA

μβ(σ )eA,B(σ ).

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2.2. Relation between crossover time and capacity. The first important ingre-dient of the potential-theoretic approach to metastability is a formula for the aver-age crossover time fromAtoB. To state this formula, we define the probability measureνAB onAwhich we already referred to in Section1, namely,

νAB(σ )=

⎧ ⎨ ⎩

μβ(σ )eA,B(σ )

CAP(A,B) , forσA,

0, forσAc. (2.9)

The following proposition is proved in, for example, Bovier [5].

PROPOSITION2.1. For any two nonempty disjoint setsA,BX,

σA

νAB(σ )Eσ(τB)=

1 CAP(A,B)

σBc

μβ(σ )hA,B(σ ).

(2.10)

REMARKS. (1) Due to (2.7) and (2.8), the probability measureνAB(σ )can be written as

νAB(σ )=μβ(σ )cβ(σ )

CAP(A,B)Pσ(τB< τA), σA,

(2.11)

and thus has the flavor of a last-exit biased distribution. Proposition2.1explains why our main results on average crossover times stated in Theorem1.2and1.4are formulated for this initial distribution. Note that

μβ(A)

σBc

μβ(σ )hA,B(σ )μβ(Bc).

(2.12)

We will see that in our setting,μβ(Bc\A)=o(μβ(A))asβ→ ∞so that the sum

in the right-hand side of (2.10) is∼μβ(A)and the computation of the crossover

time reduces to the estimation of CAP(A,B).

(2) For a fixed target setB, the choice of the starting setAis free. It is tempt-ing to choose A= {σ} for some σX. This was done for the case of a finite

β-independent box. However, in our case (and, more generally, in cases where the state space is large) such a choice would give intractable numerators and de-nominators in the right-hand side of (2.10). As a rule, to make use of the identity in (2.10),A must be so large that the harmonic functionhA,B “does not change

abruptly near the boundary ofA” for the target setBunder consideration.

As noted above, average crossover times are essentially governed by capacities. The usefulness of this observation comes from the computability of capacities, as will be explained next.

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The analog of Green’s identity relates it to a bulk quantity. Indeed, in terms of the

Dirichlet formdefined by

E(h)=1

2

σ,σX

μβ(σ )cβ(σ, σ)[h(σ )h(σ)]2, h:X → [0,1],

(2.13)

it follows, via (2.1) and (2.7) and (2.8), that

CAP(A,B)=E(hA,B).

(2.14)

Elementary variational calculus shows that the capacity satisfies theDirichlet prin-ciple.

PROPOSITION2.2. For any two nonempty disjoint setsA,BX, CAP(A,B)= min

h:X→[0,1] h|A≡1,h|B≡0

E(h).

(2.15)

The importance of the Dirichlet principle is that it yields computable upper bounds for capacities by means of suitable choices of the test function h. In metastable systems, with the proper physical insight, it is often possible to guess a reasonable test function. In our setting, this will be seen to be relatively easy.

2.4. The Berman–Konsowa principle:a variational principle for lower bounds. We will describe a little-known variational principle for capacities that is originally due to Berman and Konsowa [2]. Our presentation will follow the argument given in Bianchi, Bovier and Ioffe [3].

In the following, it will be convenient to think of X as the vertex set of a graph (X,E)whose edge setE consists of all pairs(σ, σ), σ, σX, for which

cβ(σ, σ) >0.

DEFINITION 2.3. Given two nonempty disjoint sets A,BX, a loop-free nonnegative unit flow,f, fromAtoBis a functionf:E→ [0,)such that:

(a) (f (e) >0⇒f (e)=0)eE; (b) f satisfies Kirchoff’s law, that is,

σX

f (σ, σ)= σX

f (σ, σ )σX\(AB);

(2.16)

(c) f is normalized, that is,

σA

σX

f (σ, σ)=1=

σX

σB

f (σ, σ );

(2.17)

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A natural flow is theharmonic flow, which is constructed from the equilibrium potentialhA,B as

fA,B(σ, σ)=

1

CAP(A,B)μβ(σ )cβ(σ, σ )[h

A,B(σ )hA,B(σ)]+,

(2.18)

σ, σX.

It is easy to verify thatfA,B satisfies (a)–(d). Indeed, (a) is obvious, (b) uses the

harmonicity of hA,B, (c) follows from (2.6) and (2.8), while (d) comes from the

fact that the harmonic flow only moves in directions wherehA,B decreases.

