Gibbs measures - The die de

We are now interested in haraterizing probability measures on the set of perfet mathings

M ( G )

^of^the^innite^graph

G

whihare,insomeappropriatesense,innitevolumeversionsofthe Boltzmann measureon

M ( G _n )

^. ^Reall^that ^by^denition, ^theprobabilityof amathinghosen with respet to the Boltzmann measure on

M ( G _n )

^is proportional to the produt of its edge weights. Thisdenitiondoesnot workwhenthegraph isinnite, andis replaedbythenotion

of Gibbs measure, whih is a probability measure on

M ( G )

^satisfying ^the ^DLR ¹ ^onditions:

if the perfet mathing inan annular region of

G

is xed, mathings inside and outside of the annulus are independent, and the probability of any interior mathing is proportional to the

produtof itsedge-weights.

3.3.1 Limit of Boltzmann measures

A natural way ofonstruting aGibbs measureis to take thelimit ofthe Boltzmann measures

on ylinder sets of

M ( G _n )

^, ^where â ^ylinder ^set ônsists ôf âll ^perfet ^mathings ôntaining â

xedsubset of edgesof

G _n

Theorem 2 gives an expliit expression for the Boltzmann measure on ylinder sets when the

graph is planar and nite. In the ase of toroidal graphs, a similar but more ompliated

expressionholds: itisaombinationoffourstermssimilartothose ofEquation(2.2) ,involving

thematries

K _n ⁰⁰ , . . . , K _n ¹¹

^,^and^their ^inverses.

Usingtheblokdiagonalizationofthematries

K _n ^στ

ôf^the^proofôf^Theorem^9,ôneânômpute

the elements of the inverse expliitly and obtain Riemann sums. The onvergene of these

Riemann sums is againompliated bythe zerosof

P (z, w)

^on ^the^torus

T ²

,but an be shown toonvergeonasubsequeneof

n

^'s^to^the^right^hand^sideôfÊquation^{(3.7) .} Ûsingâ^Theoremôf

Sheeld [She05 ℄ whih shows a prioriexistene of thelimit,one dedues onvergene for every

n

^. ^Then, ^by Kolmogorov's extension theorem, there exists a unique probability measure on

( M ( G ), σ( A ))

^whihôinides^with^the^limitôf^the^Boltzmann^measuresôn^ylinder ^sets,^where

σ( A )

îs^the^smallest^sigma-eldôntaining^ylinder^sets. ^This^limiting^measureîsôf^Gibbs^type

by onstrution. Wehave thus skethed theproof ofthefollowing theorem.

DLRstandsforDobrushin,LanfordandRuelle

Theorem12. [CKP01 ,KOS06 ℄Let

{ e ₁ = w ₁ b ₁ , . . . , e _k = w _k b _k }

^beâ^subset ôfêdgesôf

G

. Then there existsa unique probability measure

µ

^on

( M ( G ), σ( A ))

^suh ^that:

µ( e ₁ , . . . , e _k ) = Y k i=1

K( w _i , b _i )

!

det(K ⁻ ¹ ( b _i , w _j ) ₁ _≤ _i,j _≤ _k ),

^(3.7)

where

K

îs â ^Kasteleyn^matrix âssoiated ^to^the ^graph

G

,and assuming

b

and

w

are in a single fundamental domain:

K ⁻ ¹ ( b , w + (x, y)) = 1 (2πi) ²

Z

T ²

Q bw (z, w)

P (z, w) z ^x w ^y dw w

dz z ,

and

Q bw (z, w)

^is^the

( b , w )

êlement ôf ^the âdjugate^matrix ^(transpose ôf ^the ôfator^matrix) ôf

K ₁ (z, w)

^. Îtîs â ^polynomial ⁱⁿ

z, w, z ⁻ ¹ , w ⁻ ¹

3.3.2 Ergodi Gibbs measures

Inthe previous setion,wehave expliitlydetermined aGibbsmeasure on

M ( G )

^. ^W^e ^now ^aim

at haraterizing all of them. In order to this in a way whih is oherent with the model, we

introdue thefollowing notions.

