F. den HollanderE. OlivieriE. Scoppola
Citation: Journal of Mathematical Physics 41, 1424 (2000); doi: 10.1063/1.533193 View online: http://dx.doi.org/10.1063/1.533193
Metastability and nucleation for conservative dynamics
F. den Hollander
Mathematisch Instituut, Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
E. Olivieri
Dipartimento di Matematica, Universita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy
E. Scoppola
Dipartimento di Matematica, Universita` di Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Rome, Italy
共Received 18 November 1999; accepted for publication 7 December 1999兲 In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let ⬎0 be the inverse temperature and let ⌳¯傺⌳傺Z2 be two finite boxes. Particles perform independent random walks on ⌳⌳¯ and inside ⌳¯ feel exclusion as well as a binding energy U⬎0 with particles at neighboring sites, i.e., inside⌳¯ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that⌳¯ is empty, while a total of兩⌳兩 particles is distributed randomly over⌳⌳¯ with no exclusion. That is to say, initially the system is in equilibrium with particle densityconditioned on⌳¯ being empty. For large, the system in equilibrium has⌳¯ fully occupied because
of the binding energy. We consider the case where ⫽e⫺⌬ for some ⌬
苸(U,2U) and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet
and the time of its creation in the limit as →⬁ for fixed ⌳ and
lim→⬁(1/) log兩⌳兩⫽⬁. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice⌬ 苸(U,2U) corresponds to the situation where the critical droplet has side length
lc苸(1,⬁), i.e., the system is metastable. The side length of ⌳¯ must be much larger
than lc and independent of , but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is
more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in⌳⌳¯ can be treated as a reservoir that creates particles with rateat sites on the interior boundary of⌳¯ and annihilates particles with rate 1 at sites on the exterior boundary of⌳¯ . Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside⌳¯ in the presence of a nonconservative boundary. The dynamics inside⌳¯ is still conservative and this difficulty has to be handled via local geometric
argu-ments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For
instance, rectangular droplets tend to become square through a movement of par-ticles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maxi-mal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions 共at least for not too long a time兲 without being attracted by a square. © 2000 American Institute of Physics.
关S0022-2488共00兲01503-6兴
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1. INTRODUCTION AND MAIN RESULTS
In this paper we study metastability for conservative共C兲 dynamics. In particular, we study the transition to the liquid phase of a supersaturated vapor described by a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density.
Metastability is a relevant phenomenon for thermodynamic systems close to a first-order phase transition. Suppose the system is in a pure equilibrium phase, corresponding to a point in the phase diagram close to a first-order phase transition curve. Suppose we change the thermodynamic parameters to values associated with a different equilibrium phase, corresponding to a point on the opposite side of the curve. Then, in certain experimental situations, instead of undergoing a phase transition the system can remain in the old equilibrium, far from the new equilibrium, for a long time. This unstable old equilibrium, called metastable state, persists until an external perturbation or a spontaneous fluctuation leads the system to the stable new equilibrium.
Examples of metastable states are supersaturated vapor and solutions, supercooled liquids, and ferromagnets with a magnetization opposite to the magnetic field.
In Sec. 1.1 we recall some of the main features of metastability by describing some well-known results obtained for a nonconservative 共NC兲 dynamics, namely Ising spins with Glauber dynamics. In Sec. 1.2 we introduce a conservative model, namely the lattice gas with Kawasaki dynamics, and discuss the main differences between C and NC. In Sec. 1.3 we propose a simpli-fied model, where the interaction and the exclusion only act in a finite box, and formulate our main theorem establishing metastable behavior for this model. In Sec. 1.4 we give an outline of the key ideas needed to prove this theorem, which are further developed in the remainder of the paper. In Sec. 1.5 we collect some additional notation that is used throughout the paper.
1.1. The non-conservative case
1.1.1. Grand-canonical ensemble. Let ⌳傺Z2 be a large finite box centered at the origin. With each site x苸⌳ we associate a spin variable(x), assuming the values⫹1 or ⫺1. With each configuration苸X⫽兵⫺1,⫹1其⌳we associate an energy
H共兲⫽⫺J
2共x,y兲苸⌳
兺
* 共x兲共y兲⫺h
2x
兺
苸⌳ 共x兲, 共1.1兲where⌳* is the set of bonds between nearest-neighbor sites in⌳, J⬎0 is the pair interaction,
h⬎0 is the magnetic field, and we assume periodic boundary conditions on ⌳. The
grand-canonical Gibbs measure associated with the Hamiltonian H, describing the equilibrium properties
of the system, is given by
共兲⫽e
⫺H共兲
Z 共苸X 兲, 共1.2兲
where Z is the partition function
Z⫽
兺
苸Xe⫺H共兲 共1.3兲
and⬎0 is the inverse temperature. The qualification ‘‘grand-canonical’’ is used here because
h plays the role of a chemical potential and the total magnetization 兺x苸⌳(x) is not constant
under.
It is well known that for every J, h,⬎0 in the thermodynamic limit ⌳→Z2 a unique Gibbs state with a positive magnetization exists共see e.g., Ruelle1and Sinai2兲. We will be interested in the regime where
⌳ is large but finite, h苸共0,2J兲, →⬁. 共1.4兲
丣⫽the configuration with 共x兲⫽⫹1 for all x苸⌳,
両⫽the configuration with 共x兲⫽⫺1 for all x苸⌳. 共1.5兲
In the regime共1.4兲 the Gibbs measure will be concentrated around 丣, which is the unique ground state of H. Clearly, 両 is only a local minimum of H, and it is therefore naturally related to metastability.
For l苸N, let
E共l兲⫽H共l⫻l兲⫺H共両兲, 共1.6兲
where l⫻l is the configuration in which the 共⫹1兲-spins form an l⫻l square, centered at the origin, in a sea of 共⫺1兲-spins. Then e⫺E(l) is the ratio of the probabilities to seel⫻l, respec-tively,両 under the equilibrium. It follows from共1.1兲 that E(l)⫽4Jl⫺hl2, which is maximal for l⫽2J/h. This means that, even though an arbitrarily small nonvanishing magnetic field de-termines the phase, its effect is relevant only on sufficiently large space scales, namely l⭓lcwith
lc⫽
d
2J
h
e
. 共1.7兲Only on such scales the volume energy dominates the surface energy and a larger square of 共⫹1兲-spins is energetically favorable over a smaller square 共see Fig. 1兲. The choice h苸(0,2J) corresponds to lc苸(1,⬁), i.e., to a nontrivial critical droplet size.
This describes the metastable behavior from a static point of view.
1.1.2. Glauber dynamics. In order to describe the metastable behavior from a dynamic point
of view, we introduce a discrete-time stochastic dynamics by means of a Markov chain onX with transition probabilities P(,
⬘
) satisfying the reversibility condition共兲P共,
⬘
兲⫽共⬘
兲P共⬘
,兲 ᭙,⬘
苸X, 共1.8兲whereis the Gibbs measure in共1.2兲, and the ergodicity condition
᭙,
⬘
苸X ᭚t苸N such that Pt共,⬘
兲⬎0, 共1.9兲where Ptis the t-step transition kernel. From the ergodic theorem for reversible Markov chains it follows that Pt(,
⬘
) converges to(⬘
) as t→⬁ for all,⬘
苸X.An explicit construction of a Markov chain satisfying the above conditions can be given, for instance, by the Glauber–Metropolis algorithm, which is defined as follows. For 苸X and x 苸⌳, let
x共y兲⫽
再
共y兲 if x⫽y⫺共y兲 if x⫽y 共1.10兲
and choose as transition probabilities
⫽
⬘
: P共,⬘
兲⫽再
0 if ⬘
⫽x ᭙x苸⌳ 1 兩⌳兩e⫺关H共 x兲⫺H共兲兴 ⫹ if ⬘
⫽x ᭚x苸⌳ 共1.11兲 ⫽⬘
: P共,兲⫽1⫺兺
⬘⫽ P共,⬘
兲.This dynamics randomly selects a site from⌳ and flips the spin at this site with a probability equal to the Boltzmann weight associated with the positive part of the energy difference caused by the flip. We emphasize that the dynamics given by共1.11兲 is NC, in the sense that the total magneti-zation is not a conserved quantity.