A loop-free nonnegative unit flow f is naturally associated with a probability measurePf on self-avoiding paths,γ. To see this, defineF (σ )=σXf (σ, σ),

σX \ B. Then Pf is the Markov chain (σn)n∈N0 with initial distribution

Pf

0)=F (σ0)1A(σ0)and transition probabilities

qf(σ, σ)=f (σ, σ )

F (σ ) , σX\B,

(2.19)

such that the chain is stopped upon arrival inB. In terms of this probability mea-sure, we have the following proposition (see [3] for a proof).

PROPOSITION 2.4. LetA,BX be two nonempty disjoint sets. Then, with the notation introduced above,

CAP(A,B)= sup

f∈UA,B

Ef eγ

f (el, er) μβ(el)cβ(el, er)

1 ,

(2.20)

wheree=(el, er) and the expectation is with respect toγ.Moreover,the supre-mum is realized for the harmonic flowfA,B.

The nice feature of this variational principle is that any flow gives a computable lower bound. In this sense, (2.15) and (2.20) complement each other. Moreover, since the harmonic flow is optimal, a good approximation of the harmonic function

hA,B by a test functionhleads to a good approximation of the harmonic flowfA,B

by a test flowf after puttinghinstead of hA,B into (2.18). Again, in metastable

systems, with the proper physical insight, it is often possible to guess a reasonable flow. We will see in Sections3–4how this is put to work in our setting.

3. Proof of Theorem1.2.

3.1. Proof of Theorem 1.2(a). To estimate the average crossover time from

SLStoSc, we will use Proposition2.1. WithA=SLandB=Sc, (2.10) reads

σSL

νSSLc(σ )Eσ(τSc)= 1 CAP(SL,Sc)

σS

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The left-hand side is the quantity of interest in (1.16). In Sections3.1.1and3.1.2, we estimate σSμβ(σ )hSL,Sc(σ ) and CAP(SL,S

c). The estimates will show

that

r.h.s. (3.1)= 1

N1|β|

[1+o(1)], β→ ∞.

(3.2)

3.1.1. Estimate ofσSμβ(σ )hSL,Sc(σ ).

LEMMA3.1. σSμβ(σ )hSL,Sc(σ )=μβ(S)[1+o(1)]asβ→ ∞.

PROOF. Write, using (2.1),

σS

μβ(σ )hSL,Sc(σ )=

σSL

μβ(σ )hSL,Sc(σ )+

σS\SL

μβ(σ )hSL,Sc(σ )

(3.3)

=μβ(SL)+

σS\SL

μβ(σ )Pσ(τSL< τSc).

The last sum is bounded above by μβ(S\SL), but μβ(S\SL)=o(μβ(S)) as β→ ∞by our choice ofLin (1.9).

3.1.2. Estimate ofCAP(SL,Sc).

LEMMA 3.2. CAP(SL,Sc)=N1|β|eβμβ(S)[1+o(1)] asβ→ ∞with N1=4c.

PROOF. The proof proceeds via upper and lower bounds.

Upper bound. We use the Dirichlet principle and a test function that is equal to 1 onS to get the upper bound

CAP(SL,Sc)≤CAP(S,Sc)=

σS,σSc

cβ(σ,σ)>0

μβ(σ )cβ(σ, σ)

(3.4)

=

σS,σSc

cβ(σ,σ)>0

[μβ(σ )μβ(σ)] ≤μβ(C),

where the second equality uses (1.4) in combination with the fact thatcβ(σ, σ)cβ(σ, σ )=1, by (1.3). Thus, it suffices to show that

μβ(C)N1|β|eβ[1+o(1)] asβ→ ∞.

(3.5)

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FIG. 5. Rc−1,c(x)(shaded box)and[Rc+1,c+2(x(1,1))]

c(complement of dotted box).

yβ, then σ has a (+1)-spin at y situated on the boundary of one of these

rectangles. Let

ˆ

S(x)= {σS: supp[σ] ⊆Rc−1,c(x)}, (3.6)

ˇ

S(x)=σS: supp[σ] ⊆Rc+1,c+2

x(1,1)c.

For everyσP, we haveσ= ˆσ∨ ˇσ for someσˆ ∈ ˆS(x)andσˇ ∈ ˇS(x), uniquely decomposing the configuration into two noninteracting parts inside Rc−1,c(x) and[Rc+1,c+2(x(1,1))]

c(see Figure5). We have

Hβ(σ )Hβ()= [Hβ(σ )ˆ −Hβ()] + [Hβ(σ )ˇ −Hβ()].