A probability measure on

M ( G )

^is translation-invariant, if the measure of a subset of

M ( G )

is invariant under the translation-isomorphism ation. An ergodi Gibbs measure (EGM) is a

Gibbs measure whih is translation invariant and ergodi, i.e. translation invariant sets have

measure

0

^or

1

ForanergodiGibbsmeasure

µ

^,^dene ^the^slope

(s, t)

^to^be^the^expeted^horizontal^and^vertial

height hange in the

(1, 0)

^and

(0, 1)

^diretions, ^that ^is

s = E _µ [h( v + (1, 0)) − h( v )]

^, ^and

t = E _µ [h( v + (0, 1)) − h( v )]

Let us denote by

µ n

^the ^Boltzmann ^measure ^on

M ( G _n )

^. ^F^or ^a ^xed

(s, t) ∈ R ²

, let

M s,t ( G _n )

be thesetof mathings of

G _n

whih have height hange

( ⌊ sn ⌋ , ⌊ tn ⌋ )

^. ^Assuming ^that

M ^s,t ( G _n )

is non-emptyfor

n

^suiently ^large, ^let

µ _n (s, t)

^denote ^the^onditional ^measure ^indued ^by

µ _n

M s,t ( G _n )

^. ^Then, ^a haraterization of all ergodi Gibbsmeasures on

M ( G )

^is^given ^by^the

following theoremofSheeld.

Theorem 13. [She05 ℄For eah

(s, t)

^for ^whih

M s,t ( G _n )

^is ^non-empty^for

n

^suiently^large,

µ _n (s, t)

^onverges ^as

n → ∞

^to ^an ^EGM

µ(s, t)

^of ^slope

(s, t)

^. Furthermore

µ _n

^itself^onverges

µ(s ₀ , t ₀ )

^where

(s ₀ , t ₀ )

îs ^the ^limit ôf ^the ^slopes ôf

µ n

^. ^Finally, ^if

(s ₀ , t ₀ )

^lies ⁱⁿ ^the ^interior

of the set of

(s, t)

^for ^whih

M ^s,t ( G _n )

^is ^non-empty ^for

n

^suiently ^large, ^then ^every ^EGM

of slope

(s, t)

^is ^of ^the ^form

µ(s, t)

^for ^some

(s, t)

âs âbôve; ^that îs

µ(s, t)

îs ^the ûnique êrgodi

Gibbs measure of slope

(s, t)

Proof. TheexisteneisestablishedbytakinglimitsofBoltzmannmeasuresonlarger andlarger

tori while restriting height hange. The uniqueness is muh harder, and we won't disuss it

here.

3.3.3 Newton polygon and available slopes

Theorem13raisesthefollowingquestion: whatisthesetofpossibleslopesforGibbsmeasuresor

equivalently for limitsofonditional Boltzmannmeasures ? Theanswerisgiven bytheNewton

polygon

N (P )

^dened ^as ^follows:

N (P )

^is ^the ^losed ^onvex ^hull ⁱⁿ

R ²

of the set of integer

exponents ofthe monomials of the harateristi polynomial

P (z, w)

^,^up ^to ontribution ofthe refereneow

ω ^M ⁰

^,^that^is:

N (P ) =

^onvex^hull

{ (i, j) ∈ Z ² | z ^i+x ⁰ w ^j+y ⁰

^is ^a^monomial ⁱⁿ

P (z, w) } .

Proposition 14. [KOS06 ℄ The Newton polygon is the set of possible slopes of EGMs, that is

there existsan EGM

µ(s, t)

îf ând ônlyîf

(s, t) ∈ N (P )

Proof. Observing that hanging the referene ow merely translates the Newton polygon, we

assumethat

(x ₀ , y ₀ ) = (0, 0)

Letus rst prove thatif

(s, t) ∈ N (P )

^,^then^there îsâ ^Gibbs^measure ôf^slope

(s, t)

^,^or

equiva-lently

M s,t ( G _n )

^is ^non-empty^for

n

^large ênough. ^Fôr ônveniene, ^we ^will âssume^that ^the^set

of possible slopesis losed.

