Kawasaki dynamics with two types of particles : stable/megastable configurations and communication heights

Kawasaki dynamics with two types of particles :

stable/megastable configurations and communication heights

Citation for published version (APA):

Hollander, den, W. T. F., Nardi, F. R., & Troiani, A. (2011). Kawasaki dynamics with two types of particles :

stable/megastable configurations and communication heights. (Report Eurandom; Vol. 2011026). Eurandom.

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EURANDOM PREPRINT SERIES

2011-026

Kawasaki dynamics with two types of particles:

stable/metastable configurations and communication heights

F. den Hollander, F.R. Nardi, A. Troiani

ISSN 1389-2355

stable/metastable ongurations and ommuni ation heights F. den Hollander 1 2 F.R. Nardi 3 2 A. Troiani 1 May 24, 2011 Abstra t

Thisisthe se ondinaseriesofthreepapersinwhi hwestudyatwo-dimensionallatti e gas

onsistingof twotypesofparti lessubje tto Kawasakidynami sat lowtemperatureinalarge

niteboxwithanopenboundary. Ea hpairofparti leso upyingneighboringsiteshasanegative

bindingenergyprovidedtheirtypesaredierent,whileea hparti lehasapositivea tivationenergy

thatdependsonitstype. Thereisnobinding energybetweenparti lesofthe sametype. Atthe

boundaryof theboxparti lesare reatedand annihilatedinawaythat representsthepresen e

of an innite gas reservoir. We start the dynami s from the empty box and are interestedin

the transition time to the full box. This transition is triggered by a riti al droplet appearing

somewhereinthebox.

Intherstpaperweidentiedtheparameterrangeforwhi hthesystemismetastable,showed

thattherstentran edistributiononthesetof riti aldropletsisuniform, omputedtheexpe ted

transition time up to and in luding a multipli ative fa tor of order one, and proved that the

nu leation time divided by its expe tation is exponentially distributed, all in the limit of low

temperature. Theseresultswereprovedunderthreehypotheses,andinvolvethreemodel-dependent

quantities:theenergy,theshapeandthenumberof riti aldroplets. Inthese ondpaperweprove

therstandthese ondhypothesisandidentifytheenergyof riti aldroplets. Inthethirdpaper

wesettletherest.

Both the se ond and the third paper deal with understanding the geometri properties of

sub riti al, riti al and super riti al droplets, whi h are ru ial in determining the metastable

behaviorofthesystem,asexplainedintherstpaper. Thegeometryturnsouttobe onsiderably

more omplex than for Kawasaki dynami s with one type of parti le, for whi h an extensive

literatureexists. Themain motivationbehindourworkisto understandmetastabilityof

multi-typeparti lesystems.

1

Mathemati alInstitute,LeidenUniversity,P.O.Box9512,2300RALeiden,TheNetherlands

2

EURANDOM,P.O.Box513,5600MBEindhoven,TheNetherlands

3

Se tion1.1 denesthemodel,Se tion1.2introdu esbasi notation,Se tion 1.3statesthemain

theo-rems,whileSe tion1.4 dis ussesthemaintheoremsandprovidesfurther perspe tives.

1.1 Latti e gas subje t to Kawasaki dynami s

Let

Λ

⊂ Z

2

be alarge box entered at the origin(later it will be onvenientto hoose

Λ

rhombus-shaped). Let

∂

−

Λ =

{x ∈ Λ: ∃ y /

∈ Λ: |y − x| = 1},

∂

+

_{Λ =}

{x /

∈ Λ: ∃ y ∈ Λ: |y − x| = 1},

(1.1)

be the internal, respe tively, external boundary of

Λ

, and put

Λ

−

_{= Λ}

\∂

−

_Λ

and

Λ

+

_{= Λ}

∪ ∂

+

_Λ

.

With ea h site

x

∈ Λ

we asso iatea variable

η(x)

∈ {0, 1, 2}

indi atingthe absen e of aparti le or the presen e of aparti le of type

1

or type

2

. A onguration

η =

{η(x): x ∈ Λ}

is an elementof

X = {0, 1, 2}

Λ

. Toea h onguration

η

weasso iateanenergygivenbytheHamiltonian

H =

−U

X

(x,y)∈Λ

∗,−

1

{η(x)η(y)=2}

+ ∆

1

X

x∈Λ

1

{η(x)=1}

+ ∆

2

X

x∈Λ

1

{η(x)=2}

,

(1.2) where

Λ

∗,−

₌

{(x, y): x, y ∈ Λ

−

_,

|x − y| = 1}

is theset ofnon-orientedbondsinside

Λ

−

,

−U < 0

is thebinding energy betweenneighboringparti lesofdierent typesinside

Λ

−

,and

∆

1

> 0

and

∆

2

> 0

arethea tivation energies ofparti lesoftype

1

,respe tively,

2

inside

Λ

. W.l.o.g.wewillassumethat

∆

1

≤ ∆

2

.

(1.3)

TheGibbsmeasure asso iatedwith

H

is

µ

β

(η) =

1

Z

β

e

−βH(η)

,

η

∈ X ,

(1.4)

where

β

∈ (0, ∞)

is theinversetemperatureand

Z

β

isthenormalizingpartition sum.

Kawasakidynami sisthe ontinuous-timeMarkovpro ess,

(η

t

)

t≥0

withstatespa e

X

whose tran-sitionratesare

c

β

(η, η

′

) = e

−β[H(η

′

_)−H(η)]

+

_,

_{η, η}

′

∈ X , η 6= η

′

, η

_{↔ η}

′

,

(1.5) where

η

↔ η

′

meansthat

η

′

anbeobtainedfrom

η

byoneof thefollowingmoves:

•

inter hanging

0

and

1

or

0

and

2

betweentwoneighboringsitesin

Λ

(hoppingofparti lesin

Λ

),

•

hanging

0

to

1

or

0

to

2

in

∂

−

_Λ

( reationofparti lesin

∂

−

_Λ

),

•

hanging

1

to

0

or

2

to

0

in

∂

−

_Λ

(annihilationofparti lesin

∂

−

_Λ

), and

c

β

(η, η

′

_{) = 0}

otherwise. Notethatthisdynami spreservesparti lesin

Λ

,butallowsparti lestobe reatedandannihilatedin

∂

−

_Λ

. Thinkofthelatterasdes ribingparti lesenteringandexiting

Λ

along non-orientedbonds between

∂

+

_Λ

and

∂

−

_Λ

(the rates of these movesare asso iated with the bonds

ratherthanwiththesites). Thepairs

(η, η

′

₎

with

η

↔ η

′

are alled ommuni ating ongurations,the

transitions between them are alled allowed moves. Note that parti les in

∂

−

_Λ

do notintera t: the

intera tiononlyworksin

Λ

−

.

The dynami s dened by (1.2) and (1.5) models the behavior inside

Λ

of a latti e gas in

Z

2

,

onsisting of two types of parti les subje t to random hopping with hard- ore repulsion and with

bindingbetweendierentneighboringtypes. Wemaythinkof

Z

2

theparti ledensitiesxedat

ρ

1

= e

−β∆

1

and

ρ

2

= e

−β∆

2

. Intheabovemodelthisreservoirisrepla ed

by anopen boundary

∂

−

_Λ

, where parti les are reatedand annihilated at aratethat mat hesthese

densities. Thus, thedynami sis anite-state Markovpro ess,ergodi and reversiblewith respe tto

theGibbsmeasure

µ

β

in(1.4).

Notethatthereisnobindingenergybetweenneighboringparti lesofthesametype. Consequently,

themodeldoesnot redu e toKawasakidynami sforonetypeofparti lewhen

∆

1

= ∆

2

.

1.2 Notation

Tostateourmain theoremsinSe tion 1.3,weneedsomenotation.

Denition1.1 (a)



isthe ongurationwhere

Λ

isempty.

(b)

⊞

istheset onsistingofthetwo ongurationswhere

Λ

islledwiththelargestpossible he kerboard droplet su hthatall parti les oftype

2

aresurroundedbyparti les oftype

1

.