(3.7)

Moreover, for anyy /∈supp[CB(σ )], we have

Hβ(σy)Hβ(σ )+2Jh.

(3.8) Hence,

μβ(C)=

1

σP

xβ

σxC

eβHβ(σx)

≤ 1

N1eβ[2JhHβ()]

xβ

ˇ

σ∈ ˇS(x)

eβHβ(σ )ˇ

ˆ

σ∈ ˆS(x)

ˆ

σ∨ ˇσP

eβHβ(σ )ˆ

(3.9)

≤ [1+o(1)] 1

N1|β|eβ

ˇ

σ∈ ˇS(0)

eβHβ(σ )ˇ

= [1+o(1)]N1|β|eβμβ(Sˇ(0)),

where the first inequality uses (3.7)–(3.8), withN1=2×2c=4c counting the

number of critical droplets that can arise from a proto-critical droplet via a spin flip (see Figure1), and the second inequality uses the fact that

ˆ

σ ∈ ˆS(0), σˆ ∨ ˇσP

(3.10)

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FIG. 6. Canonical order to break down a critical droplet.

with equality in the right-hand side if and only if supp[ ˆσ] =Rc−1,c(0). Combin-ing (3.4) and (3.9) with the inclusionSˇ(0)S, we get the upper bound in (3.5).

Lower bound. We exploit Proposition2.4by making a judicious choice for the flowf. In fact, in the Glauber case, this choice will be simple: with each configu-rationσSL, we associate a configuration inCScwith a unique critical droplet

and a flow that, from each such configuration, follows a unique deterministic path along which this droplet is broken down in thecanonical order(see Figure6) until the setSLis reached, that is, a square or quasi-square droplet with labelLis left

over [recall (1.7)–(1.8)]. Letw(β)be such that

lim

β→∞w(β)= ∞, βlim→∞

1

β logw(β)=0, βlim→∞|β|/w(β)= ∞

(3.11)

and define

W= {σS:|supp[σ]| ≤ |β|/w(β)}.

(3.12)

LetCLCScbe the set of configurations obtained by picking anyσSLW

and adding somewhere inβ a critical droplet at distance≥2 from supp[σ]. Note

that the density restriction imposed on W guarantees that adding such a droplet is possible almost everywhere inβ forβ large enough. Denoting byP(y)(x)the

critical droplet obtained by adding a protuberance at y along the longest side of the rectangleRc−1,c(x), we may write

CL=

σP(y)(x):σSW, x, yβ, (x, y)σ

,

(3.13)

where (x, y)σ stands for the restriction that the critical droplet P(y)(x) is not

interacting with supp[σ], which implies thatHβ(σP(y)(x))=Hβ(σ )+ (see

Figures7and8).

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FIG. 8. Going fromSLtoCLby adding a critical dropletP(y)(x)somewhere inβ.

Now, for eachσCL, we letγσ =(γσ(0), γσ(1), . . . , γσ(K))be the canonical

path fromσ =γσ(0)toSLalong which the critical droplet is broken down, where K=v(2c−3)v(L)with

v(L)= |QL(0)|

(3.14)

[recall (1.7)]. We will choose our flow such that

f (σ, σ)=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ν0(σ ), ifσ=σ, σ=γσ(1)for someσCL,

˜

σCL

fγσ˜(k−1), γσ(k)

,

ifσ=γσ(k), σ=γσ(k+1)

for somek≥1, σCL,

0, otherwise.

(3.15)

Here,ν0is some initial distribution onCLthat will turn out to be arbitrary as long

as its support is all ofCL.

From (3.15), we see that the flow increases whenever paths merge. In our case, this happens only after the first step, when the protuberance atyis removed. There-fore, we get the explicit form

f (σ, σ)=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ν0(σ ), ifσ=σ,σ=γσ(1)for someσCL, 0(σ ), ifσ=γσ(k),σ=γσ(k+1)

for somek≥1,σCL,

0, otherwise, (3.16)

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proto-critical droplet (see Figure6). Using Proposition2.4, we therefore have

CAP(SL,Sc)=CAP(Sc,SL)≥CAP(CL,SL)

σCL

ν0(σ )

K1

k=0

f (γσ(k), γσ(k+1)) μβ(γσ(k))cβ(γσ(k), γσ(k+1))

1

(3.17)

=

σCL

1

μβ(σ )cβ(γσ(0), γσ(1))

+ K−1

k=1

C

μβ(γσ(k))cβ(γσ(k), γσ(k+1))

−1 .

