The di
e
de

Béatrie de Tilière 1

September11,2014

1

LaboratoiredeProbabilitésetModèlesAléatoires,UniversitéPierreetMarieCurie,4plaeJussieu,

F-75005Paris. beatrie.de_tiliereupm.fr. Thesenotes were writtenin partwhileatInstitut de

Mathématiques,UniversitédeNeuhâtel,RueEmile-Argand11,CH-2007Neuhâtel. Supportedinpart

bytheSwissNationalFoundationgrant200020-120218.

1 Introdution 5

1.1 Statistial mehanis and 2-dimensional models . . . 5

1.2 Thedimermodel . . . 7

2 Denitions and founding results 11 2.1 Dimermodel andtiling model . . . 11

2.2 Energy ofongurations and Boltzmann measure . . . 12

2.3 Expliitomputations . . . 13

2.3.1 Partition funtion formula . . . 13

2.3.2 Boltzmann measureformula . . . 16

2.3.3 Expliitexample . . . 17

2.4 Geometri interpretationof lozenge tilings . . . 18

3 Dimer model on innite periodi bipartite graphs 21 3.1 Height funtion . . . 22

3.2 Partition funtion,harateristi polynomial, freeenergy . . . 25

3.2.1 Kasteleyn matrix . . . 25

3.2.2 Charateristi polynomial . . . 26

3.2.3 Freeenergy . . . 30

3.3 Gibbsmeasures . . . 31

3.3.1 Limitof Boltzmann measures . . . 31

3.3.2 ErgodiGibbsmeasures . . . 32

3.3.3 Newtonpolygonand available slopes . . . 32

3.3.4 Surfaetension . . . 34

3.3.5 Construting Gibbsmeasures . . . 34

3.4 Phases ofthe model . . . 36

3.4.1 Amoebas, Harnak urvesand Ronkin funtion . . . 36

3.4.2 Surfaetension revisited . . . 38

3.4.3 Phases ofthedimermodel . . . 39

3.5 Flutuations ofthe height funtion . . . 41

3.5.2 Gaussian freeeldof theplane . . . 42

3.5.3 Convergene of the height funtion to aGaussian freeeld . . . 43

Introdution

1.1 Statistial mehanis and 2-dimensional models

Statistial mehanis isthe appliation of probability theory,whihinludes mathematial tools

fordealingwith largepopulations,totheeldofmehanis,whihisonernedwiththemotion

ofpartilesor objetswhensubjetedtoafore. Statistial mehanisprovidesaframework for

relating the mirosopi properties of individual atoms and moleules to the marosopi bulk

propertiesof materials thatan be observed ineverydaylife (soure: `Wikipedia').

In otherwords, statistial mehanis aimsat studyinglarge sale propertiesof physis system,

based on probabilisti models desribing mirosopi interations between omponents of the

system. Statistial mehanis isalso knownasstatistial physis.

Itisapriorinaturaltointrodue3-dimensionalgraphsinordertoauratelymodelthemoleular

struture of a material as for example a piee of iron, a porous material or water. Sine the

3

-dimensional version of many models turns out to be hardly tratable, muh eort has been put into thestudy of their

2

-dimensional ounterpart. The latter have been shown to exhibit rih, omplexand fasinating behaviors. Here areafew examples.

•

Perolation. This model desribes the ow of a liquid through a porous material. The system onsidered is a square grid representing the moleular struture of the material.

Eahbondofthegridiseitheropenwithprobability

p

^{,orlosed with}probability

1 − p

^,

and bondsare assumedto behave independently fromeah other. The set ofopen bonds

ina given ongurationrepresentsthe partof thematerial wetted by theliquid,and the

mainissueaddressedistheexisteneandpropertiesofinnitelusters ofopenedges. The

behavior of the system depends on the parameter

p

^{: when}

p = 0

^{, all edges are losed,}

thereisnoinnitelusterandtheliquidannotowthroughthematerial;when

p = 1

^,all

edges areopen andthere is a unique inniteluster llingthewhole grid. One an show

thatthere isa spei valueof theparameter

p

^{,known asritial}

p

^{,equal to}

1/2

^{for the}

squaregrid,belowwhihtheprobabilityofhavinganinnitelusterofopenedgesis0,and

abovewhihthe probabilityofitexistingandbeingunique is

1

^{. Onesaysthatthesystem}

undergoesa phase transition at

p = 1/2

^{. Referenes[Kes82 ,Gri99 , BR06, W}

⁺

^07,Wer09℄

arebooks or leturenotesgiving an overviewof perolation theory.