( )

ω : η

→ η

′

isanypath ofallowedmovesfrom

η

∈ X

to

η

′

_{∈ X}

.

(d)

Φ(η, η

′

₎

isthe ommuni ation height between

η, η

′

_{∈ X}

denedby

Φ(η, η

′

) =

min

ω: η→η

′

max

ξ∈ω

H(ξ),

(1.6)

and

Φ(A, B)

isitsextensiontonon-emptysets

A, B

⊂ X

denedby

Φ(A, B) =

min

η∈A,η

′

_∈B

Φ(η, η

′

_).

(1.7)

(e)

V

η

isthe stabilitylevel of

η

∈ X

denedby

V

η

= Φ(η,

I

η

)

− H(η),

(1.8)

where

I

η

=

{ξ ∈ X : H(ξ) < H(η)}

isthe setof ongurations withenergylower than

η

.

(f)

X

stab

=

{η ∈ X : H(η) = min

ξ∈X

H(ξ)

}

isthe set of stable ongurations,i.e., the set of ongu-rationswith mininal energy.

(g)

X

meta

=

{η ∈ X : V

η

= max

ξ∈X \X

stab

V

ξ

}

isthe set of metastable ongurations, i.e., the set of non-stable ongurationswith maximalstabilitylevel.

(h)

Γ = V

η

for

η

∈ X

meta

(notethat

η

7→ V

η

is onstant on

X

meta

),

Γ

⋆

_{= Φ(, ⊞)}

_{− H()}

(notethat

H() = 0

).

In[3℄ wewere interestedin thetransition of theKawasaki dynami sfrom



to

⊞

in the limitas

β

→ ∞

. This transition, whi h is viewed asa rossover from a gas phase to a liquid phase, is triggeredbytheappearan eofa riti aldroplet somewherein

Λ

. The riti aldropletsformasubsetof theset of ongurationsrealizingtheenergeti minimaxofthepathsoftheKawasakidynami sfrom



to

⊞

,whi hallhaveenergy

Γ

⋆

be ause

H() = 0

.

In [3℄ we showed that the rst entran e distribution on the set of riti al droplets is uniform,

omputed theexpe ted transition time upto and in ludingamultipli ativefa torof order one,and

proved that the nu leation time divided by its expe tation is exponentially distributed, all in the

limit as

β

→ ∞

. These results, whi h are typi al for metastablebehavior, were proved under three hypotheses: (H1)

X

stab

= ⊞

. (H2) Thereexistsa

V

⋆

_{< Γ}

⋆

su h that

V

η

≤ V

⋆

forall

η

∈ X \{, ⊞}

.

(H3) A hypothesisabouttheshapeofthe ongurationsin theessentialgateforthe transitionfrom



to

⊞

(fordetailssee[3℄).

Hypotheses(H1H3)are thegeometri input that isneeded to derive themain theorems in [3℄with

thehelp ofthe potential-theoreti approa h to metastability asoutlined in Bovier[2℄. Inthepresent

paperweprove(H1H2)andidentifytheenergy

Γ

⋆

Lemma1.2 (H1H2)imply that

V



= Γ

⋆

,andhen ethat

X

meta

= 

and

Γ = Γ

⋆

.

Proof. ByDenition 1.1(eh)and (H1),

⊞

∈ I



, whi h implies that

V



≤ Γ

⋆

. Weshow that (H2) implies

V



= Γ

⋆

. Theproof isby ontradi tion. Suppose that

V



< Γ

⋆

. Then, byDenition 1.1(h), there exists a

η

0

∈ I



\⊞

su hthat

Φ(, η

0

)

− H() < Γ

⋆

. But(H2), togetherwith thenitenessof

X

,impliesthatthereexistan

m

∈ N

andasequen e

η

1

, . . . , η

m

∈ X

with

η

m

= ⊞

su hthat

η

i+1

∈ I

η

i

and

Φ(η

i

, η

i+1

)

≤ H(η

i

) + V

⋆

for

i = 0, . . . , m

− 1

. Therefore

Φ(η

0

, ⊞)

≤

max

i=0,...,m−1

Φ(η

i

, η

i+1

)

≤

i=0,...,m−1

max

[H(η

i

) + V

⋆

_{] = H(η}

0

) + V

⋆

< H() + Γ

⋆

,

(1.9) where intherstinequalityweusethat

Φ(η, σ)

≤ max{Φ(η, ξ), Φ(ξ, σ)}

forall

η, σ, ξ

∈ X

,andin the lastinequalitythat

η

0

∈ I



and

V

⋆

_{< Γ}

⋆

. Itfollowsthat

Φ(, ⊞)

− H() ≤ max{Φ(, η

0

)

− H(), Φ(η

0

, ⊞)

− H()} < Γ

⋆

,

(1.10) whi h ontradi tsDenition1.1(h). Observethattheproofusesthat

X

meta

onsistsofasingle

ong-uration.



Hypotheses (H1H2)imply that

(

X

meta

,

X

stab

) = (, ⊞)

, and that thehighest energybarrier be-tweenanytwo ongurationsin

X

istheoneseparating



and

⊞

,i.e.,

(, ⊞)

istheuniquemetastable pair. Hypothesis(H3)isneededonlytondtheasymptoti softheprefa toroftheexpe tedtransition

time inthelimitas

Λ

→ Z

2

. Themain theoremsin [3℄involvethreemodel-dependent quantities: the

energy,theshapeandthenumberof riti aldroplets.

1.3 Main theorems

In [3℄ it was shown that

∆

1

+ ∆

2

< 4U

is the metastable region, i.e., the region of parameters for whi h



isalo alminimumbutnotaglobalminimumof

H

. Moreover,itwasarguedthatwithinthis region the subregionwhere

∆

1

, ∆

2

< U

is of no interestbe ausethe riti al droplet onsists of two freeparti les,oneoftype

1

andoneoftype

2

. Thereforethepropermetastable region is

0 < ∆

1

≤ ∆

2

,

∆

1

+ ∆

2

< 4U,

∆

2

≥ U,

(1.11) asindi atedin Fig.1.

Figure1: Propermetastableregion.

Inthispresentpaper,theanalysiswill be arriedoutforthesubregionwhere

0 < ∆

1

< U,

∆

2

− ∆

1

> 2U,

∆

1

+ ∆

2

< 4U,

(1.12) asindi atedinFig.2. Note: These ondandthirdrestri tionimplytherstrestri tion. Nevertheless,

wewrite allthreebe auseea hplaysanimportantrolein thesequel.

The following three theorems are the main result of the present paper and are valid subje t to

Theorem1.3

X

stab

= ⊞

. Theorem1.4 Thereexistsa

V

⋆

≤ 10U − ∆

1

su hthat

V

η

≤ V

⋆

forall

η

∈ X \{, ⊞}

. Consequently, if

Γ

⋆

_{> 10U}

_{− ∆}

1

,then

X

meta

= 

and

Γ = Γ

⋆

. Theorem1.5

Γ

⋆

₌

_−[ℓ

⋆

_(ℓ

⋆

_{− 1) + 1](4U − ∆}

1

− ∆

2

) + (2ℓ

⋆

+ 1)∆

1

+ ∆

2

with

ℓ

⋆

₌



∆

1

4U

_{− ∆}

1

− ∆

2



∈ N.

(1.13)

Theorem1.3 settleshypothesis (H1)in [3℄, Theorem1.4 settleshypothesis (H2) in[3℄when

Γ

⋆

_>

10U

_{− ∆}

1

,while Theorem1.5identies

Γ

⋆

.

Assoonas

V

⋆

_{< Γ}

⋆

,theenergylands apedoesnot ontainwellsdeeperthanthosesurrounding



and

⊞

. Theorems1.3and1.4implythatthiso ursatleastwhen

Γ

⋆

_{> 10U}

_{− ∆}

1

,whileTheorem1.5 identies

Γ

⋆

andallowsustoexhibitafurthersubregionof(1.12)wherethelatterinequalityissatised.