ByLemma8,theabsolutevalueoftheoeient

z ⁱ w ^j

ⁱⁿ

P (z, w)

^is^the^weighted^sum^of

math-ings of

G ₁

with height hange

(i, j)

^, ^thus ^there ^is ^a ^mathing orresponding to eah extremal pointof

N (P )

^,^i.e. ^if

(s, t)

îsânêxtremal^pointôf

N (P)

^,^then

M s,t ( G ₁ ) 6 = ∅

^. ^It^sues^to^show

thatif

M s 1 ,t 1 ( G _n ₁ )

^and

M s 2 ,t 2 ( G _n ₂ )

^are^non-empty^for ^some

n ₁

^and

n ₂

^,^then

M ^s 1+ s 2

2 , ^t ¹⁺ ₂ ^t ² ( G _m )

is also non-empty for some

m

^. Îndeed, ^by îndution, ^this âllows ^to ^prove êxistene ôf â ^Gibbs

measure of slope

(s, t)

^for â ^dense ^subset ôf ^the ^Newton^polygon. ^The ^proofîs ênded ^by ûsing

theassumption thatthe set ofpossibleslopesis losed.

Withoutlossof generality,weanassumethat

n ₁ = n ₂

^,^otherwise^take ^the^lm^of^their ^periods.

Considertwomathingsof

M s 1 ,t 1 ( G _n )

^and

M s 2 ,t 2 ( G _n )

^,respetively. Thesuperimpositionofthe two mathings being a set of disjoint alternating yles, one an hange from one mathingto

theotherbyrotatingalongtheyles. Iftheheight hanges

( ⌊ s ₁ n ⌋ , ⌊ t ₁ n ⌋ )

( ⌊ s ₂ n ⌋ , ⌊ t ₂ n ⌋ )

^of^the

two mathings areunequal, some ofthese yles have non-zero homology in

H 1 ( T ² , Z )

^,^so^that

rotating along them will hange the height hange. On the toroidal graph

G _2n

, onsider four opies ofthetwo mathings andshift halfofthenon-trivial yles;thisreates anewmathing

with height hange

( ⌊ (s 1 + s 2 )n ⌋ , ⌊ (t 1 + t 2 )n ⌋ ) = ⌊ ^s ¹ ^+s ₂ ² 2n ⌋ , ⌊ ^t ¹ ^+t ₂ ² 2n ⌋

, thus proving that

M ^s ¹⁺ ^s ²

2 , ^t ¹⁺ ₂ ^t ² ( G _m )

^is^non-empty ^for

m = 2n

Let us now suppose that there exists a Gibbs measure

µ(s, t)

^of ^slope

(s, t)

^and ^prove ^that

(s, t) ∈ N (P )

^. ^Denote^by

~ E ₁

thesetofdiretededgesofthefundamentaldomain

G ₁ = ( V ₁ , E ₁ )

Realling thatthe divergene

div

îs â^linear^funtionôf ôws,^the^setôf non-negative, white-to-blakunit ows denes apolytopeof

R ^~ ^E ¹

{ ω ∈ R ^~ ^E ¹ : ∀ wb ∈ E ₁ , ω( b , w ) = 0, 0 ≤ ω( w , b ) ≤ 1; ∀ w ∈ W ₁ , div ω( w ) = 1, ∀ b ∈ B ₁ , div ω( b ) = − 1 } .

The mapping

ψ

^whih âssigns ^to â ôw

ω

^the ^total ^ux^aross

γ x

^and

γ y

^is^a ^linear^mapping

fromthepolytopeto

R ²

,implyingthattheimage ofthepolytopeunder

ψ

îs^theônvex^hull ôf

theimagesof the extremalpointsof thepolytope.

Now, from Setion 3.1, we know that every dimer onguration of

G ₁

denes a non-negative, white-to-blakunitowtakingvaluesin

{ 0, 1 }

ônêvery^diretedêdgeôf

~ E ₁

. Theonversebeing also true, this implies that extremal points of thepolytope aregiven by dimer ongurations.