•

^{TheIsingmodel. Thesystemonsideredisamagnetmadeofpartilesrestritedtostayon}

agrid. Eahpartilehasaspinwhihpointseitherupor down (spin

± 1

^{). Eahong-}

uration

σ

^{ofspinsonthewholegridhasanenergy}

E (σ)

^{,whihisthesumofaninteration}

Figure1.1: An inniteluster ofopen edges, when

p = ¹ ₂

^{. Courtesyof V.Beara.}

energy between pairs of neighboring spins, and of an interation energy of spins with an

external magneti eld. The probabilityof a onguration

σ

^is proportional to

e ^{− kT 1 E (σ)}

^,

where

k

^{is the Boltzmannonstant, and}

T

^{is theexternal} temperature. When there isno magneti eldand thetemperature is lose to

0

^{,spins tendto align withtheir neighbors}

and a typial onguration onsists of all

+1

^{or all}

− 1

^{. When the} temperature is very high,all ongurationshave thesame probability ofourringanda typialonguration

onsistsofamixture of

+1

^and

− 1

^{. Again,there isaritial} temperature

T c

^{at whihthe}

Isingmodel undergoesaphase transitionbetween theordered and disorderedphase. The

literature on the Ising model is huge, as an introdutory reading we would suggest the

bookbyBaxter[Bax89 ℄,theonebyMCoyandWu[MW73℄,theleturenotesbyVelenik

[Vel ℄ and referenestherein.

Figure1.2: An Isingonguration, when

1

T = 0.9

^{. Courtesyof V.Beara.}

These two examples illustrate some of the prinipal hallenges of

2

-dimensional statistial me- hanis,whih are:

•

^{Find theritialparameters of themodels.}

•

^{Understandthe behavior ofthe model inthesub-ritial and}super-ritial regimes.

•

^{Understandthebehaviorofthesystematritiality. Critialsystemsexhibitsurprisingfea-}

tures,andarebelievedto be universalinthesalinglimit, i.e. independent ofthespei

featuresofthe lattie on whih themodelis dened. Verypreise preditionswereestab-

lished by physiistsin the last 30-50 years, in partiular by Nienhuis, Cardy, Duplantier

and many others. On the mathematis side, a huge step forward was the introdution

of the Shramm-Loewner evolution by Shramm in [Sh00℄, a proess onjetured to de-

sribe the limiting behavior of well hosen observables of ritial models. Manyof these

onjetures weresolved inthe following years,inpartiular byLawler, Shramm,Werner

[LSW04 ℄ andSmirnov [Smi10 ℄,Chelkak-Smirnov [CS12 ℄. The importane of these results

was aknowledged with the two Fields medals awarded to Werner (2006) and Smirnov

(2010). Interesting ollaborations between thephysis andmathematis ommunities are

emerging, withfor example thework ofDuplantier andSheeld [DS11 ℄.

The general framework for statistial mehanis is the following. Consider an objet

G

(most often a graph) representing the physial system, and dene all possible ongurations of the

system. To every onguration

σ

^{, assignan energy}

E (σ)

^{, thenthe}probability of ourrene of theonguration

σ

^{isgiven bytheBoltzmann measure}

µ

^:

µ(σ) = e ^{−E (σ)} Z ( G ) .

Note thatthe energy is oftenmultiplied by aparameter representing theinverse external tem-

perature. Thedenominator

Z ( G )

^isthenormalizing onstant, knownasthepartition funtion:

Z ( G ) = X

σ

e ^{−E (σ)} .

When the systemis innite, the above denition does not hold, but we do not want to enter

into these onsiderations here.

Thepartitionfuntionisoneofthekeyobjetsofstatistialmehanis. Indeeditenodesmuh

of the marosopi behavior of the system. Hene, its omputation is the rst question one

addresses when studying suh amodel. It turns out thatthere are very fewmodels where this

omputation an be done exatly. Having a losed form for the partition funtion opens the

waytondingmanyexat results,andto having avery deepunderstanding ofthemarosopi

behaviorof the system.

Twofamousexamplesarethe

2

-dimensionalIsingmodel,wheretheomputationofthepartition funtion is due to Onsager [Ons44 ℄, and the dimer model where it is due to Kasteleyn [Kas61 ,

Kas67 ℄,andindependently toTemperleyandFisher[TF61 ℄. Thedimermodelisthemaintopi

of theseletures and isdened inthe next setion.