Thisfurther subregion ontainstheshadedregionin Fig.3.

Figure3: Theparameterregionwhere

Γ

⋆

_{> 10U}

_{− ∆}

1

ontainstheshadedregion.

1.4 Dis ussion

1. InSe tion4wewillseethatthe riti aldropletsforthe rossoverfrom



to

⊞

onsistofa rhombus-shaped he kerboard with a protuberan e plus a free parti le, asindi atedin Fig.4. A moredetailed

des riptionwillbegivenin [4℄.

2. Abbreviate

Figure 4: A riti al droplet. Light-shaded squares are parti les of type

1

, dark-shaded squares are parti les of type

2

. The parti les of type

2

form an

ℓ

⋆

_{× (ℓ}

⋆

_{− 1)}

quasi-square with aprotuberan e

atta hedtooneofitslongestsides,andareallsurroundedbyparti lesoftype

1

. Inaddition,thereis afreeparti leoftype

2

. Assoonasthisfreeparti leatta hesitselfproperly toaparti leoftype

1

thedynami sisoverthehill (see[3℄, Se tion2.3,item3).

andwrite

ℓ

⋆

_{= (∆}

1

/ε) + ι

with

ι

∈ [0, 1)

. Thenaneasy omputationshowsthat

Γ

⋆

_{= (∆}

1

)

2

/ε + ∆

1

+

4U + ει(1

− ι)

. Fromthisweseethat

ℓ

⋆

∼ ∆

1

/ε,

Γ

⋆

∼ (∆

1

)

2

/ε,

ε

↓ 0.

(1.15)

The limit

ε

↓ 0

orresponds to the weakly supersaturated regime, where the latti e gas wants to ondensatebuttheenergeti thresholdtodosoishigh(be ausethe riti aldropletislarge). Fromthe

viewpointof metastability thisregime is the mostinteresting. Theshaded region in Fig.3 aptures

thisregimeforall

0 < ∆

1

< U

. Thisregion ontainsthesetofparameterswhere

(∆

1

)

2

_{/ε + ∆}

1

+ 4U >

10U

_{− ∆}

1

,i.e.,

ε/U < (∆

1

/U )

2

_/[6

− 2(∆

1

/U )]

.

3. Thesimplifying features of(1.12)over(1.11)are thefollowing:

∆

1

< U

impliesthat ea htime a parti leoftype

1

enters

Λ

andatta hesitselftoaparti leoftype

2

inadroplettheenergygoesdown, while

∆

2

− ∆

1

> 2U

implies that no parti le of type

2

sits on the boundary of adroplet that has minimal energygiventhenumberofparti les oftype

2

in thedroplet. In[3℄ we onje turedthat the metastabilityresultspresentedtherea tuallyholdthroughouttheregiongivenby(1.11),eventhough

the riti aldropletswillbedierent when

∆

1

≥ U

.

Aswill be ome learin Se tion 3, the onstraint

∆

1

< U

hastheee t that in all ongurations that are lo al minimaof

H

allparti les ontheboundaryof adropletare oftype

1

. It will turn out that su h ongurations onsist of a singlerhombus-shaped he kerboard droplet. We expe t that as

∆

1

in reasesfrom

U

to

2U

thereisagradualtransitionfromarhombus-shaped he kerboard riti al droplettoasquare-shaped he kerboard riti aldroplet. Thisis oneof thereasonswhyitis di ult

togobeyond(1.12).

4. WhatmakesTheorem1.4hardtoproveisthattheestimateon

V

η

hastobeuniformin

η /

∈ {, ⊞}

. In ongurations ontainingseveraldropletsand/ordroplets loseto

∂

−

_Λ

theremaybeala koffree

spa emakingthemotionofparti lesinside

Λ

di ult. Theme hanismsdevelopedinSe tion5allowus torealizeanenergyredu tiontoa ongurationthatliesonasuitablereferen epathforthenu leation

within anenergybarrier

10U

− ∆

1

alsoin theabsen eof freespa earoundea hdroplet.

Wewill seein Se tion 5that fordropletssu ientlyfar awayfromother dropletsandfrom

∂

−

_Λ

aredu tionwithin anenergybarrier

≤ 4U + ∆

1

ispossible. Thus, ifwewould beableto ontrol the ongurationsthat fail to havethisproperty,then wewouldhave

V

⋆

_{≤ 4U + ∆}

1

and, onsequently, wouldhave

X

meta

= 

and

Γ = Γ

⋆

throughoutthesubregiongivenby(1.12)be ause

Γ

⋆

_{> 4U + ∆}

1

.

Anotherwayof phrasingthelast observationis the following. Weviewtheliquid phase asthe

of ongurationsllingmostof

Λ

but stayingawayfrom

∂

−

_Λ

, thenthemetastabilityresultsderived

in [3℄wouldapplythroughoutthesubregiongivenby(1.12).

5. Theorems 1.3 and 1.5 an a tually be proved without the restri tion

∆

2

− ∆

1

> 2U

. However, removalofthisrestri tionmakesthetaskofshowingthatindropletswithminimalenergyallparti les

oftype

2

aresurroundedbyparti lesoftype

1

moreinvolvedthanwhatisdoneinSe tion3. Weomit thisextension, sin etherestri tion

∆

2

− ∆

1

> 2U

isneededforTheorem1.4anyway.

Outline. Se tion2 ontainspreparations. Theorems1.31.5areprovedinSe tions35,respe tively.

The proofs are purely ombinatorial, and are rather involved due to the presen e of two types of

parti les rather than one. Se tions 34 deal with stati s and Se tion 5with dynami s. Se tion 5is

te hni allythehardest andtakesupabouthalf ofthepaper. Moredetailed outlinesaregivenat the

beginningofea hse tion.

2 Coordinates, denitions and polyominoes

Se tion 2.1 introdu estwo oordinate systemsthat are used to des ribe the parti le ongurations:

standardanddual. Se tion 2.2liststhemain geometri denitions that areneededin therestof the

paper. Se tion2.3provesalemmaaboutpolyominoes(niteunionsofunitsquares)andSe tion2.4a

lemmaabout

2

tiled lusters( he kerboard ongurationswhereallparti lesoftype

2

aresurrounded by parti les of type

1

). These lemmas are needed in Se tion 3 to identify the droplets of minimal energygiventhenumberofparti lesoftype

2

in

Λ

.

2.1 Coordinates

1.Asite

i

∈ Λ

isidentiedbyitsstandard oordinates

(x

1

(i), x

2

(i))

,andis alledoddwhen

x

1

(i)+x

2

(i)

isoddandevenwhen

x

1

(i) + x

2

(i)

iseven. Thestandard oordinatesofaparti le

p

in

Λ

aredenoted by

x(p) = (x

1

(p), x

2

(p))

. Theparity ofaparti le

p

isdenedas

x

1

(p) + x

2

(p) + η(x(p))

modulo2,and

p

issaidtobeoddwhentheparityis

1

andevenwhentheparityis

0

. 2.Asite

i

∈ Λ

isalsoidentiedbyitsdual oordinates

u

1

(i) =

x

1

(i)

− x

2

(i)

2

,

u

2

(i) =

x

1

(i) + x

2

(i)

2

.

(2.1)

Twosites

i

and

j

aresaidtobeadja ent, written

i

∼ j

, when

|x

1

(i)

− x

1

(j)

| + |x

2

(i)

− x

2

(j)

| = 1

or, equivalently,

|u

1

(i)

− u

1

(j)

| = |u

2

(i)

− u

2

(j)

| =

1

2

(seeFig.5).

3.For onvenien e,wetake

Λ

tobethe

(L +

3

2

)

× (L +

3

2

)

dualsquare enteredat theoriginforsome

L

_{∈ N}

with

L > 2ℓ

⋆

(to allowfor

H(⊞) < H()

;see Se tion 3.1). Parti lesintera t onlyinside

Λ

−

, whi h is the

(L +

1

2

)

× (L +

1

2

)

dual square entered at the origin. This dual square, a rhombus in standard oordinates, is onvenientbe ausethe lo al minima of

H

arerhombus-shaped aswell(see Se tion 3).