Thus, all we have to do is to control the sum between square brackets.

Becausecβ(γσ(0), γσ(1))=1 (removing the protuberance lowers the energy),

the term with k=0 equals 1/μβ(σ ). To show that the terms with k≥1 are of

higher order, we argue as follows. Abbreviate=h(c−2). For everyk≥1 and σ (0)CL, we have [see Figure9and recall (1.2) and (1.3)]

μβ(γσ(k))cβ

γσ(k), γσ(k+1)

= 1

eβ[Hβ(γσ(k))Hβ(γσ(k+1))] (3.18)

μβ(σ0)eβ[2Jh]=μβ(σ )eβδ,

whereδ=2Jh=2Jh(c−1) >0 [recall (1.6)]. Therefore, K−1

k=1

C

μβ(γσ(k))cβ(γσ(k), γσ(k+1))

1

μβ(σ )

CKeδβ

(3.19)

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and so, from (3.17), we get

CAP(SL,Sc)

σCL

μβ(σ )

1+CKeβδ =

μβ(CL)

1+CKeβδ = [1+o(1)]μβ(CL).

(3.20)

The last step is to estimate, with the help of (3.13),

μβ(CL)=

1

σCL

eβHβ(σ )= 1

σSLW

x,yβ

(x,y)σ

eβHβ(σP(y)(x))

=eβ 1

σSLW

eβHβ(σ )

x,yβ

(x,y)σ

1 (3.21)

eβμβ(SLW)N1|β|[1−(c+1)2/w(β)].

The last inequality uses the fact that|β|(c+1)2/w(β)is the maximal number of

sites inβwhere it is not possible to insert a noninteracting critical droplet [recall

(3.12) and note that a critical droplet fits inside anc×c square]. According to

LemmaA.1in AppendixA, we have

μβ(SLW)=μβ(SL)[1+o(1)],

(3.22)

while conditions (1.8) and (1.9) imply thatμβ(SL)=μβ(S)[1+o(1)]. Combining

the latter with (3.20) and (3.21), we obtain the desired lower bound.

3.2. Proof of Theorem1.2(b). We use the same technique as in Section3.1, which is why we only give a sketch of the proof.

To estimate the average crossover time from SLS to Sc\C, we will use

Proposition2.1. WithA=SLandB=Sc\C, (2.10) reads

σSL

νSSc\C

L (σ )Eσ(τSc\C)=

1

CAP(SL,Sc\C)

σSC

μβ(σ )hSL,Sc\C(σ ). (3.23)

The left-hand side is the quantity of interest in (1.17).

In Sections 3.2.1 and 3.2.2 we estimate σSCμβ(σ )hSL,Sc\C(σ ) and CAP(SL,Sc\C). The estimates will show that

the right-hand side of (3.23)= 1

N2|β|

[1+o(1)], β→ ∞.

(3.24)

3.2.1. Estimate ofσSCμβ(σ )hSL,Sc\C(σ ).

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PROOF. Write, using (2.1),

σSC

μβ(σ )hSL,Sc\C(σ )

(3.25)

=μβ(SL)+

σ(S\SL)C

μβ(σ )Pσ(τSL< τSc\C).

The last sum is bounded above byμβ(S\SL)+μβ(C). As before,μβ(S\SL)= o(μβ(S)) as β → ∞. But (1.35) and (3.9) imply that μβ(C) =o(μβ(S)) as β→ ∞.

3.2.2. Estimate ofCAP(SL,Sc\C).

LEMMA 3.4. CAP(S,Sc\C)=N2|β|eβμβ(S)[1+o(1)] as β → ∞ withN2=43(2c−1).

PROOF. The proof is similar to that of Lemma3.2, except that it takes care of the transition probabilities away from the critical droplet.