1.2 The dimer model

The dimer model was introdued in the physis and hemists ommunities to represent the

adsorption of di-atomi moleules on the surfae of a rystal. It is part of a larger family of

modelsdesribingtheadsorptionofmoleulesofdierentsizesonalattie. Itwasrstmentioned

inapaperbyFowlerandRushbrooke[FR37℄in1937. Asmentionedintheprevioussetion,the

rst major breakthrough inthe study of the dimer model is the omputation of the partition

funtion byKasteleyn [Kas61 ,Kas67 ℄and independently byTemperley andFisher [TF61 ℄.

Itisinteresting to observe thatfora longtime, thephysis andmathematis ommunities were

unawareof their respetive advanes. Mathematiians studied related questionsasfor example

geometri andombinatorial propertiesoftilings ofregionsoftheplanebydominoesorrhombi.

To the best of our knowledge, the latter problem was rst introdued in a paper by David

and Tomei [DT89 ℄. A major breakthrough was ahieved inthe paper[Thu90℄Thurston,where

theauthor interprets rhombus tilings as

2

-dimensional interfaes ina

3

-dimensional spae. An example ofrhombus tilingis given inFigure1.3.

Figure1.3: Rhombus tiling. Courtesyof R.Kenyon.

Inthelate

90

^'sandearly

00

^{'s,alotofprogressesweremadein}understandingthemodel,seethe papersofKenyon[Ken97 , Ken00 ℄, Cohn-Kenyon-Propp [CKP01℄, Kuperberg[Kup98 ℄, Kenyon-

Propp-Wilson [KPW00 ℄. In

2006

Kenyon-Okounkov-Sheeld [KOS06 ℄, followed by Kenyon- Okounkov [KO06 , KO07 ℄ wrote breakthrough papers, whih give a full understanding of the

model on innite, periodi, bipartite graphs. Suh deep understanding of phenomena isa real

treasurein statistialmehanis.

My goal for these letures is to present the results of Kasteleyn, Temperley and Fisher, of

Thurston, and of the paper of Kenyon, Okounkov and Sheeld. As you will see, the dimer

model has ramiations to many elds of mathematis: probability, geometry, ombinatoris,

analysis, algebraigeometry. Iwilltryto beasthoroughaspossible,but ofoursesome results

addressing the eld of algebrai geometry reah the limit of my knowledge, so thatI will only

Inspiration for these notes omes in large parts from the letures given by R. Kenyon on the

subjet [Ken04,Ken℄. The main otherreferenes are[Kas67 ,Thu90,KOS06 , KO06 ℄.

Definitions and founding results

2.1 Dimer model and tiling model

Inthissetion, wedenethedimermodel andtheequivalenttiling model,usingtheterminology

of statistial mehanis. The system onsidered is a graph

G = ( V , E )

^{satisfying the following:}

itis planar,simple(no loopsandno multiple edges),nite orinnite.

Congurationsofthesystemareperfetmathingsofthegraph

G

. Aperfetmathing isasubset of edgeswhih overseah vertexexatlyone. Inthephysisliterature, perfet mathings are

also referredto asdimer ongurations, adimer being a di-atomi moleule represented by an

edge of theperfet mathing. Letus denoteby

M ( G )

^{thesetof all dimer}ongurations ofthe graph

G

.

Figure 2.1gives an example of a dimer onguration when thegraph

G

is a nite subgraph of thehoneyomb lattie

H

.

Figure2.1: Dimeronguration ofa subgraph ofthehoneyomb lattie.

In order to dene theequivalent tiling model, we onsider a planar embedding of the graph

G

, and suppose that it is simply onneted, i.e. that it is the one-skeleton of a simply onneted

union of faes. From now on, when we speak of a planar graph

G

, we atually mean a graph witha partiular planarembedding.

The tiling model is dened on the dual graph

G ∗

of

G

. An embedding of the dualgraph

G ∗

is

obtained by assigning a vertex to every faeof

G

and joining two verties of

G ^∗

by an edge if andonly ifthe orrespondingfaesof

G

areadjaent. Thedualgraph willalso be thoughtofas anembeddedgraph. Whenthegraphisnite,wetakeaslightlydierentdenitionofthedual:

we take

G

to be the dual of

G ∗

and remove thevertex orresponding to the outer fae, aswell

asedges onnetedto it, seeFigure 2.2.

A tile of

G ∗

is a polygon onsisting of two adjaent inner faes of

G ∗

glued together. A tiling of

G ∗

is a overing of the graph

G ∗

with tiles, suh that there are no holes and no overlaps.