1.1.3. Metastability. Suppose we consider the typical paths of the Markov chain defined by
共1.11兲, starting from 両, in the regime 共1.4兲. We can use a computer simulation and perform a large number of independent runs共see e.g., Tomita and Miyashita3兲. What we see is that in the sea of 共⫺1兲-spins small droplets of 共⫹1兲-spins appear, which however shrink and disappear before they are able to become large. Only after a very long time, and under the effect of a large fluctuation, a large enough droplet appears that grows without hesitation.
In order to understand this behavior, let us compare the probabilities of shrinking, respectively growing for a connected cluster of共⫹1兲-spins in a sea of 共⫺1兲-spins. First of all, each cluster of 共⫹1兲-spins becomes rectangular after a finite time 共independent of兲 with a probability of order one following a sequence of transitions with H(x)⫺H()⬍0. Indeed, the rectangle is the only shape such that:共i兲 all 共⫹1兲-spins have ⭐2 nearest-neighbor 共⫺1兲-spins; 共ii兲 all 共⫺1兲-spins have ⬍2 nearest-neighbor 共⫹1兲-spins. Hence for the rectangle there are no spins that can be flipped with H(x)⫺H()⬍0.
Starting from a rectangular cluster of共⫹1兲-spins, to remove a row or column of length l costs (l⫺1)h:
关H(x)⫺H()兴
⫹⫽关h兴⫹⫽h for each of the sites except the last one.
关H(x)⫺H()兴
⫹⫽关h⫺2J兴⫹⫽0 for the last site.
On the other hand, to add a row or column of length l costs 2J⫺h: 关H(x)⫺H()兴
⫹⫽关2J⫺h兴⫹⫽2J⫺h for the first site.
关H(x)⫺H()兴
⫹⫽关⫺h兴⫹⫽0 for each of the sites except the first one.
This means that if the minimal side length l of the rectangular cluster is such that h(l⫺1)⬎2J ⫺h, i.e., l⭓lcwith lcgiven by共1.7兲, then it tends to grow, while if l⬍lcthen it tends to shrink.
The above heuristic argument has been developed in a rigorous way by Neves and Schonmann4,5共see also Schonmann6–8兲. Let (t)t苸N0 be the Markov process onX with transition
probabilities as in共1.11兲. Write P,Efor its probability law and expectation on path space given
0⫽. Let
⫽min兵t苸N0:t⫽其 共1.12兲
be the first hitting time of the configuration. The main result for metastability reads:
Theorem 1.13: (Neves and Schonmann4,5) Fix h苸(0,2J), with 2J/h not integer, put lc
共a兲 Let R be the set of configurations where the 共⫹1兲-spins form a rectangle in a sea of 共⫺1兲-spins. For 苸R, let l1()⫻l2() be the rectangle of 共⫹1兲-spins in , and let l() ⫽min兵l1(),l2()其. Then, for any苸R,
l共兲⬍lc: lim
→⬁P共両⬍丣兲⫽1,
共1.14兲
l共兲⭓lc: lim
→⬁P共丣⬍両兲⫽1.
共b兲 Let R* be the set of configurations where the 共⫹1兲-spins form an lc⫻(lc⫺1) or (lc
⫺1)⫻lc rectangle with a protuberance attached anywhere to one of the sides of length lc. Let 両,丣⫽max兵t⬍丣:t⫽両其and 両,R*,丣⫽min兵t⬎両,丣:t苸R*其. Then
lim →⬁P両共両,R*,丣⬍丣兲⫽1. 共1.15兲 共c兲 Let ⌫⫽⌫(J,h)⫽4Jlc⫺h(lc 2⫺l c⫹1). Then lim →⬁P両共e 共⌫⫺␦兲⬍ 丣⬍e共⌫⫹␦兲兲⫽1 ᭙␦⬎0. 共1.16兲
R*is the set of critical droplets, i.e., the set of saddle points between両 and 丣, and⌫(J,h) is the formation energy of a critical droplet under the Hamiltonian in 共1.1兲. Theorem 1.13 not only identifies the size and shape of critical droplets共see Fig. 2兲, it also shows that R* is the ‘‘gate’’ of the transition from両 to丣 and it identifies the transition time up to logarithmic equivalence in
.
1.1.4 Nucleation. The problem of identifying the typical path of nucleation, i.e., the path
between 両,丣 and 丣, corresponds to the problem of the typical first exit of (t)t苸N0 from a
suitable region in the state spaceX. This problem is discussed in detail in Freidlin and Wentzell,9 Chap. 6, Schonmann,7Olivieri, and Scoppola,10,11Catoni and Cerf12under rather general hypoth-eses on the Markov chain. We recall here the main result for the case of the Glauber Ising model.
A sequence of configurations1,...,n(n苸N) is called standard when
共1兲 the 共⫹1兲-spins ofi form a rectangular droplet Ri⫽l1,i⫻l2,i; 共2兲 Ri⫹1Riis a single row or a single column;
共3兲 if min兵l1,i,l2,i其⬍lc, then兩l1,i⫺l2,i兩⭐1; 共4兲 R1⫽2⫻2 and Rn⫽丣.
The configurations in such a sequence are stable, since they are local minima of H, i.e., H(i)
⬍min⬘⬃iH(
⬘
), where⬘
⬃if and only if P(,⬘
)⬎0. With eachiit is possible to associatea permanence set Qi 共a suitable ‘‘environment’’ of i: a generalized basin of attraction of i
w.r.t. the dynamics at⫽⬁) and a permanence time Ti⫽EiQ
i
c共the mean exit time of Qistarting
from i). In this way we obtain a standard sequence of permanence sets 共see Olivieri and
Scoppola13for more precise definitions兲.
For each standard sequence of permanence sets and each⑀⬎0 we can introduce a tube of trajectories T⑀,(Q1,...,Qn), defined as the set of paths of configurations visiting the ordered
sequenceQ1,...,Qnand spending in each setQia time that falls in the interval关Tie⫺⑀,Tie⫹⑀兴.
In terms of these quantities the main result for the path of nucleation reads:
Theorem 1.17: (Schonmann,7Olivieri and Scoppola13) For every,⑀⬎0 there exists a 0 ⫽0(,⑀) such that for all⬎0:
P両„共t兲t苸关両,丣,丣兴苸T⑀,共Q1,...,Qn兲 for some standard
sequence of permanence sets Q1,...,Qn…⬎1⫺e⫺. 共1.18兲
Theorem 1.17 shows that the transition from両 to 丣 takes place in a narrow tube around rectan-gular droplets that are squares or quasi-squares when the droplet is subcritical.