Upper bound. Recalling (2.13)–(2.15) and noting that Glauber dynamics does not allow transitions withinC, we have, for allh:C→ [0,1],

CAP(SL,Sc\C)≤CAP(S,Sc\C)

(3.26)

σC μβ(σ )

ˆ

h(σ )−12+ ˇ

h(σ )−02,

wherecˆσ =ηScβ(σ, η)andcˇσ=ηSc\Ccβ(σ, η). The quadratic form in the

right-hand side of(3.26)achieves its minimum forh(σ )= ˆcσ/(cˆσ+ ˇcσ), so

CAP(SL,Sc\C)

σC

Cσμβ(σ )

(3.27)

with= ˆcσcˇσ/(cˆσ + ˇcσ). We have

σC

Cσμβ(σ )=

1

σP

xβ

σxC

CσxeβHβ(σ x)

=eβ(2Jh) 1

σP

eβHβ(σ )2 1

24+ 2

3(2c−4) (3.28)

=eβ(2Jh)μβ(P)N2=

1

N1

μβ(C)N2,

where, in the second line, we use the fact that =12 ifσ has a protuberance in a

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FIG. 10. Canonical order to break down a proto-critical droplet plus a double protuberance.In the first step,the double protuberance has probability 12to be broken down in either of the two possible ways.The subsequent steps are deterministic,as in Figure6.

Lower bound. In analogy with (3.13), denoting byP(y)2 (x)the droplet obtained by adding a double protuberance at y along the longest side of the rectangle

Rc−1,c(x), we define the setDLS

c\Cby DL=

σP(y)2 (x):σSLW, x, yβ, (x, y)σ

.

(3.29)

As in (3.15), we may choose any starting measure onDL. We choose the flow as

follows. For the first step, we choose

f (σ, σ )=12ν0(σ ), σDL, σCL,

(3.30)

which reduces the double protuberance to a single protuberance [compare (3.13) and (3.29)]. For all subsequent steps, we follow the deterministic paths γσ used

in Section3.1.2, which start fromγσ(0)=σ (see Figure10). Note, however, that

we get different values for the flows f (γσ(0), γσ(1)) depending on whether or

not the protuberance sits in a corner. In the former case, it has only one possible antecedent, and so

f (γσ(0), γσ(1))=12ν0(σ ),

(3.31)

while in the latter case it has two antecedents, and so

f (γσ(0), γσ(1))=ν0(σ ).

(3.32)

This time, the termsk=0 andk=1 are of the same order, while, as in (3.19), all of the subsequent steps give a contribution that is a factorO(eδβ)smaller. Indeed, in analogy with (3.17), we obtain, writingσσ whencβ(σ, σ ) >0,

CAP(SL,Sc\C)

=CAP(Sc\C,SL)≥CAP(DL,SL)

σDL 1 2

σCL

σσ

f (σ, σ ) μβ(σ ) +

f (σ, γσ(1)) μβ(σ )

+ K−1

k=1

f (γσ(k), γσ(k+1)) μβ(γσ(k))cβ(γσ(k), γσ(k+1))

−1

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σDL

1 2

σCL

σσ

μβ(σ )[f (σ, σ )+f (σ, γσ(1))+CKeβδ]−1

= [1+o(1)]μβ(CL)

2c−4

2c

1 1+1/2+

1 2

4 2c

1 1/2+1/2

= [1+o(1)]μβ(CL) N2 N1 .

Using (3.21) and the remarks following it, we get the desired lower bound.

3.3. Proof of Theorem1.2(c). Write

σDcM

μβ(σ )hSL,DM(σ )

=

σSL

μβ(σ )hSL,DM(σ )+

σDcM\SL

μβ(σ )hSL,DM(σ ) (3.34)

=μβ(SL)+

σDcM\SL

μβ(σ )Pσ(τSL< τDM).

The last sum is bounded above byμβ(S\SL)+μβ(DMc \S). Butμβ(S\SL)= o(μβ(S)), asβ→ ∞by our choice ofLin (1.9), whileμβ(DcM\S)=o(μβ(S))

asβ→ ∞because of the restrictioncM2c−1. Indeed, under that restriction,

the energy of a square droplet of sizeMis strictly larger than the energy of a critical droplet.

PROOF OF THEOREM 1.2(c). The proof follows along the same lines as that of Theorems 1.2(a) and (b) in Sections 3.1 and 3.2. The main point is to prove that CAP(SL,DM)= [1+o(1)]CAP(SL,Sc\C). Since CAP(SL,DM)

CAP(SL,Sc\C), which was estimated in Section 3.2, we need only prove a lower

bound on CAP(SL,DM). This is done by using a flow that breaks down anM×M

droplet to a square or quasi-square dropletQLin the canonical way, which takes M2−v(L)steps [recall Figure6and (3.14)]. The leading terms are still the proto-critical droplet with a single and a double protuberance. With eachM×Mdroplet is associated a unique critical droplet so that the pre-factor in the lower bound is the same as in the proof of Theorem1.2(b).