Sine therefereneow issuh that

(x ₀ , y ₀ ) = (0, 0)

^,^theîmage ôfâ ^dimerônguration ûnder

ψ

îs îts ^height ^hange. ^This ^means ^that^the îmage ôf êxtremal ^pointsôf ^the ^polytopeôntains

the extremal points of the Newton polygon

N (P )

^; ^the îmage ôf ^the ^polytope ûnder

ψ

^is ^thus

N (P )

TheGibbs measure

µ(s, t)

^of ^slope

(s, t)

^denes^a non-negative, white-to-blak ow

ω ^µ(s,t)

∀ e = wb ∈ E ₁ , ω ^µ(s,t) ( w , b ) = µ(s, t)( e ), ω ^µ(s,t) ( b , w ) = 0.

Sine

µ(s, t)

^is^aprobabilitymeasure,theow

ω ^µ(s,t)

^has^divergene¹âtêvery^white^vertexând

-1ateveryblakvertex. Itthusbelongstothepolytopeanditsimageunder

ψ

^belongs^to

N (P )

Theproof isonluded byobserving thatthe image of

µ(s, t)

^under

ψ

^is ^the^slope

(s, t)

Example 3.2. Figure3.7showstheNewtonpolygonofthedimermodelonthesquare-otagon

graph with weights 1 on the edges. Marked points represent monomials of the harateristi

polynomial

P (z, w) = 5 − z − ¹ z − w − w ¹

1 1

Figure 3.7: Newton polygon of the dimer model on the square-otagon graph with uniform

weights.

3.3.4 Surfae tension

For every

(s, t) ∈ N (P )

^,^let

Z s,t ( G _n )

^be^the ^partition ^funtion ^of

M s,t ( G _n )

^,^that^is:

Z s,t ( G _n ) = X

M ∈M ^s,t ( G _n )

ν(M ).

Then, by denition, thefree energy ofthemeasure

µ(s, t)

^is:

σ(s, t) = − lim

n →∞

1 n ² log Z s,t ( G _n ).

The funtion

σ : N (P ) → R

is knownas thesurfae tension. Sheeld [She05 ℄ provesthatit is stritly onvex.

Asa onsequene of this denition and of Theorem 13, one dedues thatthe measure

µ(s 0 , t 0 )

of Theorem13 isthe onewhih hasminimal freeenergy. Moreover, sine thesurfae tensionis

stritly onvex, the surfaetension minimizing slope isunique and equal to

(s ₀ , t ₀ )

3.3.5 Construting Gibbs measures

Theorem12ofSetion3.3.1provesanexpliitexpressionfortheGibbsmeasure

µ(s ₀ , t ₀ )

^of^slope

(s ₀ , t ₀ )

^. Ôur ^goal^now îs^to ôbtainânêxpliit êxpression^for ^the^Gibbs^measures

µ(s, t)

^with^all

possibleslopes

(s, t)

ReallthatbyTheorem 13,theGibbsmeasure

µ(s, t)

îs^the^limit ôf^theônditional^Boltzmann

measures

µ n (s, t)

^on

M s,t ( G _n )

^. ^The ^problem îs ^that ônditional ^measures âre ^hard ôbjets

to work with in order to obtain expliit expressions. But we know how to handle the full

Boltzmann measure, whih onverges to the Gibbs measure of slope

(s ₀ , t ₀ )

^. ^So ^the ^idea ^to

avoidhandling onditionalmeasures isto modifytheweightfuntionon theedgesof

G _n

insuh

a waythatmathings withanother slope than

(s ₀ , t ₀ )

^get ^favored. ^Hene, ^we ^are^looking ^for ^a

weight funtion whih satisesthe following: thenew weight of a mathingis equal to theold

weight multipliedbyaquantity whih dependsonly on its height hange. This anbe done by

introduing magneti eld oordinates asfollows.