The main idea behind Theorem 1.17共which is actually valid in a much more general context兲 is the following. The Markov chain (t)t苸N0 is in the Freidlin–Wentzell regime, i.e., its state
space is finite and its transition probabilities satisfy the following estimates:
e⫺关V共,⬘兲⫹␥兴⭐P共,
⬘
兲⭐e⫺关V共,⬘兲⫺␥兴 ᭙⬃⬘
, 共1.19兲 where V(•,•) is a non-negative function, and lim→⬁␥⫽0. Indeed, this property trivially fol-lows from共1.11兲, because ⌳ is fixed and V(,⬘
)⫽关H(⬘
)⫺H()兴⫹. With the help of共1.19兲 it is standard to obtain estimates onEiQi candP
i(Qic⫽
⬘
)共see Freidlin and Wentzell,9
Chap. 6兲. The main steps in the proof of Theorem 1.17 are the following:
共1兲 One must solve a certain sequence of variational problems defined in terms of the energy function H. These variational problems are minimax problems necessary to find the minimal saddle point energy between pairs of states,
⬘
defined byH共,
⬘
兲⫽ min:→⬘
max 苸
H共兲, 共1.20兲
where :→
⬘
denotes a path from to ⬘
. The output of this first step is a standard sequence of configurations.共2兲 One must associate with each stable configuration a permanence set and a permanence time. This can be done by using a so-called cycle decomposition: indeed, the permanence sets are generalized cycles. Cycles can be defined in the Freidlin–Wetzell regime 共see Freidlin and Wentzell,9Chap. 6, Olivieri and Scoppola,13Trouve´14兲. In the case of the Glauber Ising model cycles turn out to be connected sets of configurations with energy below a given value.
1.2. The conservative case
1.2.1. Canonical ensemble. In the present paper we want to study the metastable behavior of
conservative systems. To that end we consider a lattice gas model defined as follows. Let⌳傺Z2 be a large finite box centered at the origin, with periodic boundary conditions. With each x 苸⌳ we associate an occupation variable (x), assuming the values 0 or 1. A lattice gas
H共兲⫽⫺U
兺
共x,y兲苸⌳*共x兲共y兲,
共1.21兲 where⌳* denotes the set of bonds in ⌳, i.e., there is a binding energy U⬎0 between neigh-boring occupied sites. For A傺⌳, we let
NA共兲⫽
兺
x苸A 共x兲. 共1.22兲
We fix the particle density in⌳ at 1
兩⌳兩x苸⌳
兺
 共x兲⫽⫽e⫺⌬, 共1.23兲
where⌬⬎0 is an activity parameter. This corresponds to a total number of particles in ⌳equal to
N⫽兩⌳兩. 共1.24兲
On the set of configurations with N particles
NN⫽兵苸X: N⌳共兲⫽N其, 共1.25兲
we define the canonical Gibbs measure
N共兲⫽ e⫺H共兲1N N共兲 ZN 共 苸X 兲, 共1.26兲 where ZN⫽
兺
苸NN e⫺H共兲. 共1.27兲We see from共1.23兲 and 共1.24兲 that in order to have particles at all we must pick 兩⌳兩 at least exponentially large in. This means that the regime where⌳is fixed, considered in the NC-case, has no relevance here. We will in fact be interested in the regime
⌬苸共U,2U兲, →⬁, lim
→⬁
1
log兩⌳兩⫽⬁, 共1.28兲
which takes over the role that共1.4兲 played in the NC-case.
1.2.2. Kawasaki dynamics. We define a stochastic dynamics in terms of a continuous-time
Markov chain (t)t⭓0 with state spaceNN, given by the following generator:
and
c共共x,y兲,兲⫽e⫺关H共共x,y兲兲⫺H共兲兴⫹. 共1.31兲
It is easily verified that the reversibility condition holds:
N共兲c共共x,y兲,兲⫽N共共x,y兲兲c共共x,y兲,共x,y兲兲. 共1.32兲
The Markov chain (t)t⭓0 can be represented as follows. With each bond b⫽(x,y)苸⌳*we associate a random clock ringing at exponential times. When the clock at b rings, we consider the configuration with the particles swapped along b. This configuration is accepted with a Metropolis rate given by the Boltzmann factor in共1.31兲. More formally, for each bond b putb,0⫽0 and let b,i,i苸N, be the sequence of random times whose increments are i.i.d. exponentially distributed
with mean 1. Since兩⌳兩⬍⬁, we have
P共᭚b,b
⬘
,i,i⬘
:b,i⫽b⬘,i⬘兲⫽0. 共1.33兲Now, if t⫽b,i for some b and i, then we define t⫽
再
t⫺ with probability 1⫺e⫺关H共t⫺
b 兲⫺H共 t⫺兲兴⫹
t⫺ b
with probability e⫺关H共tb⫺兲⫺H共t⫺兲兴⫹, 共1.34兲
while between ringing times the configuration stays fixed.
1.2.3. Metastability. In order to see that for the regime in共1.28兲 one can expect metastable
behavior, let us consider the grand-canonical Gibbs measure associated with the model, i.e.,
共兲⫽ e⫺H共兲 Z , 共1.35兲 where H共兲⫽H共兲⫺N⌳ 共兲, 共1.36兲
苸R is an activity parameter, and
Z⫽
兺
苸Xe
⫺H共兲. 共1.37兲
It turns out that if⫽⫺⌬, then for the description of metastability the canonical Gibbs measure is equivalent to the grand-canonical Gibbs measure in the limit of large , provided they are suitably restricted in the following way.
Consider the lattice gas at low temperature at its condensation point. Let
l共兲⫽
1⫹m*共兲
2 , g共兲⫽
1⫺m*共兲
2 共1.38兲
denote the density of the liquid, respectively, gas phase. Here m*() is the spontaneous magne-tization in the spin language关see 共1.44兲兴. Since
g共兲⫽e⫺2U关1⫹o共1兲兴 共→⬁兲, 共1.39兲
be described in terms of a restricted ensemble 共see Lebowitz and Penrose15 and Capocaccia, Cassandro, and Olivieri16兲, namely, the grand-canonical Gibbs measure restricted to a suitable subset of configurations, for instance, where all sufficiently large clusters are suppressed. At low temperature this supersaturated gas will stay rarified, so that its metastable state can be described as a pure gas phase with strong mixing properties.
In these conditions, let us make a rough calculation of the probability to see an l⫻l droplet of occupied sites centered at the origin. Under the restricted ensemble, which we denote by*, we have
*共l⫻l droplet兲⬇l2e2l共l⫺1兲U, 共1.40兲
since is the probability to find a particle at a given site and U is the binding energy between particles at neighboring sites. Substituting ⫽e⫺⌬ we obtain
*共l⫻l droplet兲⬇e⫺E共l兲, 共1.41兲
where
E共l兲⫽2Ul⫺共2U⫺⌬兲l2. 共1.42兲
The maximum of E(l) is at l⫽U/(2U⫺⌬). This means that droplets with side length l⬍lchave
a probability decreasing in l and droplets with side length l⭓lca probability increasing in l, where
lc⫽
d
U2U⫺⌬
e
共1.43兲共see Fig. 3兲. The choice ⌬苸(U,2U) corresponds to lc苸(1,⬁), i.e., to a non-trivial critical droplet
size.
Another way of understanding our choice of⌬ is the following. In the grand-canonical Gibbs measure the configuration can be represented in terms of spin variables. Indeed, after we make the substitution(x)⫽关„1⫹(x)…/2兴, where(x)苸兵⫺1,⫹1其 is a spin variable, we can write
H共兲⫽⫺U
兺
共x,y兲苸⌳* 1⫹共x兲 2 1⫹共y兲 2 ⫺x苸⌳兺
 1⫹共x兲 2 ⫽⫺U4兺
共x,y兲苸⌳* 共x兲共y兲⫺2U2⫹兺
x苸⌳ 共x兲⫹const. 共1.44兲So if ⫽⫺⌬, then we have a spin Hamiltonian like 共1.1兲 with pair interaction J⫽U/2 and magnetic field h⫽2U⫺⌬. By the discussion developed in Sec. 1.1.3, we therefore expect meta-stable behavior with a critical droplet size given by共1.43兲 关compare with 共1.7兲兴. The metastable behavior for the NC-case in the spin language occurs when h苸(0,2J). This corresponds precisely
to⌬苸(U,2U).