Figure2.2 gives an exampleof a tilingof a nite subgraphof thetriangular lattie

T

,thedual graph of the honeyomb lattie. Tiles of the triangular lattie are

60 ^◦

^{-rhombi, and are also}

knownaslozenges or alissons.

Figure 2.2: Dual graph of a nite subgraph of the honeyomb lattie (left). Tiling of this

subgraph (right).

Another lassial example is thetiling modelon the graph

Z ²

,the dualof thegraph

Z ²

. Tiles aremade oftwo adjaent squares,and areknownasdominoes.

Dimer ongurations of the graph

G

are in bijetion with tilings of the graph

G ∗

through the

followingorrespondene, seealsoFigure2.3: dimeredgesofperfetmathingsonnetpairs of

adjaent faesforming tilesofthe tiling. It isan easy exeriseto provethat thisindeed denes

abijetion.

Figure2.3: Bijetion between dimerongurations of thegraph

G

and tilings ofthe graph

G ^∗

.

2.2 Energy of onfigurations and Boltzmann measure

Welet

G

beaplanar,simplegraph. Inthissetion, andfortheremainder ofChapter2,wetake

G

to be nite. Supposethatedges areassigned apositive weight funtion

ν

^{,that iseveryedge}

e

of

G

hasweight

ν( e )

^.

The energy of a dimeronguration

M

^of

G

,is

E (M) = − P

e ∈ M log ν( e ).

^{The weight}

ν(M )

^of

adimer onguration

M

^of

G

,isexponential of minus its energy:

ν(M) = e ^{−E (M)} = Y

e ∈ M

ν( e ).

Note that by the orrespondene between dimer ongurations and tilings, the funtion

ν

^an

beseenasweighting tilesof

G ∗

,

ν(M )

^{isthenthe weight of thetiling} orrespondingto

M

^.

µ

^{is a}probabilitymeasure onthe set ofdimerongurations

M ( G )

^{,dened by:}

∀ M ∈ M ( G ), µ(M) = e ^{−E (M )}

Z( G ) = ν(M ) Z( G ) .

The term

Z( G )

^{is the}normalizing onstant known asthepartition funtion. It isthe weighted sumof dimerongurations, thatis,

Z ( G ) = X

M ∈M ( G )

ν (M ).

When

ν ≡ 1

^{,the partition funtion ounts the number ofdimer} ongurations of thegraph

G

, or equivalently thenumberoftilings ofthe graph

G ^∗

,andtheBoltzmann measureissimplythe uniform measureon the setof dimerongurations.

Whenanalyzingamodelofstatistialmehanis,therstquestionaddressedisthatofomput-

ingthefree energy, whih isminus theexponential growth rateofthepartitionfuntion,asthe

size of the graphinreases. The most natural way ofattaining this goal, ifthe modelpermits,

istoobtain anexpliitexpressionforthepartitionfuntion. Reallthatthedimermodelisone

of the rare

2

-dimensional models where a losed formula an be obtained. This is the topi of thenext setion.

2.3 Expliit omputations

The expliit omputation of the partition funtion is due to Kasteleyn [Kas61 , Kas67 ℄ and

independentlytoTemperleyandFisher[TF61 ℄. AproofofthisresultisprovidedinSetion2.3.1,

inthe asewhere the underlying graph

G

isbipartite. Thisisone ofthefoundingresults ofthe dimer model, paving the way to obtaining other expliit expressions as for example Kenyon's

losedformulafortheBoltzmannmeasure[Ken97 ℄,seeSetion2.3.2. InSetion2.3.3,weprovide

an exampleof omputation ofthepartition funtion and oftheBoltzmann measure.

2.3.1 Partition funtion formula

We restrit ourselves to the asewhere thegraph

G

is bipartite, theproof inthe non-bipartite ase issimilar inspirit although a little moreinvolved. The simpliationin thebipartite ase

isdue to Perus [Per69℄.

Agraph

G = ( V , E )

^{isbipartite ifthesetofverties}

V

anbesplitintotwosubsets

W ∪ B

,where

W

denotes white verties,

B

blak ones, and verties in

W

are only adjaent to verties in

B

. We suppose that

| W | = | B | = n

^{, for otherwise there are no perfet mathings of the graph}

G

; indeeda dimeredgealwaysovers ablakanda whitevertex.