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4. Proof of Theorem1.4.

4.1. Proof of Theorem1.4(a).

4.1.1. Estimate ofσS(C˜\C+)μβ(σ )hSL,(Sc\ ˜C)C+(σ ).

LEMMA 4.1. σS(C˜\C+)μβ(σ )hSL,(Sc\ ˜C)C+(σ ) = μβ(S)[1 + o(1)] as β→ ∞.

PROOF. Write, using (2.1),

σS(C˜\C+)

μβ(σ )hSL,(Sc\ ˜C)C+(σ )

(4.1)

=μβ(SL)+

σ(S\SL)(C˜\C+)

μβ(σ )Pσ

τSL< τ(Sc\ ˜C)C+.

The last sum is bounded above byμβ(S\SL)+μβ(C˜\C+), butμβ(S\SL)= o(μβ(S))asβ→ ∞by our choice ofLin (1.32). In LemmaB.3in AppendixB.3,

we will show thatμβ(C˜\C+)=o(μβ(S))asβ→ ∞.

4.1.2. Estimate ofCAP(SL, (Sc\ ˜C)C+).

LEMMA 4.2. CAP(SL, (Sc\ ˜C)C+)=N|β|β4πeβμβ(S)[1+o(1)]as β→ ∞withN=132c(2c−1).

PROOF. The argument is in the same spirit as that in Section3.1.2. However, a number of additional obstacles that arise from the conservative nature of Kawasaki dynamics need to be overcome. The proof proceeds via upper and lower bounds and takes up quite a bit of space.

Upper bound. The proof consists of seven steps.

1.Proto-critical droplet (see Figure11) and free particle.We have CAPSL, (Sc\ ˜C)C+

≤CAP(SC, (Sc\ ˜C)C+)

(4.2)

= min

h:Xβ(nβ )→[0,1]

h|SC−≡1,h|(Sc\ ˜C)C+≡0

1 2

σ,σXβ(nβ )

μβ(σ )cβ(σ, σ)[h(σ )h(σ)]2.

Split the right-hand side into a contribution coming fromσ, σ∈ ˜Cand the rest: right-hand side of (4.2)=I+γ1(β),

(28)

FIG. 11. Schematic picture of the setsS,C−,C+defined in Definition1.3and the setC˜ interpo-lating betweenCandC+.

where

I= min

h:C˜→[0,1]

h|C−≡1,h|C+≡0

1 2

σ,σ∈ ˜C

μβ(σ )cβ(σ, σ)[h(σ )h(σ)]2

(4.4)

and γ1(β) is an error term that will be estimated in Step 7. This term will turn

out to be small because μβ(σ )cβ(σ, σ) is small when either σX (nβ)

β \ ˜C or σXβ(nβ)\ ˜C. Next, partitionC˜, C−,C+into sets C˜(x),C(x),C+(x), xβ,

by requiring that the lower-left corner of the proto-critical droplet is in the center of the boxBLβ,Lβ(x). Then, becausecβ(σ, σ)=0 whenσ∈ ˜C(x)andσ∈ ˜C(x) for somex=x, we may write

I = |β| min

h:C˜(0)→[0,1]

h|C(0)≡1,h|C+(0)≡0 1 2

σ,σ∈ ˜C(0)

μβ(σ )cβ(σ, σ)[h(σ )h(σ)]2.

(4.5)

2.Decomposition of configurations.Define [cf. (3.6)]

ˆ

C(0)=σ1BLβ ,Lβ(0):σ∈ ˜C(0)

,

(4.6)

ˇ

C(0)=σ1[BLβ ,Lβ(0)]c:σ ∈ ˜C(0).

Everyσ∈ ˜C(0)can then be uniquely decomposed asσ = ˆσ∨ ˇσ for someσˆ ∈ ˆC(0)

andσˇ ∈ ˇC(0). Note thatCˆ(0)hasK=c(c−1)+2 particles andCˇ(0)hasK

particles [and recall that, by the first half of (1.35),→ ∞asβ→ ∞]. Define Cfp(0)= {σ∈ ˜C(0):H

β(σ )=Hβ(σ )ˆ +Hβ(σ )ˇ },

(4.7)

Figure

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