In physical terms,⌬苸(0,U) corresponds to the unstable gas, ⌬⫽U to the spinodal point, ⌬苸(U,2U) to the metastable gas, ⌬⫽2U to the condensation point, and ⌬苸(2U,⬁) to the stable gas.
The above describes the metastable behavior from a static point of view. A comparison of Glauber vs Kawasaki dynamics in the spin language is indicated in Fig. 4. In Fig. 4 the boldface dashed lines represent the ‘‘metastable branches.’’ In the description with the restricted ensemble there is a specific value of h that corresponds to a canonical metastable state with magnetization m. The horizontal dashed line 共labeled with K兲 represents a Kawasaki transition towards a stable equilibrium with the same global magnetization but with a ‘‘segregation’’ of the two stable pure phases in the equilibrium grand-canonical ensemble at h⫽0: the saturated gas and the condensed gas共or liquid兲 at the condensation point.
1.2.4. Local description. Let us now consider the metastable behavior from a dynamic point
of view and see what happens locally. As discussed in the NC-case, we want to compare the probabilities of growing, respectively, shrinking for a rectangular cluster of particles. Again the argument will be very rough. Suppose we pick a large finite box ⌳¯ , centered at the origin, and start with an l⫻l droplet inside ⌳¯ . Suppose that the effect on ⌳¯ of the gas in ⌳⌳¯ may be described in terms of the creation of new particles with rate ⫽e⫺⌬ at sites on the interior boundary of⌳ and the annihilation of particles with rate 1 at sites on the exterior boundary of ⌳¯ . In other words, suppose that inside⌳¯ the Kawasaki dynamics may be described by a Metropolis algorithm with energy given by the local grand-canonical Hamiltonian:
H¯共兲⫽H共兲⫹⌬N⌳¯共兲. 共1.45兲
Then the energy barriers for adding, respectively, removing a row or column of length l are given in terms of the local saddles of H¯ 共see Fig. 5兲:
energy barrier for adding ⫽2⌬⫺U,
energy barrier for removing ⫽共2U⫺⌬兲共l⫺2兲⫹2U, 共1.46兲
and the balance of the two barriers indeed gives the critical size lc in共1.43兲.
Let us briefly discuss the main difficulties arising in the attempt to develop the above idea rigorously and underline the main differences with the NC-case. As we already remarked, in the C-case the Markov chain (t)t⭓0 is not in the Freidlin–Wentzell regime, so we need new ideas.
The real difficulty is to find the correct way to treat the gas in⌳⌳¯ . The heuristic discussion
given above was based on the assumption that the dynamics inside⌳¯ is effectively described by the local grand-canonical Hamiltonian H¯ in 共1.45兲. However, unlike the NC-dynamics, the C-dynamics is not really local: Particles must arrive from or return to the gas, which acts as a
reservoir. It is therefore not possible to decouple the dynamics of the particles inside⌳¯ from the
dynamics of the gas in⌳⌳¯ . This means that the gas must be controlled in some detail in order to prove that the above assumption is indeed a good enough approximation.
A second consequence of the non-local behavior of the C-dynamics is that the argument used in the NC-case, based on the stability of configurations and on the corresponding partition into cycles of the state space 共see Sec. 1.1.4兲, is completely lost in the C-case. In other words, we cannot define the stability of a configuration inside ⌳¯ , since it depends on the configuration in ⌳⌳¯ . A different aspect of the same problem is the following: What is the mechanism by which the gas remains in or close to equilibrium, so that its description in terms of H¯ is correct, even over long time intervals during which exchange of many particles occurs?
1.3. A simplified model
Unfortunately, we are unable to handle the model described in Sec. 1.2. Instead, in the present paper we solve the problem of metastability for a simplified model. Namely, we remove the
interaction outside the box⌳¯0⫽⌳¯⫺⌳¯ , with ⫺⌳¯ the interior boundary of⌳¯ , i.e., we replace the interaction energy共1.21兲 by
H共兲⫽⫺U
兺
共x,y兲苸⌳¯0*共x兲共y兲. 共1.47兲
Moreover, we also remove the exclusion outside⌳¯ , i.e., the dynamics of the gas outside⌳¯ is that of independent random walks共IRWs兲. These two simplifications will allow us to control the gas and to overcome the difficulties outlined in Sec. 1.2.4.
Our state space is
NN⫽兵苸X:N⌳共兲⫽N其, 共1.48兲
where X⫽兵0,1其⌳¯⫻N0⌳⌳¯, N⌳
()⫽⌺x苸⌳(x), and N⫽兩⌳兩 共with ⫽e⫺⌬). The local
grand-canonical Hamiltonian is
H¯共兲⫽H共兲⫹⌬N⌳¯共兲, 共1.49兲
where H is the Hamiltonian in共1.47兲. Throughout the remainder of this paper we assume that we are in the regime共1.28兲.
Our main theorem reads as follows. Let
䊏⫽兵苸X:共x兲⫽1 ᭙x苸⌳¯0其,
共1.50兲 䊐⫽兵苸X:共x兲⫽0 ᭙x苸⌳¯其.
For¯苸X¯⫽兵0,1其⌳¯, let ¯ denote the canonical Gibbs measure conditioned on the configuration
inside⌳¯ being¯ , i.e.,
¯共兲⫽
共兲1I¯共兲
共I¯兲 共苸X 兲, 共1.51兲
where I¯⫽兵苸X:兩⌳¯⫽¯其, with 兩⌳¯ the restriction of to ⌳¯ , and is the canonical Gibbs
measure defined in共1.24兲–共1.27兲. For¯苸X¯ ⫽兵0,1其⌳¯, writeP
¯,E¯ to denote the probability law
and expectation for the Markov process (t)t⭓0 on X following the Kawasaki dynamics with
Hamiltonian 共1.47兲 given that 0 is chosen according to ¯. Write 䊐¯ to denote the empty
configuration in ⌳¯ , i.e., 䊐⫽I䊐¯ . ForA傺X, let
A⫽min兵t⭓0:t苸A其 共1.52兲
be the first hitting time of the setA.
Theorem 1.53: Fix⌬苸(32U,2U), with U/(2U⫺⌬) not integer, put lc⫽关U/(2U⫺⌬)兴, and suppose that lim→⬁(1/)log兩⌳兩⫽⬁.
共a兲 Let R¯傺X¯ be the set of configurations inside ⌳¯ where the particles form a square or quasi-square contained in⌳¯0. For¯苸R¯ , let l1(¯ )⫻l2(¯ ) with兩l1(¯ )⫺l2(¯ )兩⭐1 be the square
or quasi-square of particles in¯ , and let l(¯ )⫽min兵l1(¯ ),l 2(¯ )其. Then, for any¯苸R¯ , l共¯兲⬍lc: lim
→⬁P¯共䊐⬍䊏兲⫽1
共1.54兲
l共¯兲⭓lc: lim
→⬁P¯共䊏⬍䊐兲⫽1.
共b兲 Let C¯* be the set of configurations defined in 共4.21兲 (see Fig. 6 for an example). Let 䊐,䊏⫽max兵t⬍䊏:t苸䊐其and 䊐,C¯*,䊏⫽min兵t⬎䊐,䊏:t苸C¯*其. Then
Theorem 1.53 is the analogue of Theorem 1.13. There are, however, a number of important differences.