2.2 Denitions

1.A site

i

∈ Λ

is saidto be latti e- onne ting in the onguration

η

ifthere exists alatti e path

λ

from

i

to

∂

−

_Λ

su hthat

η(j) = 0

forall

j

∈ λ

with

j

6= i

. Wesaythataparti le

p

islatti e- onne ting if

x(p)

isalatti e- onne tingsite.

2.Twoparti lesin

η

atsites

i

and

j

are alled onne ted if

i

∼ j

and

η(i)η(j) = 2

. Iftwoparti les

p

1

and

p

2

are onne ted,thenwesaythatthere isana tivebond

b

betweenthem. Thebond

b

issaidto bein ident to

p

1

and

p

2

. Aparti le

p

issaidtobesaturated ifitis onne tedtofourotherparti les, i.e.,therearefoura tivebondsin identto

p

. Thesupportofthe onguration

η

,i.e.,theunionofthe

Figure5: A ongurationrepresentedin: (a)standard oordinates;(b)dual oordinates. Light-shaded

squaresareparti lesoftype

1

,dark-shadedsquaresareparti lesoftype

2

. Indual oordinates,parti les of type

2

are representedbylargersquaresthanparti les oftype

1

toexhibit thetiled stru ture of the onguration.

unit squares entered at theo upiedsites of

η

, isdenoted by supp

(η)

. Fora onguration

η

,

n

1

(η)

and

n

2

(η)

denotethenumberofparti lesoftype

1

and

2

in

η

,and

B(η)

denotesthenumberofa tive bonds. Theenergyof

η

equals

H(η) = ∆

1

n

1

(η) + ∆

2

n

2

(η)

− UB(η)

.

3.Let

G(η)

be thegraph asso iatedwith

η

, i.e.,

G(η) = (V (η), E(η))

, where

V (η)

is theset of sites

i

_{∈ Λ}

su h that

η(i)

6= 0

, and

E(η)

is theset of the pairs

{i, j}

,

i, j

∈ V (η)

, su h that theparti les at sites

i

and

j

are onne ted. A onguration

η

′

is alled asub onguration of

η

, written

η

′

≺ η

, if

η

′

_{(i) = η(i)}

forall

i

∈ Λ

su hthat

η

′

_{(i) > 0}

. Asub onguration

c

≺ η

isa luster ifthegraph

G(c)

is amaximal onne ted omponentof

G(η)

. Thesetofnon-saturatedparti lesin

c

is alledtheboundary of

c

, andis denoted by

∂c

. Clearly,allparti les in thesame luster havethe sameparity. Therefore the on eptofparityextendsfrom parti lesto lusters.

4.Forasite

i

∈ Λ

,thetile enteredat

i

,denotedbyt

(i)

,isthesetofvesites onsistingof

i

andthe four sites adja entto

i

. If

i

is anevensite, thenthetile issaidto beeven, otherwisethetileis said to beodd. Thevesites ofatilearelabeled

a

,

b

,

c

,

d

,

e

asin Fig.6. Thesites labeled

a

,

b

,

c

,

d

are alledjun tion sites. Ifaparti le

p

sitsatsite

i

,thent

(i)

isalsodenotedbyt

(p)

andis alledthetile asso iatedwith

p

. Instandard oordinates,atile isasquare of size

√

2

. In dual oordinates, itis a unit square.

5.Atilewhose entralsiteiso upiedbyaparti leoftype

2

andwhosejun tionsitesareo upiedby parti lesoftype

1

is alleda

2

tile(seeFig.6). Two

2

tilesaresaidtobeadja entiftheirparti lesof type

2

havedualdistan e 1. A horizontal(verti al)

12

baris amaximalsequen eof adja ent

2

tiles allhavingthesamehorizontal(verti al) oordinate. If thesequen ehaslength

1

,then the

12

baris alleda

2

tiledprotuberan e. A luster ontainingatleastoneparti leoftype

2

su hthatallparti les of type

2

aresaturated is saidto be

2

tiled. A

2

tiled ongurationis a onguration onsistingof

2

tiled lustersonly.

(a) (b) ( ) (d)

Figure 6: Tiles: (a) standard representationof the labels of atile; (b) standardrepresentation of a

6.Thetile support of a onguration

η

isdened as

[η] =

[

p∈̟

2

(η)

t

(p),

(2.2)

where

̟

2

(η)

isthe set of parti les oftype

2

in

η

. Obviously,

[η]

is theunion ofthe tilesupportsof the lustersmakingup

η

. Forastandard luster

c

thedual perimeter,denoted by

P (c)

, isthelength of theEu lidean boundaryofits tilesupport

[c]

(whi h in ludes aninner boundarywhen

c

ontains holes). Thedualperimeter

P (η)

ofa

2

tiled onguration

η

is thesumofthedualperimetersof the lustersmakingup

η

.

7.

V

⋆,n

2

istheset of ongurationssu hthatin

Λ

−−

thenumberofparti lesoftype

2

is

n

2

.

V

4n

2

⋆,n

2

is thesetof ongurationssu hthatin

Λ

−−

thenumberofparti lesoftype

2

is

n

2

,thenumberofa tive bonds is

4n

2

, and there is no isolated parti le of type

1

. In other words,

V

4n

2

⋆,n

2

is the set of

2

tiled ongurationswith

n

2

parti lesoftype

2

. Thelowerindex

⋆

isusedtoindi ate that ongurationsin these sets anhaveanarbitrarynumberofparti lesoftype

1

. A onguration

η

is alledstandard if

η

∈ V

4n

2

⋆,n

2

, and itstile support isastandardpolyomino in dual oordinates (seeDenition 2.1 below forthedenitionofastandardpolyomino).

8.Aunithole isanemptysitesu hthatallfourofitsneighborsareo upiedbyparti lesofthesame

type (either all of type

1

or all of type

2

). An empty site with three neighboring sites o upied by a parti le oftype

1

is alled agood dual orner. In thedual representationa good dual orner is a on ave orner(seeFig.7).

2.3 A lemma on polyominoes

Thetilesupport ofa luster

c

anberepresentedbyapolyomino,i.e.,aniteunionof unitsquares. Thefollowingnotationisused:

ℓ

1

(c) =

width of

c

(= numberof olumns).

ℓ

2

(c) =

heightof

c

(= numberofrows).

v

i

(c) =

numberofverti aledgesin the

i

-thnon-emptyrowof

c

.

h

j

(c) =

numberofhorizontaledgesinthe

j

-thnon-empty olumnof

c

.

P (c) =

lengthoftheperimeterof

c

.

Q(c) =

numberofholesin

c

.

ψ(c) =

numberof onvex ornersof

c

.

φ(c) =

numberof on ave ornersof

c

. Notethat

ψ(c) =

P

N(c)

i=1

ψ(i)

and

φ(c) =

P

N

(c)

i=1

φ(i)

,where

N (c)

is thenumberofverti esin the polyominorepresenting

c

. Iftwoedges

e

1

and

e

2

arein identto vertex

i

atarightanglewithaunit squareinsideandnounitsquaresoutside,then

ψ(i) = 1

and

φ(i) = 0

(Fig.7(a)). Ontheotherhand, ifthereisnounit squareinsideandthreeunitsquaresoutside,then

ψ(i) = 0

and

φ(i) = 1

(Fig.7(b)). If four edges

e

1

,

e

2

,

e

3

,

e

4

are in ident to vertex

i

, with two unit squares in opposite angles, then

ψ(i) = 0

and

φ(i) = 2

(Fig.7( )).

Denition2.1 [Alonso and Cerf [1℄.℄ A polyomino is alled monotone if its perimeter is equal to

the perimeter of its ir ums ribing re tangle. A polyomino whose support is a quasi-square (i.e., a

re tangle whose side lengthsdier by atmost one), with possibly a bar atta hedto one of its longest

Figure 7: Corners of polyominoes: (a) one onvex orner; (b) one on ave orner; ( ) two on ave

orners. Shadedmeano upiedbyaunitsquare.