Reallthat

γ _x,n

γ _y,n

âreôriented^horizontal ând ^vertial ^yles ⁱⁿ^the ^dual^graph

G ∗

n

^obtained

by taking

n

^times ^the ^basis^vetor

e x

^of ^the ^underlying ^lattie

Z ²

n

^times ^the ^basis^vetor

e y

respetively,embeddedonthetorus. Then,on

G ∗

n

^there^are

n

^horizontal^opies^of^the^yle

γ x,n

and

n

^vertial ^opies ^of ^the ^yle

γ _y,n

^. ^Let

(B _x , B _y )

^be ^two ^real ^numbers ^known ^as ^magneti

eld oordinates. Multiply all edges rossing the

n

^opies ^of ^the ^horizontal ^yle

γ x,n

^by

e ^± ^B ^x

depending on whether the white vertex is on the left or on the right. In a similar way, edges

rossing the

n

^opies ^of ^the^vertial ^yle

γ _y,n

^are ^multiplied^by

e ^± ^B ^y

^. ^This ^denes ^a^magneti

altered weight funtion,denoted by

ν _(B _x _,B _y ₎

^satisfying ^our requirement. Indeed, let

M 0

^be ^the

periodireferenemathingof

G _n

,anddenoteby

x ⁿ ₀

y ₀ ⁿ

^the^total^ux^of

ω ^M ⁰

^through

γ x,n

γ y,n

Then, arguing ina way similarto the proofof Lemma 8,one an express themagneti altered

weight funtion

ν _(B _x _,B _y ₎

^as^the^weight ^funtion

ν

^,^multiplied ^by^a ^quantity^whih ^only^depends

on theheight hange:

∀ M ∈ M ( G _n ), ν _(B _x _,B _y ₎ (M ) = ν(M)e ^nB ^x ^(h ^M ^x ^+x ⁿ ⁰ ⁾ e ^nB ^y ^(h ^M ^y ^+y ⁿ ⁰ ⁾ .

^(3.8)

Let

P _(B _x _,B _y ₎ (z, w)

^be^theharateristipolynomialofthegraph

G ₁

orrespondingtothemagneti altered weight funtion. The key fat is that

P _(B _x ,B y ) (z, w)

ân êasily ^be êxpressed ûsing ^the

harateristipolynomial

P (z, w)

^of^the^graph

G ₁

: expressing

P _(B _x _,B _y ₎ (z, w)

^using^Lemma⁸^and

replaing

ν _(B _x _,B _y ₎ (M )

^by^the ^right ^hand^side ^of ^(3.8)ⁱⁿ^the^ase ^where

n = 1

^,^yields:

P _(B _x _,B _y ₎ (z, w) = P(e ^B ^x z, e ^B ^y w).

Let

Z _(B _x _,B _y ₎ ( G _n )

^be^the ^partition^funtion^and

µ ^(B n ^x ^,B ^y ⁾

^be^the^Boltzmann^measure^of^the^graph

G _n

withthemagneti alteredweight funtion. Denote by

µ _(B _x _,B _y ₎

^the^Gibbs^measure^obtained

asweak limit of the Boltzmann measures

µ ^(B n ^x ^,B ^y ⁾

^. ^Then, âsâ ^diret ôrollaryôf ^Theorems ¹¹

and 12,we have:

So,itisquiteremarkablethattheresultsobtainedfortheweightfuntion

ν

^also^yield^the^results

for themagneti alteredweight funtion

ν _(B _x _,B _y ₎

Note that we have not yet related the magneti eld oordinates to the slope of the Gibbs

measure

µ _(B _x _,B _y ₎

^. ^This^is ^postponed^until ^Setion ^3.4.2.

Dans le document The die de (Page 31-36)

Gibbs measures

M ( G )

G

M ( G n )

M ( G n )

M ( G )

G

M ( G n )

G n

K n 00 , . . . , K n 11

K n στ

P (z, w)

T 2

n

n

( M ( G ), σ( A ))

σ( A )

{ e 1 = w 1 b 1 , . . . , e k = w k b k }

G

µ

( M ( G ), σ( A ))

µ( e 1 , . . . , e k ) = Y k i=1

K( w i , b i )

!

det(K − 1 ( b i , w j ) 1 ≤ i,j ≤ k ),

K

G

b

w

K − 1 ( b , w + (x, y)) = 1 (2πi) 2

Z

T 2

Q bw (z, w)

P (z, w) z x w y dw w

dz z ,

Q bw (z, w)

( b , w )