The mechanisms for the evolution of clusters under the Kawasaki dynamics and the Glauber dynamics are different. In particular, under the Kawasaki dynamics there is a movement of par-ticles along the border of a rectangular droplet, leading to a共more stable兲 square droplet on a time scale much shorter than the one needed to grow or shrink共of order e⌬). Moreover, the subcriti-cality vs supercritisubcriti-cality of a rectangle共i.e., its tendency to reach 䊐 before 䊏 or vice versa兲 is related to its area. In contrast, under the Glauber dynamics the subcriticality vs supercriticality is related to its minimal side length: a subcritical rectangle is attracted by the maximal square contained in its interior, while a supercritical rectangle does not manifest any tendency towards a square shape.
Let us comment on Theorem 1.53:
Theorem 1.53共a兲: We only identify the subcriticality vs supercriticality of squares and quasi-squares. We believe that it is possible to show that, starting from an l1⫻l2 rectangle that is not square or quasi-square, the system forms a square or quasi-square with volume⬎l1l2 in a time of order e⌬ and from there proceeds as described in共1.54兲.
Theorem 1.53共b兲: C¯*is the set of critical droplets, i.e., the set of saddle points between䊐 and 䊏, that form the ‘‘gate’’ of the transition from 䊐 to 䊏. Let R¯*傺X¯ be the set of configurations
inside⌳¯ where the particles form an lc⫻(lc⫺1) or (lc⫺1)⫻lcquasi-square with a protuberance
attached anywhere to one of the sides of length lc and with a free particle anywhere else, all
contained in⌳¯0共see Fig. 6兲. We will see in Sec. 4.2 that C¯*consists of all configurations that are ‘‘U-equivalent’’ to some configuration inR¯*, i.e., have the same energy and can be connected via a path with a ‘‘maximal saddle U.’’ In particular,C¯*傻R¯*, but the full set is more complex共see
Fig. 9 in Sec. 5.2兲. This complexity comes from the fact that under the Kawasaki dynamics particles can move along the border of a rectangular droplet at a cost U.
Theorem 1.53共c兲: ⌫(U,⌬) is the energy of a critical droplet under the local grand-canonical Hamiltonian in 共1.49兲.
The critical configuration in Fig. 6 has the same shape as in the NC-case共see Fig. 2兲, but with an extra free particle. This particle signals that the ‘‘gate’’ of the transition from䊐 to 䊏 has been passed and that the droplet starts to grow without hesitation.
It is certainly feasible to also prove the analogue of Theorem 1.17 for the simplified model. However, in the present paper we will not address this issue for reasons of space.
Remarks:
共1兲 Our proof of Theorem 1.53 shows that the convergence in 共1.54兲–共1.56兲 is exponentially fast in.
共2兲 As explained above, the removal of the interaction outside ⌳¯0and the exclusion outside⌳¯
allows us to mathematically control the gas. From a physical point of view this approximation seems very reasonable, because→⬁ corresponds to a low density limit (⫽e⫺⌬) in which the gas essentially behaves like an ideal gas.
共3兲 In the simplified model we are focusing on the local aspects of metastability and nucle-ation: the removal of the interaction outside⌳¯0 forces the critical droplet to appear inside⌳¯0. In the original model with interaction and exclusion throughout ⌳, if lim inf→⬁(1/)log兩⌳兩 is large enough, then the decay from the metastable to the stable state is driven by the formation of many droplets far away from the origin, which subsequently grow, coalesce and reach⌳¯0. This is a much harder problem, which we hope to tackle in the future共see Deghampour and Schonmann17 for a description of this behavior for Ising spins under Glauber dynamics兲. Also, in the original model the question of the growth of large supercritical droplets comes up, which is absent for the simplified model because⌳¯0is finite. For Kawasaki dynamics this poses new problems compared to Glauber dynamics, because large droplets deplete the gas.
1.4. Outline of the paper
Our strategy to prove Theorem 1.53 will be the following. In Sec. 2 we show that, under the measure ¯(¯苸X¯ ), particle densities in suitable regions around ⌳¯ are not too far from their
expected value. With the help of large deviation estimates we show that these density properties are preserved under the dynamics over very long time intervals with a very large probability. In Sec. 3 we use this fact to control the gas, essentially via a series of mixing propositions. Once the gas behavior is under control, we start to tackle the metastability problem inside⌳¯ . This is done in Secs. 6–7 via recurrence and reduction. Namely, in Sec. 6 we show that certain subsets of configurations of increasing ‘‘regularity’’
X1傻X2傻X3 共1.57兲
are visited by the process on certain basic time scales
T1⫽e0ⰆT2⫽eUⰆT3⫽e⌬. 共1.58兲
This fact leads us in Sec. 7 to define a reduced Markov chain with state space X3, whose transition probabilities we can estimate in a way that allows us to control the metastable behavior. In essence, we show that this reduced chain is ‘‘equivalent’’ in its metastable behavior to a local
Markov chain with state spaceX¯⫽兵0,1其⌳¯ that is reversible w.r.t. the local grand-canonical Hamil-tonian H¯ defined in共1.49兲. This approximation is what drives the argument. In Sec. 5 we study the local Markov chain using general ideas from renormalization. The dynamics inside ⌳¯ is still conservative, and this difficulty has to be handled via local geometric arguments, as explained in Sec. 4. Here we also show that the Kawasaki dynamics has its own peculiarities, which need to be understood in order to describe the evolution of droplets. The proof of our main result in Theorem 1.53 comes in Sec. 8. Here the fact that the full Markov chain is reversible w.r.t. the canonical
Gibbs measure plays an important role. In the Appendix we prove the equivalence of the canonical
and the grand-canonical ensemble for the simplified model in the regime共1.28兲. This equivalence is used in some of the calculations.
1.5. Additional notation
We use capital letters for subsets ofZ2, calligraphic capital letters for subsets of the configu-ration space X, and boldface capital letters for events involving the Markov process and the clocks. This style is used consistently in order to keep different types of quantities apart. We use the symbols t,T for time,for the canonical Gibbs measure with particle density⫽e⫺⌬ 关recall 共1.24兲–共1.27兲兴, and for a generic positive constant.
For A傺Z2, the set of共nearest-neighbor兲 bonds in A is
A*⫽兵b⫽共x,y兲: x,y苸A其. 共1.59兲
ForA傺X, the base of A is
BASE共A兲⫽min兵A傺Z2:苸A⇒共苸A ᭙ such that 兩A⫽兩A兲其, 共1.60兲
i.e., the minimal set of sites on which the configuration determines the eventA. For A傺Z2, the interior resp. exterior boundary of A are
⫺A⫽兵x苸A: ᭚b⫽共x,y兲: y苸A其,
共1.61兲
⫹A⫽兵x苸A: ᭚b⫽共x,y兲: y苸A其.
For l苸N, the box with side length l centered at the origin is denoted by ⌳l. The side length
of⌳¯0, the local box appearing in the Hamiltonian H in共1.47兲, is l0. We assume that l0Ⰷlc, the
critical droplet size defined in共1.43兲.
All quantities that live on⌳¯ are written with a bar on top, in order to distinguish them from quantities that live on⌳or other boxes. A function哫 f () is called superexponentially small 共SES兲 if
lim
→⬁
1
log f共兲⫽⫺⬁. 共1.62兲
We frequently round off large integers, in order to avoid a plethora of brackets liked•e. 2. LD-ESTIMATES FOR CLOCKS AND EQUILIBRIUM
In this section we formulate several large deviation estimates that will be needed later on. 2.1. LD-estimates for clocks
Let b,i,i苸N, denote the ringing times of the clock at bond b. For t⬎0, let rb(t)⫽max兵i
苸N :b,i⭐t其 denote the number of rings prior to time t. For m,n苸N, put rb(n,n⫹m)⫽rb(n
⫹m)⫺rb(n). For A傺Z2, T⭓0 and␦⬎0, define
RT␦共A兲⫽兵᭙b苸A* ᭙n⭐T ᭙m⭓e␦: rb共n,n⫹m兲苸关
1 2m,
3
2m兴其. 共2.1兲
Proposition 2.2 below shows that clocks ring regularly over long time intervals. This proposition will be needed to switch from continuous to discrete time.