Inthesequel,akeyrolewillbeplayedbythequantity

T (c) = 2P (c) + [ψ(c) − φ(c)] = 2P (c) + 4 − 4Q(c).

(2.3) Lemma2.2 (i) All polyominoes

c

with a xed number of monominoes minimizing

T (c)

are single- omponent monotonepolyominoesof minimal perimeter, whi hin lude the standardpolyominoes.

(ii) If the number of monominoesis

ℓ

2

,

ℓ

2

_{− 1}

,

ℓ(ℓ

− 1)

or

ℓ(ℓ

− 1) − 1

for some

ℓ

∈ N\{1}

,then the standardpolyominoes arethe onlyminimizers of

T (c)

.

Proof. Intheproofweassumew.l.o.g.that thepolyomino onsistsofasingle luster

c

.

(i)Theproofusesproje tion. Pi kanynon-monotone luster

c

. Let

˜

c = (π

2

◦ π

1

)(c),

(2.4)

where

π

2

and

π

1

denotetheverti al,respe tively,thehorizontalproje tionof

c

. Theee tofverti al andhorizontalproje tionisillustratedin Fig.8. By onstru tion,

c

˜

isamonotonepolyomino(seee.g. thestatementonFerrersdiagramsin theproofof AlonsoandCerf[1℄,Theorem2.2).

Figure8: Ee tofverti alandhorizontalproje tion.

Supposerstthat

Q(c) = 0

. Then

T (c) = 2P (c)+4

. Sin e

c

isnotmonotone,wehave

P (˜

c) < P (c)

, andso

c

isnotaminimizerof

T (c)

.

Supposenextthat

Q(c)

≥ 1

. Sin e

P (c) =

ℓ

2

(c)

X

i=1

v

i

(c) +

ℓ

1

(c)

X

j=1

h

j

(c)

(2.5)

andeveryholebelongstoatleastonerowand one olumn,wehave

P (c)

_{≥ 2[ℓ}

1

(c) + ℓ

2

(c)] + 4Q(c).

(2.6)

Ontheotherhand,sin e

˜

c

isamonotonepolyomino,wehave

v

i

(˜

c) = h

j

(˜

c) = 2

forall

i

and

j

,andso

P (˜

c) = 2[ℓ

1

(˜

c) + ℓ

2

(˜

c)].

(2.7)

Moreover,sin e

ℓ

1

(˜

c)

≤ ℓ

1

(c)

and

ℓ

2

(˜

c)

≤ ℓ

2

(c)

, we an ombine (2.62.7)toget

T (˜c) − T (c) = [2P (˜c) + 4] − [2P (c) + 4 − 4Q(c)] = 2[P (˜c) − P (c)] + 4Q(c) ≤ −4Q(c) ≤ −4 < 0,

(2.9) andso

c

isnotaminimizerof

T (c)

.

(ii)Wesawintheproofof(i)thatif

c

isaminimizerof

T (c)

,then

c

ismonotone,andhen edoesnot ontainholesandminimizes

P (c)

. The laimthereforefollowsfromAlonsoandCerf[1℄,Corollary3.7, whi hstatesthatifthenumberofmonominoesis

ℓ

2

,

ℓ

2

_{− 1}

,

ℓ(ℓ

− 1)

or

ℓ(ℓ

− 1) − 1

forsome

ℓ

∈ N\{1}

,

thenthestandardpolyominoesaretheonlyminimizersof

P (c)

.



2.4 Relationbetween

T

and thenumberofmissingbondsin

2

tiled lusters

Inthisse tionwe onsider

2

tiled lustersandlinkthenumberofparti lesoftype

1

andtype

2

tothe numberofa tivebondsand thegeometri quantity

T

onsideredin Se tion2.3.

Lemma2.3 Forany

2

tiled luster

c

(i.e.,

c

∈ V

4n

2

⋆,n

2

forsome

n

2

),

4n

1

(c) = B(c)+

T (c)

and

4n

2

(c) =

B(c)

.

Proof. The laim of the lemma is equivalent to the armation that

T (c) = M(c)

with

M (c)

the number of missing bonds in

c

. Indeed, informally, for everyunit perimeter two bonds are lost with respe t tothefourbondsthat would bein identto ea h parti leoftype

1

ifitweresaturated,while onebondislost atea h onvex ornerandonebondis gainedatea h on ave orner.

Formally,let

p

beaparti leoftype

1

,

B(p)

thenumberofbondsin identto

p

,and

M (p) = 4

−B(p)

thenumberofmissingbondsof

p

. Considerthesetofparti lesoftype

1

attheboundaryofa

2

tiled luster,i.e.,theset ofnon-saturatedparti lesoftype

1

. Ea hoftheseparti lesbelongstooneoffour lasses(seeFig.9):

lass

1

:

p

hastwoneighboringparti lesoftype

2

belongingto thesame

12

bar.

lass

2

:

p

hastwoneighboringparti lesoftype

2

belongingto dierent

12

bars.

lass

3

:

p

hasthreeneighboring parti lesoftype

2

.

lass

4

:

p

hasoneneighboringparti leoftype

2

.

(a) (b) ( ) (d)

Figure 9: The ir led boundaryparti leof type

1

belongsto: (a) lass

1

; (b) lass

2

; ( ) lass

3

;(d) lass

4

.

Let

M

k

(c)

bethenumberofmissingbondsofparti lesof lass

k

in luster

c

,and

A

k

(c)

thenumber ofedgesin identtoparti lesof lass

k

in luster

c

. Then

M

1

(c) = 2, A

1

(c) = 2;

M

2

(c) = 2, A

2

(c) = 4;

M

3

(c) = 1, A

3

(c) = 2;

M

4

(c) = 3, A

4

(c) = 2.

(2.10)

Let

N

k

(c)

be thenumberof parti les of lass

k

of type

1

in luster

c

. Observing that a luster has two on ave ornersperparti leof lass

2

,one on ave orner perparti le of lass

3

and one onvex ornerperparti leof lass

4

,we anwrite

2P (c) =

4

X

k=1

A

k

(c)N

k

(c) = 2N

1

(c) + 4N

2

(c) + 2N

3

(c) + 2N

4

(c)

(2.12)

(thesum ountsea hedgeofthe

2

tiletwi e). Thetotalnumberofmissingbonds,ontheotherhand, is

M (c) =

4

X

k=1

M

k

(c)N

k

(c) = 2N

1

(c) + 2N

2

(c) + N

3

(c) + 3N

4

(c).

(2.13) Combining(2.112.13),wearriveat

T (c) = M(c)

.



3 Proof of Theorem 1.3: identi ation of

X

stab

Re all that

Λ

−

(the part of

Λ

where parti les intera t) is an

(L +

1

2

)

× (L +

1

2

)

dual square with

L > 2ℓ

⋆

. Let

η

stab

, η

′

stab

bethe ongurations onsisting ofa

2

tileddualsquare ofsize

L

witheven parity,respe tively,oddparity. Thesetwo ongurationshavethesameenergy. Theorem1.3saysthat

X

stab

=

{η

stab

, η

′

stab

} = ⊞

. Se tion3.1 ontainstwolemmasabout

2

tiled ongurationswithminimal energy. Se tion3.2usesthesetwolemmastoproveTheorem1.3.

3.1 Standard ongurations are minimizersamong

2

tiled ongurations

Lemma3.1 Within

V

4n

2

⋆,n

2

,thestandard ongurationsa hieve the minimalenergy.

Proof. Re allfromitem2inSe tion2.2that

H(η) = ∆

1

n

1

(η) + ∆

2

n

2

(η)

− UB(η).

(3.1)

In

V

4n

2

⋆,n

2

both

n

2

and

B = 4n

2

are xed, and hen e

min

η∈V

_⋆,n2

4

n2

H(η)

is attained at a onguration minimizing

n

1

. ByLemma 2.3,if

η

∈ V

4n

2

⋆,n

2

,then

n

1

(η) =

1

₄

[B(η) +

T (η)],

n

2

(η) =

1

₄

B(η).