K 1 (z, w)

z, w, z − 1 , w − 1

M ( G )

M ( G )

M ( G )

0

1

µ

(s, t)

(1, 0)

(0, 1)

s = E µ [h( v + (1, 0)) − h( v )]

t = E µ [h( v + (0, 1)) − h( v )]

µ n

M ( G n )

(s, t) ∈ R 2

M s,t ( G n )

G n

( ⌊ sn ⌋ , ⌊ tn ⌋ )

M s,t ( G n )

n

µ n (s, t)

µ n

M s,t ( G n )

M ( G )

(s, t)

M s,t ( G n )

n

µ n (s, t)

n → ∞

µ(s, t)

(s, t)

µ n

µ(s 0 , t 0 )

(s 0 , t 0 )

µ n

(s 0 , t 0 )

(s, t)

M s,t ( G n )

n

(s, t)

µ(s, t)

(s, t)

M ( G _n )

M ( G _n )

M ( G _n )

G _n

K _n ⁰⁰ , . . . , K _n ¹¹

K _n ^στ

T ²

{ e ₁ = w ₁ b ₁ , . . . , e _k = w _k b _k }

µ( e ₁ , . . . , e _k ) = Y k i=1

K( w _i , b _i )

det(K ⁻ ¹ ( b _i , w _j ) ₁ _≤ _i,j _≤ _k ),

K ⁻ ¹ ( b , w + (x, y)) = 1 (2πi) ²

T ²

P (z, w) z ^x w ^y dw w

K ₁ (z, w)

z, w, z ⁻ ¹ , w ⁻ ¹

s = E _µ [h( v + (1, 0)) − h( v )]

t = E _µ [h( v + (0, 1)) − h( v )]

M ( G _n )

(s, t) ∈ R ²

M s,t ( G _n )

G _n

M ^s,t ( G _n )

µ _n (s, t)

µ _n

M s,t ( G _n )

M s,t ( G _n )

µ _n (s, t)

µ _n

µ(s ₀ , t ₀ )

(s ₀ , t ₀ )

(s ₀ , t ₀ )

M ^s,t ( G _n )

R ²

ω ^M ⁰

{ (i, j) ∈ Z ² | z ^i+x ⁰ w ^j+y ⁰

(x ₀ , y ₀ ) = (0, 0)

M s,t ( G _n )

z ⁱ w ^j

G ₁

M s,t ( G ₁ ) 6 = ∅

M s 1 ,t 1 ( G _n ₁ )

M s 2 ,t 2 ( G _n ₂ )

n ₁

n ₂

M ^s 1+ s 2

2 , ^t ¹⁺ ₂ ^t ² ( G _m )

n ₁ = n ₂

M s 1 ,t 1 ( G _n )

M s 2 ,t 2 ( G _n )

( ⌊ s ₁ n ⌋ , ⌊ t ₁ n ⌋ )

( ⌊ s ₂ n ⌋ , ⌊ t ₂ n ⌋ )

H 1 ( T ² , Z )

G _2n

( ⌊ (s 1 + s 2 )n ⌋ , ⌊ (t 1 + t 2 )n ⌋ ) = ⌊ ^s ¹ ^+s ₂ ² 2n ⌋ , ⌊ ^t ¹ ^+t ₂ ² 2n ⌋

M ^s ¹⁺ ^s ²

2 , ^t ¹⁺ ₂ ^t ² ( G _m )

~ E ₁

G ₁ = ( V ₁ , E ₁ )

R ^~ ^E ¹

{ ω ∈ R ^~ ^E ¹ : ∀ wb ∈ E ₁ , ω( b , w ) = 0, 0 ≤ ω( w , b ) ≤ 1; ∀ w ∈ W ₁ , div ω( w ) = 1, ∀ b ∈ B ₁ , div ω( b ) = − 1 } .

R ²

G ₁

~ E ₁

(x ₀ , y ₀ ) = (0, 0)

ω ^µ(s,t)

∀ e = wb ∈ E ₁ , ω ^µ(s,t) ( w , b ) = µ(s, t)( e ), ω ^µ(s,t) ( b , w ) = 0.