Proposition 2.2: For all A傺Z2, T⭓0 and␦⬎0:
P共RT␦共A兲c兲⭐T兩A*兩SES. 共2.3兲
Proof: Write
P共RT␦共A兲c兲⫽兵᭚b苸A* ᭚n⭐T ᭚m⭓e␦: rb共n,n⫹m兲苸关
where b0 is any given bond. We have rb0共0,m兲⬍ 1 2m⇒b0,d12me⬎m, 共2.5兲 rb0共0,m兲⬎ 3 2m⇒b0,d32me⬍m.
Since b0,m⫽X1⫹¯⫹Xm, with (Xi)i苸N i.i.d. exponential random variables with mean 1, a
standard LD-estimate gives that the summand of the last term in 共2.4兲 is ⭐e⫺m for some ⬎0. Hence the claim follows. QED
2.2. LD-estimates for equilibrium
2.2.1. Hitting times. Proposition 2.6 below gives us an estimate on the hitting time, under the
dynamics starting in equilibrium, of sets that have a small probability under the equilibrium measure.
Proposition 2.6. LetA傺X andA⫽inf兵s⭓0:s苸A其. Then, for any t⭓0,
P共A⬍t兲⫽
兺
苸X共兲P共A⬍t兲⭐3t兩BASE共A兲*兩共A兲. 共2.7兲
The same holds when is replaced byB⫽1B/(B) for any B傺X.
Proof: FixA. For s⭓0, let
Fs⫽兵s苸A,u苸A᭙0⭐u⬍s其. 共2.8兲
Fix⑀⬎0 and define
Rs⫽兵some clock in BASE共A兲* rings during 关s,s⫹⑀兲其. 共2.9兲
Then we have
P共A⬍t兲⫽P共᭚s苸关0,t兲: Fs兲⫽P共᭚s苸关0,t兲: Fs艚Rs兲⫹P共᭚s苸关0,t兲: Fs艚Rs
c兲. 共2.10兲
The first term equalsP(A⬍t)关1⫺e⫺⑀兩BASE(A)*兩兴, because clocks have no memory. The second term is bounded above by
P共᭚0⬍i⭐t/⑀:i⑀苸A兲⭐ t
⑀ 共A兲, 共2.11兲
where we use that P(i⑀苸A) does not depend on i because is the equilibrium measure.
Combining the latter two observations with共2.10兲 we get P共A⬍t兲⭐t共A兲
冋
1
⑀e⑀兩BASE共A兲*兩
册
. 共2.12兲Optimize over⑀, i.e., pick⑀⫽1/兩BASE(A)*兩, to arrive at the claim. QED
2.2.2. Recurrence times. Proposition 2.13 gives us control over the successive times at which
the dynamics hits a certain set. This proposition will be needed later on to establish recurrence properties to certain special sets.
Then
P共᭚t苸关0,T
⬙
兲:s苸A ᭙s苸关t,t⫹T⬘
兲兲⭐T⬙
关3兩BASE共Bc兲*兩共Bc兲⫹共1⫺p兲T⬘/T兴. 共2.15兲 Proof: Pick any t苸关0,T⬙
). Split the time interval 关t,t⫹T⬘
) into pieces of length T. By 共2.14兲 共i兲–共ii兲, on the event兵Bc⭓T⬙
其, if at the beginning of a piece the process is not inA, thenit has a probability at most 1⫺p not to enter A during this piece. Hence the probability not to enterA during the time interval 关t,t⫹T
⬘
) is at most (1⫺p)T⬘/Tby the Markov property. Conse-quently,P共᭚t苸关0,T
⬙
兲:s苸A ᭙s苸关t,t⫹T⬘
兲兲⭐P共Bc⬍T⬙
兲⫹T⬙
共1⫺p兲T⬘/T. 共2.16兲Now use Proposition 2.6 to get the claim. QED
2.2.3. Particle density in annuli around⌳¯ . Propositions 2.17, 2.20, and 2.23 below give us
control over the number of particles in annuli around⌳¯ with a side length that is close to the mean particle distance on an exponential scale. In the proofs we compute the estimates using the grand-canonical Gibbs measureonZ2 with particle density, rather than the canonical Gibbs measure on⌳ with total particle number 兩⌳兩. However, by the equivalence of ensembles proved in the Appendix, the difference is SES under our assumption that lim→⬁(1/)log兩⌳兩 ⫽⬁ 共see the remark at the end of the Appendix兲.
Proposition 2.17: Let␥⬎0 and l⫹⫽e(1/2)(⌬⫹␥). Then, for all␥
⬘
苸(0,␥),共兵苸X:N⌳
l⫹⌳¯0共兲⭐e
␥⬘其兲⫽SES. 共2.18兲
Proof: Abbreviate M⫽e␥⬘. LetA⫽兵苸X:N⌳
l⫹⌳¯()⭐M其. Then
共A兲⭐eM
兺
苸X共兲e
⫺N⌳l⫹⌳¯共兲⫽eM关e⫺共1⫺e⫺1兲兴兩⌳l
⫹⌳¯ 兩⫽eM共1⫹o共1兲兲exp关⫺共1⫺e⫺1兲e␥兴,
共2.19兲 where we use thatoutside⌳¯ places particles according to a Poisson random field with density
, and we note that兩⌳l⫹兩⫽e␥/. QED
Proposition 2.20: Let␥⬎0 and l⫺⫽e(1/2)(⌬⫺␥). Then
共兵苸X:N⌳
l⫺⌳¯0共兲⭓log其兲⫽SES. 共2.21兲
Proof: Abbreviate M⫽log. LetA⫽兵苸X:N⌳
l⫺⌳¯()⭓M其. Then 共A兲⭐e⫺␥M
兺
苸X共兲e ␥N⌳ l⫺⌳ ¯共兲⫽e⫺␥M关e共e␥⫺1兲兴兩⌳l⫺⌳¯ 兩⫽e⫺␥M共1⫹o共1兲兲,
共2.22兲 where we note that兩⌳l⫺兩⫽e⫺␥/. QED
Proposition 2.23: Let␥⬎0 and l⫺⫽e(1/2)(⌬⫺␥). Then, for all n苸N,
共兵苸X:N2n⫹1⌳
l⫺2n⌳l⫺共兲⭓共2
n⫹1⫺2n兲2log其兲⫽共SES兲22n. 共2.24兲
Proof: Abbreviate M⫽log. For n苸N, let An⫽兵苸X:N2n⫹1⌳
l⫺2n⌳l⫺()⭓(2
n⫹1
共An兲⭐e⫺␥共2
n⫹1⫺2n兲2 log 
共1⫹o共1兲兲 共n苸N兲 共2.25兲
with the error term uniform in n. QED
Define X0 1⫽ 兵苸X:N⌳ l⫹\⌳ ¯共兲⬎e␥⬘其, X0 2⫽ 兵苸X:N⌳ l⫺⌳¯共兲⬍log其, 共2.26兲 X0 3,n⫽ 兵苸X:N2n⫹1⌳ l⫺2n⌳l⫺共兲⬍共2 n⫹1⫺2n兲2log其, and put X0⫽X0 1艚X 0 2艚
再
艚
n苸N X0 3,n冎
. 共2.27兲Proposition 2.28: Let AT⫽兵t苸X0 ᭙t苸关0,T)其. Then P共AT
c兲⫽SES for all T⭐eC with C arbitrarily large. 共2.29兲
Proof: Estimate P共AT c 兲⭓P共共X01兲c⬍T兲⫹P共共X 0 2兲c⬍T兲⫹
兺
n苸NP共共X0 3,n兲c⬍T兲 共2.30兲and use Proposition 2.6 in combination with Propositions 2.17, 2.20, and 2.23. Here note that
兩BASE((X0 1 )c)*兩, 兩BASE((X0 2 )c)*兩 and 2⫺2n兩BASE((X0 3,n
)c)*兩 grow only exponentially fast
with. QED
Proposition 2.28 will be crucial later on. Namely, it says that over the exponentially long intervals we are considering for the metastable behavior we may as well assume that the process (t)t⭓0 never leavesX0. The setX0 consists of those configurations where the gas outside⌳¯ is ‘‘close to equilibrium.’’