(3.2) Hen e,tominimize

n

1

(η)

wemustminimize

T (η)

. The laimthereforefollowsfromLemma2.2(i).



Forastandard ongurationthe omputationoftheenergyisstraightforward. For

ℓ

∈ N

,

ζ

∈ {0, 1}

and

k

∈ N

0

with

k

≤ ℓ + ζ

, let

η

ℓ,ζ,k

denote the standard onguration onsisting of an

ℓ

× (ℓ + ζ)

(quasi-)squarewithabaroflength

k

atta hedto oneofitslongestsides(seeFig.10).

Figure10: Astandard ongurationwith

ℓ = 7, ζ = 1

and

k = 5

.

Lemma3.2 Theenergyof

η

ℓ,ζ,k

is(re all (1.14))

Proof. Notethat

P (η

ℓ,ζ,k

_{) = 2[ℓ + (ℓ + ζ) + 1}

{k>0}

]

and

Q(η

ℓ,ζ,k

_{) = 0}

,sothat

T (η

ℓ,ζ,k

_{) = 4[ℓ + (ℓ + ζ) + 1 + 1}

{k>0}

].

(3.4) Alsonotethat

B(η

ℓ,ζ,k

_{) = 4[ℓ + (ℓ + ζ) + k],}

(3.5)

be auseallparti lesoftype

2

aresaturated. However,by(3.13.2),wehave

H(η

ℓ,ζ,k

_{) =}

−

1

4

εB(η

ℓ,ζ,k

_{) +}

1

4

T (η

ℓ,ζ,k

_)∆

1

,

(3.6)

andsothe laimfollowsby ombining(3.43.6).



Notethattheenergyin reasesby

∆

1

− ε

(whi h is

> 0

ifandonlyif

ℓ

⋆

_{≥ 2}

by(1.13))whenabar

oflength

k = 1

isadded, andde reasesby

ε

ea htimethebarisextended. Notefurtherthat

H(η

ℓ,1,0

)

_{− H(η}

ℓ,0,0

) = ∆

1

− ℓε,

H(η

ℓ+1,0,0

)

− H(η

ℓ,1,0

) = ∆

1

− (ℓ + 1)ε,

(3.7) whi hshowthattheenergyofagrowingsequen eofstandard ongurationsgoesupwhen

ℓ < ℓ

⋆

and

goesdownwhen

ℓ

≥ ℓ

⋆

. The highestenergy is attained at

η

ℓ

⋆

_−1,1,1

, whi h is the riti al dropletin

Fig.4.

Itisworthnotingthat

H(η

2ℓ

⋆

_,0,0

s

) < 0

,i.e.,theenergyofadualsquareof sidelength

2ℓ

⋆

islower

thantheenergyof



. Thisiswhyweassumed

L > 2ℓ

⋆

,to allowfor

H(⊞) < H()

.

3.2 Stable ongurations

Inthisse tionweuseLemmas3.13.2toproveTheorem1.3.

Proof. Let

η

denote any ongurationin

X

stab

. Belowwewillshowthat: (A)

η

doesnot ontainanyparti lein

∂

−

_Λ

.

(B)

η

is a

2

tiled onguration,i.e.,

η

∈ V

4n

2

⋆,n

2

forsome

n

2

(

= n

2

(η)

).

On e we have (A) and (B), we observe that

η

annot ontain a number of

2

tiles larger than

L

2

.

Indeed, onsiderthetilesupportof

η

. Sin e

Λ

−

isan

(L +

1

2

)

× (L +

1

2

)

dualsquare,ifthetilesupport of

η

tsinside

Λ

−

,thensodoesthedual ir ums ribingre tangleof

η

. Butanyre tangleofarea

≥ L

2

has atleast oneside oflength

L + 1

. Hen e

n

2

(η)

≤ L

2

, and thereforethe numberof

2

tiles in

η

is at most

L

2

. ByLemmas 3.13.2, the global minimum of the energy is attained at the largestdual

quasi-square that tsinside

Λ

−

, sin e

L > 2ℓ

⋆

. Wetherefore on ludethat

η

∈ {η

stab

, η

′

stab

}

, whi h

provesthe laim.

Proofof(A) . Sin ein

∂

−

_Λ

parti lesdonotfeelanyintera tionbuthaveapositiveenergy ost,removal

ofaparti lefrom

∂

−

_Λ

alwayslowerstheenergy.

Proofof(B). Wenotethefollowingthreefa ts:

(1)

η

doesnot ontainisolatedparti lesoftype

1

.

(2)

∂

−

_Λ

−

doesnot ontainanyparti leoftype

2

.

(3) Allparti lesoftype

2

in

η

havealltheirneighboringsiteso upiedbyaparti le.

For(1), simplynote that the ongurationobtainedfrom

η

by removingisolated parti leshaslower energy. For (2), note that parti les in

∂

−

_Λ

−

haveat most twoa tivebonds. Therefore, if

η

would haveaparti leoftype

2

in

∂

−

_Λ

−

, thentheremovalof thatparti le would lowertheenergy,be ause

hasanemptyneighboringsite,thentheadditionofaparti leoftype

1

atthissite lowerstheenergy, be ause

∆

1

< U

(re all(1.12)).

We an now omplete the proof of (B) as follows. The onstraint

∆

2

− ∆

1

> 2U

implies that anyparti le of type

2

in

η

must haveat least three neighboring sites o upied by aparti le of type

1

. Indeed, the removal of a parti le of type

2

with at most two a tive bonds lowers the energy. But thefourth neighboring sitemust also be o upiedbya parti leof type

1

. Indeed,suppose that this site would be o upied by a parti le of type

2

. Then this parti le would have at most three a tivebonds. Considerthe onguration

η

˜

obtainedfrom

η

afterrepla ing thisparti le byaparti le of type

1

. Then

B(˜

η)

− B(η) ≥ −2

,

n

1

(˜

η)

− n

1

(η) = 1

and

n

2

(˜

η)

− n

2

(η) =

−1

. Consequently,

H(˜

η)

− H(η) ≤ ∆

1

− ∆

2

+ 2U < 0

. Hen e,anyparti leoftype

2

in

η

mustbesaturated.



4 Proof of Theorem 1.5: identi ation of

Γ

⋆

_{= Φ(, ⊞)}

InSe tion4.1weproveTheorem1.5subje ttothefollowinglemma.

Lemma4.1 Forany

n

2

≤ L

2

,the ongurations ofminimal energy with

n

2

parti les oftype

2

belong to

V

4n

2

⋆,n

2

,i.e.,are

2

tiled ongurations.

Theproofofthislemmaisgivenin Se tion4.2.

4.1 Proof of Theorem 1.5 subje t to Lemma 4.1

Proof. For

Y ⊂ X

, dene theexternal boundary of

Y

by

∂

Y = {η ∈ X \Y : ∃η

′

∈ Y, η ↔ η

′

}

and thebottom of

Y

by

F(Y) = arg min

η∈Y

H(η)

. A ordingtoManzo, Nardi,Olivieriand S oppola[5℄, Se tion 4.2,

Φ(, ⊞) = min

η∈∂B

H(η)

for

B ⊂ X

any(!) setwiththefollowingproperties:

(I)

B

is onne tedviaallowedmoves,



∈ B

and

⊞ /

∈ B

.

(II) Thereisapath

ω

⋆

_{: }

_{→ ⊞}

su hthat

{arg max

η∈ω

⋆

H(η)

} ∩ F(∂B) 6= ∅

. Thus,ourtaskistondsu ha

B

and omputethelowestenergyof

∂

B

.