3. LD-ESTIMATES FOR INDEPENDENT RANDOM WALKS
In this section we formulate several large deviation estimates that involve hitting times for particles performing independent random walks. We do the estimates pretending that the random walks live on Z2 instead of⌳
. However, this only causes an error that is SES because of our
assumption that lim→⬁(1/)log兩⌳兩⫽⬁. 3.1. LD-estimates for a single random walk
3.1.1. Hitting times. Let
共t兲t⭓0 共3.1兲
be a simple random walk onZ2 with jump rate 1. LetPx denote its law on path space given 0 ⫽x. Let ⌳¯⫽min兵t⭓0:t苸⌳¯其. Proposition 3.2 below gives us control over⌳¯ when the random
walk starts from x苸⫹⌳¯ .
Proposition 3.2: There exist⬎0 and t0⬎0 such that, for all t⬎t0,
min
x苸⫹¯⌳
Px共⌳¯⬎t兲⭓
Proof: We begin by proving the analogous estimate for discrete time.
共1兲 Let (n)n苸N0 be a simple random walk onZ2. Let⌳¯⫽min兵n⬎0:n苸⌳¯其共which does not
include n⫽0), and put
n⌳ ¯
⫽max兵0⭐m⭐n:m苸⌳¯其. 共3.4兲
Pick x苸⌳¯ and write
1⫽
兺
m⫽0 n Px共n⌳ ¯ ⫽m兲⫽兺
m⫽0 n兺
y苸⫺⌳¯ Px共m⫽y兲Py共⌳¯⬎n⫺m兲. 共3.5兲Split the sum over m into two parts: 0⭐m⭐n关1⫺(1/log n)兴 and the rest. The first part can be bounded above by 兩⫺⌳¯兩
冋
1⫹兺
m⫽1 n关1⫺共1/log n兲兴 1 m册
max y苸⫺⌳¯ Py冉
⌳¯⬎ n log n冊
, 共3.6兲where we use that
max
z苸Z2
P0共m⫽z兲⭐ 1
m ᭙m⭓1 共3.7兲
共see Spitzer18Sec. 7兲. The second part can be bounded above by
兩⫺⌳¯兩
兺
m⫽n关1⫺共1/log n兲兴⫹1 n
1
m . 共3.8兲
Combining the two bounds in共3.6兲 and 共3.8兲 with 共3.5兲, we obtain, for n large enough,
max y苸⫺⌳¯ Py
冉
⌳¯⬎ n log n冊
⭓ 2 log n. 共3.9兲共2兲 Since any two sites in ⫺⌳¯ can be connected by a path outside ⌳¯ of length at most
2(l0⫹2), it follows that, uniformly in n,
min y苸⫺⌳¯ Py
冉
⌳¯⬎ n log n冊
⭓3 max y苸⫺⌳¯ Py冉
⌳¯⬎ n log n⫺4冊
. 共3.10兲Together with共3.9兲 this gives
min x苸⫺⌳¯ Px
冉
⌳¯⬎ n log n冊
⭓ 5 log n, 共3.11兲which implies共3.3兲 for discrete time after replacing n by n log n.
共3兲 The extension to continuous time is trivial, via a standard LD-estimate on the clock of the
random walk. QED
The bound in Proposition 3.2 decays very slowly with t because SRW onZ2is only margin-ally recurrent. This slow decay will be useful later on in estimates of probabilities of various events where we want to keep particles away from ⌳¯ .
共ˆ t A兲
t⭓0 共3.12兲
be a simple random walk onZ2\A with jump rate 1 with the property that when it hits⫹A it gets
‘‘trapped,’’ in the sense that a step from⫹A to⫹⫹A, the exterior boundary of A艛⫹A, occurs
at rate e⫺U. Proposition 3.13 below gives us control over the time this random walk spends in the trap⫹A starting from x苸⫹⫹A.
Proposition 3.13: There exist⫽(A)⬎0 and0⬎0 such that, for all␦⬎0, all⬎0and
all t苸关eU,eC兴 with U⬍C⬍⬁,
min x苸⫹⫹A Px共ˆt A苸⫹A兲⭓1 t e 共U⫺␦兲 2共C兲2. 共3.14兲
Proof: Again, we first prove the analogous estimate for discrete time. The proof uses the
following asymptotic result for simple random walk (n)n苸N0 on Z2. Let 0⫽min兵n⬎0:n⫽0其.
Then there exists a 1⬎0 such that Px共0⫽n兲⬃ 1 n log2n 共᭙x苸Z 2,n→⬁兲 共3.15兲 共see Spitzer18 Sec. 7兲.
共1兲 From 共3.15兲 it is easily deduced that for all rectangular A傺Z2 there exists a⫽(A) ⬎0 such that min x苸⫹⫹A Px共⫹A⫽n兲⭓ n log2n 共n→⬁兲, 共3.16兲 where⫹A⫽min兵n⬎0:n苸⫹A其. 共2兲 Let (ˆ n A
)n苸N0 be the discrete-time version of 共3.12兲. Let n0⫽e(U⫺␦)Ⰶn. Then, for x 苸⫹⫹A,
Px共ˆn A苸⫹A
兲⭓共1⫹o共1兲兲Px共n⫺n0⬍⫹A⭐n兲. 共3.17兲
Here we throw away all the first hits of⫹A at or prior to time n⫺n0and require the random walk to stay trapped for a time at least n0. The latter costs not more than (1⫺e⫺U)n0⫽1 ⫺e⫺␦⫹o(). But, by共3.16兲, we have
min x苸⫹⫹A Px共n⫺n0⬍⫹A⭐n兲⭓
兺
m⫽n⫺n0⫹1 n m log2m⬃ n0 n log2n 共n→⬁兲, 共3.18兲and so for all n苸关eU,eC兴 andsufficiently large, min x苸⫹⫹A Px共ˆn苸⫹A兲⭓ 1 ne 共U⫺␦兲 2共C兲2. 共3.19兲
共3兲 The extension to continuous time is again trivial, via a standard LD-estimate on the clock
of the random walk. QED
Proposition 3.13 will be used to control the time that particles arriving from the gas stay attached to a droplet inside⌳¯ .
3.2. Mixing propositions for independent random walks
For0苸X0, let C1␥(0) denote the event that no particle in 0艚(⌳⌳¯ ) enters ⌳¯ within time T⫽e关⌬⫺(␥/2)兴 and all of them are outside⌳l⫺ at time T. We recall that l⫺⫽e(1/2)(⌬⫺␥).
Proposition 3.20: For all ␥⬎0 there exist ⬎0 and 0⬎0 such that for all  ⬎0: min0苸X0P0(C␥1(0))⭓(/)log.
Proof: Because outside⌳¯ particles perform independent simple random walks关see 共3.1兲兴, we
have min 0苸X0 P0共C1 共 0兲兲⭓关 min x苸⫹⌳¯ Px共⌳¯⬎T,T苸⌳l⫺兲兴log, 共3.21兲
where we use that N⌳
l⫺⌳¯(0)⭐log for all 0苸X0, and that the probability between square
brackets is minimal in x苸⌳l⫺⌳¯ when x苸⫹⌳¯ . We have
Px共⌳¯⬎T,T苸⌳l⫺兲⭓Px共⌳¯⬎T兲⫺Px共T苸⌳l⫺兲. 共3.22兲
But, by Proposition 3.2, we know that
min x苸⫹⌳¯ Px共⌳¯⬎T兲⭓ 1 log T⬃ 1
冉
⌬⫺␥ 2冊
 , 共3.23兲 while共3.7兲 gives max ⫹⌳¯ Px共T苸⌳l⫺兲⭐ 2兩⌳l⫺兩 T ⫽2e ⫺␥. 共3.24兲Insert共3.23兲–共3.24兲 into 共3.22兲 to get the claim. QED
For0苸X0, let C2␥,␦(0) denote the event that no particle in0艚(Z2\⌳l⫺) enters⌳¯ within
time T⫽e(⌬⫺␦).
Proposition 3.25: For all␦⬎␥⬎0: min
0苸X0P0(C2 ␥,␦( 0))⫽1⫺SES. Proof: We have P0共C2 ␥,␦共 0兲兲⫽
兿
x苸Z2\⌳l ⫺ Px共⌳¯⬎T兲0共x兲⫽兿
x苸Z2\⌳l ⫺ 关1⫺Px共⌳¯⭐T兲兴0共x兲. 共3.26兲But, by Brownian approximation, we have
Px共⌳¯⭐T兲⭐exp关⫺兩x兩2/T兴Ⰶ1 共3.27兲
uniformly in x苸Z2\⌳l
⫺. Hence, for sufficiently large,
P0共C2 ␥,␦共 0兲兲⭓exp
再
⫺ 1 2x苸Z兺
2⌳ l⫺ 0共x兲exp关⫺兩x兩2/T兴冎
. 共3.28兲The sum in the exponent can be estimated from above by
兺
n⫽0 ⬁ N2n⫹1⌳ l⫺2n⌳l⫺共0兲exp关⫺ 1 22 2ne共␦⫺␥兲兴. 共3.29兲For0苸X0, t1⭓e(⌬⫺2␥) and x1苸⫺⌳¯ , let
C3␥共0;t1;x1兲 共3.30兲
denote the event that some particle from 0艚(Z2\⌳¯ ) enters ⌳¯ during the time interval关t1,t1 ⫹1) at site x1 without having entered ⌳¯ during the time interval 关t1⫺T1,t1) with T1 ⫽e(⌬⫺2␥).
Proposition 3.31: For all␥⬎0 there exist⬎0 and0⬎0 such that for all ⬎0: min
0苸X0
P0共C3
␥共
0;t1;x1兲兲⭐e⫺共⌬⫺2␥兲log 共3.32兲
uniformly in T1⭐t1⭐T⫽eC and x1苸⫺⌳¯ , with C arbitrarily large.
Proof: Let us look at the particle configuration at time t1⫺T1. By Proposition 2.28 we know that with a probability 1⫺SES this configuration falls in X0. Hence, using the Markov property at time t1⫺T1, we get max 0苸X0 P0共C3 ␥共 0;t1;x1兲兲⫽SES⫹ max 0苸X0
兺
苸X0 P0共t1⫺T1⫽兲P0共C3 ␥共 0;t1;x1兲兩t1⫺T1⫽兲. 共3.33兲 But, by Proposition 3.25 and共3.7兲, we havemax 苸X0 P0共C3 ␥共 0;t1;x1兲兩t1⫺T1⫽兲 ⭐SES⫹ max 苸X0 x苸⌳
兺
l⫺⌳¯ 共x兲Px共T1⫽x1兲 ⭐SES⫹1 T1 max 苸X0 N⌳ l⫺⌳¯共兲⭐SES⫹1e ⫺共⌬⫺2␥兲log. 共3.34兲Substitution into共3.33兲 gives the claim. QED
Propositions 3.20, 3.25, and 3.31 will be needed to control the dynamics of the gas outside⌳¯ . 4. LOCAL MARKOV CHAIN: DEFINITIONS AND SADDLE POINTS
In this section we introduce the local Markov chain that approximates our dynamics inside⌳¯ , and we study its geometric properties. In Sec. 5 we will study the recurrence properties of this Markov chain, which will be needed in Secs. 6–7 to study the metastable behavior of the full Markov chain.
4.1. Definition of the local Markov chain
We denote by b⫽(x,y) an oriented bond, i.e., an ordered pair of nearest-neighbor sites, and define
*⌳¯out⫽兵b⫽共x,y兲: x苸⌳¯ ,y苸⌳¯其,
共4.1兲
*⌳¯in⫽兵b⫽共x,y兲: x苸⌳¯ ,y苸⌳¯其,
and*⌳¯⫽*⌳¯out艛*⌳¯in. Two configurations¯ ,¯
⬘
苸X¯⫽兵0,1其⌳¯ with¯⫽¯⬘
are calledcommu-nicating states if there exists a bond b苸⌳¯*艛*⌳¯ such that ¯
⬘
⫽Tb¯ , where Tb¯ is the•b苸⌳¯*: Tb¯ denotes the configuration obtained from¯ by interchanging particles along b;
•b苸*⌳¯out共i.e., b is exiting from ⌳¯ ):
Tb¯共z兲⫽
再
¯共z兲 ᭙z⫽x, 0 z⫽x; 共4.2兲 •b苸*⌳¯in共i.e., b is entering ⌳¯ ): Tb¯共z兲⫽再
¯共z兲 ᭙z⫽y, 1 z⫽y. 共4.3兲Definition 4.4: The local Markov chain (¯t)t⭓0 is the Markov chain on X¯⫽兵0,1其⌳ ¯ with generator 共Lf 兲共¯兲⫽
兺
b苸⌳¯*艛*⌳¯ c共b,¯兲关 f 共Tb¯兲⫺ f 共¯兲兴, 共4.5兲where H¯ is defined in共1.49兲 and
c共b,¯兲⫽e⫺关H¯共Tb¯兲⫺H¯ 共¯兲兴⫹. 共4.6兲 Note that
b苸*⌳¯in: c共b,¯兲⫽e⫺⌬,
共4.7兲
b苸*⌳¯out: c共b,¯兲⫽1.
These rates do not depend on¯ because there is no interaction between particles in ⌳¯⌳¯0 and particles in⌳¯0.
In a standard way the above dynamics can be realized with the help of Poisson clocks. To study the transitions of the local Markov chain, we consider the discrete-time version that is obtained from the continuous-time version by looking at the process when some clock in ⌳¯*艛*⌳¯ rings. We denote by P¯ (¯ ,¯
⬘
) the corresponding transition probabilities, i.e.,P
¯共¯ ,¯
⬘
兲⫽P¯共¯¯1⫽¯
⬘
兲 共4.8兲with¯1 the first ringing time of a clock in ⌳¯*艛*⌳¯ . It is easy to verify that the stochastic dynamics defined by 共4.5兲–共4.6兲 and 共4.8兲 is reversible w.r.t. H¯ . In particular, the transition probabilities P(¯ ,¯
⬘
) can be written in the formP
¯共¯ ,¯
⬘
兲⫽q共¯ ,¯⬘
兲e⫺关H¯共¯⬘兲⫺H¯ 共¯兲兴⫹, 共4.9兲
where q(¯ ,¯
⬘
) is an irreducible symmetric Markov kernel living on the set of communicating states.4.2. Geometric definitions
Let us recall some definitions from Olivieri and Scoppola.10
共1兲 A pathis a sequence ⫽1,...,n(n苸N,i苸X¯ ) with P¯ (i,i⫹1)⬎0 for i⫽1,...,n