For(I), hoose

B

tobethesetofall ongurations

η

su hthat

n

2

(η)

≤ ℓ

⋆

_(ℓ

⋆

_{− 1) + 1}

. Clearlythis

set is onne ted, ontains



anddoesnot ontain

⊞

. For(II), hoose

ω

⋆

asfollows. Aparti le oftype

2

isbroughtinside

Λ

(

∆H = ∆

2

),movedto the originandissaturatedbyfourtimesbringingaparti leof type

1

(

∆H = ∆

1

)and atta hingitto the parti le of type

2

(

∆H =

−U

). After this rst

2

tile hasbeen ompleted,

ω

⋆

followsa sequen eof

in reasing

2

tileddual quasi-squares. Thepassagefrom onequasisquareto thenextis obtainedby adding a

12

barto oneofthe longestsides, asfollows. First aparti leof type

2

isbroughtinside

Λ

(

∆H = ∆

2

)andisatta hedtooneofthelongestsidesofthequasi-square(

∆H =

−2U

). Next,twi e aparti le oftype

1

is broughtinside thebox(

∆H = ∆

1

)and isatta hed tothe (not yet saturated) parti le of type

2

(

∆H =

−U

) in order to omplete a

2

tiled protuberan e. Finally, the

12

bar is ompleted by bringing a parti le of type

2

inside

Λ

(

∆H = ∆

2

), moving it to a on ave orner (

∆H =

−3U

), andsaturatingitwith aparti leof type

1

(

∆H = ∆

1

,respe tively,

∆H =

−U

). Itis obviousthat

ω

⋆

eventuallyhits

⊞

. Thepath

ω

⋆

isreferredtoasthereferen epathfor the nu leation.

Call

η

⋆

the ongurationin

ω

⋆

onsistingof an

ℓ

⋆

_{× (ℓ}

⋆

_{− 1)}

quasi-square,a

2

tiledprotuberan e atta hed tooneofits longestsides,and afree parti leoftype

2

(seeFig.11; there aremany hoi es for

ω

⋆

dependingonwherethe

2

tiledprotuberan esareadded;allthese hoi esareequivalent. Note that,inthenotationofLemma3.2,

η

⋆

_{= η}

ℓ

⋆

_−1,1,1

+ fp[2]

,where

+fp[2]

denotestheadditionofafree parti leoftype

2

. Observethat:

(a)

ω

⋆

exits

B

viathe onguration

η

⋆

;

(b)

η

⋆

_{∈ F(∂B)}

( )

η

⋆

∈ {arg max

η∈ω

⋆

H(η)

}

.

Observation(a)isobvious,while(b)followsfromLemmas 3.1and4.1. Tosee( ),note thefollowing:

(1)Thetotalenergydieren eobtainedbyaddinga

12

baroflength

ℓ

onthesideofa

2

tiled luster is

∆H(

addinga

12

bar

) = ∆

1

− εℓ

,whi h hangessignat

ℓ = ℓ

⋆

(re all(3.7));(2)The ongurations

of maximal energy in a sequen e of growing quasi-squares are those where a free parti le of type

2

enters the box after the

2

tiled protuberan e has been ompleted. Thus, within energy barrier

2∆

1

+ 2∆

2

− 4U = 4U − ε

the

12

bar is ompleted downwards in energy. This means that, after onguration

η

⋆

ishit, thedynami s anrea hthe

2

tileddual squareof

ℓ

⋆

_{× ℓ}

⋆

whilestayingbelow

theenergy level

H(η

⋆

₎

. Sin eall

2

tileddual quasi-squareslargerthan

ℓ

⋆

_{× (ℓ}

⋆

_{− 1)}

havean energy

smallerthanthatofthe

2

tileddualquasi-square

ℓ

⋆

_{× (ℓ}

⋆

_{− 1)}

itself,thepath

ω

⋆

doesnotagainrea h

theenergylevel

H(η

⋆

).

Be auseof(a ),wehave

Φ(, ⊞) = H(η

⋆

₎

. To ompletetheproof,useLemma3.2to ompute

H(η

⋆

_{) = H(η}

ℓ

⋆

_−1,1,1

+ fp[2]) =

_−ε[ℓ

⋆

_(ℓ

⋆

− 1) + 1] + ∆

1

(2ℓ

⋆

+ 1) + ∆

2

.

(4.1)



Figure11: A riti al onguration

η

⋆

. Thisisthedual versionofthe riti aldropletin Fig.4.

4.2 Proof of Lemma 4.1

TheproofofLemma4.1is arriedoutin twosteps. InSe tion4.2.1weshowthatthe laimholdsfor

single- luster ongurationswithaxednumberofparti lesoftype

2

. InSe tion4.2.2weextendthe laimtogeneral ongurationswithaxednumberofparti lesoftype

2

.

4.2.1 Single lustersof minimalenergyare

2

tiled lusters

Lemma4.2 Foranysingle- luster onguration

η

∈ V

⋆,n

2

\V

4n

2

⋆,n

2

thereexistsa onguration

η

˜

∈ V

4n

2

⋆,n

2

su hthat

H(˜

η) < H(η)

.

Proof. Pi kany

η

∈ V

⋆,n

2

\V

4n

2

⋆,n

2

. Everyneighboringsiteofaparti leoftype

2

inthe lusteriseither emptyoro upiedby aparti leoftype

1

, andthere is atleast one non-saturatedparti leof type

2

. Sin e

η

onsistsofasingle luster,

η

˜

anbe onstru tedinthefollowingway:

• ˜η(i) = η(i)

forall

i

∈

supp

(η)

.

H(η) = ∆

1

n

1

(η) + ∆

2

n

2

(η)

− UB(η),

H(˜

η) = ∆

1

n

1

(˜

η) + ∆

2

n

2

(˜

η)

− UB(˜η),

(4.2)

and

n

2

(η) = n

2

(˜

η)

,wehave

H(˜

η)

_{− H(η) = ∆}

1

[n

1

(˜

η)

− n

1

(η)]

− U[B(˜η) − B(η)].

(4.3) By onstru tion,

B(˜

η)

− B(η) ≥ n

1

(˜

η)

− n

1

(η) > 0

. Sin e

0 < ∆

1

< U

(re all(1.12)), itfollowsfrom

(4.3)that

H(˜

η) < H(η)

.



4.2.2 Congurations ofminimalenergywith xed numberof parti lesof type

2

Lemma4.3 For any

n

2

and any onguration

η

∈ V

⋆,n

2

onsisting of at least two lusters, any onguration

η

⋆

su hthat

η

⋆

isasingle luster,

η

⋆

_{∈ V}

4n

2

⋆,n

2

and

η

⋆

isastandard onguration satises

H(η

⋆

_{) < H(η)}

.

Proof. Let

η

∈ V

⋆,n

2

bea onguration onsisting of

k > 1

lusters, labeled

c

1

, . . . , c

k

. Let

η

n

2

(c

i

)

denoteanystandard ongurationwith

n

2

(c

i

)

parti lesoftype

2

. ByLemmas 3.1and4.2,wehave

H(η) =

k

X

i=1

H(c

i

)

≥

k

X

i=1

H(η

n

2

(c

i

)

_).

(4.4)

ByLemma2.3,wehave(re all(1.14))

k

X

i=1

H(η

n

2

(c

i

)

_{) =}

k

X

i=1

∆

1

n

1

(η

n

2

(c

i

)

) + ∆

2

n

2

(η

n

2

(c

i

)

)

− UB(η

n

2

(c

i

)

)



=

k

X

i=1

∆

1

n

2

(η

n

2

(c

i

)

) +

1

₄

T (η

n

2

(c

i

)

) + ∆

2

n

2

(η

n

2

(c

i

)

)

− U4n

2

(η

n

2

(c

i

)

)



=

k

X

i=1

 − εn

2

(η

n

2

(c

i

)

) +

₄

1

∆

1

T (η

n

2

(c

i

)

).

(4.5)

ButfromLemma2.2 itfollowsthat

k

X

i=1

T (η

n

2

(c

i

)

_{) >}

_{T η}

P

k

i=1

n

2

(c

i

)

,

(4.6) where

η

P

k

i=1

n

2

(c

i

)

denotes any standard onguration with

P

k

i=1

n

2

(c

i

) = n

2

(η)

parti les of type

2

. Combining(4.44.6),wearriveat

H(η) >

_−εn

2

(η) +

1

₄

∆

1

T (η

n

2

(η)

) = H(η

n

2

(η)

).

(4.7)



5 Proof of Theorem 1.4: upper bound on

V

η

for

η /

∈ {, ⊞}

Inthis se tionwe showthat forany onguration

η /

∈ {, ⊞}

it ispossible tondapath

ω : η

→ η

′

with

η

′

∈ {, ⊞}

su h that

max

ξ∈ω

H(ξ)

≤ H(η) + V

⋆

with

V

⋆

≤ 10U − ∆

1

and

η

′

∈ I

η

. By Denition 1.1( e),thisimpliesthat

V

η

≤ V

⋆

forall

η /

∈ {, ⊞}

andthereforesettlesTheorem1.4. Se tion5.3des ribesanenergyredu tionalgorithm tond

ω

. Roughly,theideaisthatif

η

ontains onlysub riti al lusters,thenthese lusters anberemovedonebyonetorea h



,whileif

η

ontains

⊞

that goes via a sequen e of in reasing re tangles. In parti ular, the super riti al luster is rst extendedtoa

2

tiledre tangletou hingthenorth-boundaryof

Λ

,afterthatitisextendedtoa

2

tiled re tangletou hingthewest-boundaryandtheeast-boundaryof

Λ

,andnally itisextended to

⊞

.

To arry out this task, six energy redu tion me hanisms are needed, whi h are introdu ed and

explainedinSe tion 5.2:

•

Movingunit holesinside

2

tiled lusters (Se tion5.2.1).

•

Addingandremoving

12

barsfromlatti e- onne tingre tangles(Se tion5.2.2).

•

Changingbridgesinto

12

bars(Se tion5.2.3).

•

Maximallyexpanding

2

tiledre tangles(Se tion 5.2.4).

•

Mergingadja ent

2

tiledre tangles(Se tion5.2.5).

•

Removingsub riti al lusters(Se tion 5.2.6).

Ea h of Se tions 5.2.15.2.6states adenition and a lemma, and uses these to provea proposition

abouttherelevantenergyredu tionme hanism. Thesixpropositionsthusobtainedwillbe ru ialfor

theenergyredu tionalgorithmin Se tion5.3.

InSe tion 5.1webeginbydeningbeamsandpillars,whi hareneededthroughoutSe tion5.2.

5.1 Beams and pillars

Lemma5.1 Let

η

bea onguration ontaining atile

t

that has atleast threejun tion siteso upied by a parti le of type

1

. Then the onguration

η

′

obtained from

η

by turning

t

into a

2

tile satises

H(η

′

₎

≤ H(η)

.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Figure12: Possibletileswithatleastthreejun tion siteso upiedbyaparti leoftype

1

.

Proof. W.l.o.g. we may assume that

η(t

a

) = η(t

b

) = η(t

d

) = 1

, and that

η

′

is the onguration in Fig.6(d),i.e.,

η

′

_(t

a

) = η

′

(t

b

) = η

′

(t

c

) = η

′

(t

d

) = 1

,

η

′

_(t

e

) = 2

. Thefollowingeight asesarepossible (seeFig.12andre all(1.12) ):

(i)

(η(t

c

), η(t

e

)) = (0, 0)

. Oneparti leof type

1

and oneparti le of type

2

areadded, and at least four newbondsarea tivated:

∆H

≤ ∆

1

+ ∆

2

− 4U < 0

.

(ii)

(η(t

c

), η(t

e

)) = (0, 2)

. Oneparti le of type

1

is added, and one newbond is a tivated:

∆H =

∆

1

− U < 0

.

(iii)

(η(t

c

), η(t

e

)) = (2, 0)

. Oneparti leof type

2

ismovedto anothersite withoutdea tivating any bonds,after whi h ase(ii)applies.

(iv)

(η(t

c

), η(t

e

)) = (2, 2)

. Oneparti leof type

2

with atmostthreea tivebondsis repla edbyone parti leoftype

1

with atleastonea tivebond:

∆H

≤ ∆

1

− ∆

2

+ 2U < 0

.

(v)

(η(t

c

), η(t

e

)) = (1, 0)

. Oneparti leoftype

2

isadded,and fournewbondsarea tivated:

∆H =

(vi)

(η(t

c

), η(t

e

)) = (0, 1)

. One parti le of type

1

is moved to another site without dea tivating any a tive bond, one parti le of type

2

is added, and at least four new bonds are a tivated:

∆H

_{≤ ∆}

2

− 4U < 0

.

(vii)

(η(t

c

), η(t

e

)) = (2, 1)

. Twoparti lesareex hangedwithoutdea tivatinganybonds:

∆H

≤ 0

. (viii)

(η(t

c

), η(t

e

)) = (1, 1)

. One parti le oftype

1

is repla edby aparti le of type

2

, and four new

bondsarea tivated:

∆H = ∆

2

− ∆

1

− 4U < 0

.



Denition5.2 A beam of length

ℓ

isa row (or olumn) of

ℓ + 1

parti les of type

1

atdual distan e

1

of ea h other. A pillar is a parti le of type

1

at dual distan e

1

of the beam not lo ated at one of the twoendsof the beam. The parti lein thebeamsitting nexttothe pillardivides the beamintotwo

se tions. The lengthsof thesetwose tionsare

≥ 0

andsum upto

ℓ

. Thesupportofapillared beamis the unionof allthe tile supports. Thesupport onsists ofthree rows(or olumns)of sitesanupper,

middleandlower row(or olumn) whi h arereferredtoasroof, enter andbasement (seeFig.13).

Figure 13: A south-pillared horizontal beam of length

10

with a west-se tion of length 4 and an east-se tionoflength6.

Notethatabeam anhavemorethanonepillar. Lemma5.1 impliesthefollowing.

Corollary 5.3 Let

η

bea onguration ontaining apillared beam

˜b

su hthat supp

(˜b)

isnot

2

tiled . Then the onguration

η

′

obtainedfrom

η

by

2

tilingsupp

(˜b)

satises

H(η

′

₎

≤ H(η)

.

5.2 Six energy redu tion me hanisms

5.2.1 Moving unitholesinside

2

tiled lusters

Inthisse tionweshowhowaunithole anmoveinsidea

2

tiled luster. Inparti ular,weshowthat su hmotionis possiblewithinanenergybarrier

6U

by hangingthe ongurationonlylo ally.

Denition5.4 Asetof sites

S

inside

Λ

obtainedfroma

4

× 4

squareafter removingthe four orner sitesis alledaslot.

Givenaslot

S

,weassignalabeltoea hofthe

12

sitesin

S

asinFig.14(a): rst lo kwiseinthe enter of

S

andthen lo kwiseonthe boundaryof

S

. We allthepairs

(S

1

, S

3

)

and

(S

2

, S

4

)

slot- onjugate sites.

Lemma5.5 Let

S

beaslot,andlet

η

0

beany ongurationsu hthatallparti lesin

S

havethe same parity. W.l.o.g. this parity may be taken tobe even, sothat

η(S

1

) = 0

and

η(S

3

) = 2

. Let

η

1

be the onguration obtained from

η

by inter hanging the states of

S

1

and

S

3

. Then

H(η

0

) = H(η

1

)

, and thereexistsapath

ω : η

0

→ η

1

thatnever ex eedsthe energylevel

H(η

0

) + 6U

.

Proof. W.l.o.g.wetake

η

0

asinFig.14(b ). Let

a

→ b

denotethemotionof aparti lefrom site

a

to site

b

. Forthepath

ω

we hoose thefollowingsequen e of moves:

S

4

→ S

1

;

S

3

→ S

4

;

S

2

→ S

3

;

S

1

→ S

2

;

S

4

→ S

1

;

S

3

→ S

4

. Therstthreemovesandthese ondthreemovesea harearotationby

π

2

ofthesub ongurationat the sites

S

1

, S

2

, S

3

, S

4

. Note that all ongurationsin

ω

havethesame numberof parti les ofea h typeand hen ethe hanges in energyonly depend on the hange in the