Labelthewhiteverties

w ₁ , . . . , w _n

andtheblakones

b ₁ , . . . , b _n

,andsupposethatedgesof

G

are oriented. Thehoie oforientation will be speiedlater inthe proof. Then theorresponding

oriented, weighted, adjaeny matrix is the

n × n

^matrix

K

^{whose lines are indexed by white}

verties,whose olumns areindexedby blakones, andwhose entry

K( w _i , b _j )

^is:

K( w _i , b _j ) =

 



 

ν( w _i b _j )

^if

w _i ∼ b _j

,and

w _i → b _j

− ν( w _i b _j )

^if

w _i ∼ b _j

,and

w _i ← b _j

0

^iftheverties

w _i

and

b _j

arenot adjaent.

By denition,the determinant of the matrix

K

^is:

det(K) = X

σ ∈S ⁿ

sgn

(σ)K( w ₁ , b _σ(1) ) . . . K( w _n , b _σ(n) ),

where

S n

^{is the set of} permutations of

n

^{elements. Let us rst observe that eah non-zero}

term intheexpansionof

det(K)

^{orrespondsto theweightof adimer}onguration, uptosign.

Thus, the determinant of

K

^{seems to be the} appropriate objet for omputing the partition funtion, theonly problem beingthat not all terms may be ounted withthe same sign. Note

that reversing the orientation of an edge

w _i b _j

hanges thesign of

K( w _i , b _j )

^{. The remainder of}

theproofonsistsinhoosingan orientation ofthe edgesof

G

allowingto ompensate signature of permutations, so that all terms in the expansion of the determinant of

K

^{indeed have the}

same sign.

Let

M ₁

^and

M ₂

^{be two perfet mathings of}

G

drawn one on top of the other. Dene an alternatingyle to be ayle of

G

whose edgesalternate between edges of

M ₁

^and

M ₂

^{. Then,}

analternating ylehaseven length,and ifthelength isequal to 2,theyleisa doubled edge,

thatis an edgeovered byboth

M ₁

^and

M ₂

^{. Thesuperimpositionof}

M ₁

^and

M ₂

^{is aunion of}

disjoint alternating yles,see Figure2.4. Thisis beause,bydenition ofa perfet mathing,

eah vertex is adjaent to exatlyone edge of themathing, sothat in the superimposition of

two mathings

M 1

^and

M 2

^{,eah vertex is adjaent to exatly one edgeof}

M 1

^{and one edge of}

M ₂

^.

M M ₁

2

Figure 2.4: Superimposition of two dimer ongurations

M ₁

^and

M ₂

^{of a subgraph}

G

of the honeyomb lattie

H

.

One an transformthe mathing

M ₁

^{into the mathing}

M ₂

^{,byreplaing edges of}

M ₁

^{by those}

of

M ₂

^inallalternating ylesoflength

≥ 4

^ofthesuperimposition. Thus,arguingbyindution, itsuesto showthat thesign ofthe ontributions of

M 1

^and

M 2

^to

det(K)

^{isthesame when}

M ₁

^and

M ₂

^{dieralong asingle} alternating yleof length

≥ 4

^{. Letus assume thatthis isthe}

ase, denote the unique yle by

C

^{and by}

w _i

1 , b _j

1 , . . . , w _i

k , b _j

k

^{its verties inlokwise order,}

seeFigure 2.5.

Let

σ

^(resp.

τ

^)bethepermutationorrespondingto

M 1

^(resp.

M 2

^{). Thenbythe}orrespondene between enumeration of mathingsand termsintheexpansion ofthedeterminant, we have:

j ₁ = σ(i ₁ ) = τ (i ₂ ), j ₂ = σ(i ₂ ) = τ (i ₃ ), . . . , j _k = σ(i _k ) = τ (i ₁ ).

Ifwe let

c

^bethe permutation yle

c = (i k . . . i ₁ )

^{,thenwe dedue:}

τ (i ℓ ) = σ(i ℓ − 1 ) = σ ◦ c(i ℓ ).

^(2.1)

...

w _i

1 b _j

1

w _i

2 b _j w _i 2

k

b _j

k

M 1

M ₂ C

Figure 2.5: Labeling ofthe vertiesof anexample of superimpositionyle of

M ₁

^and

M ₂

^.

Inorderto hekthatthe ontributions of

M ₁

^and

M ₂

^to

det(K)

^{have thesamesign,itsues}

tohekthatthe signoftheratiooftheontributionsispositive. Thesignofthisratio,denoted

by

Sign(M ₁ /M ₂ )

^,is:

Sign(M ₁ /M ₂ ) = Sign

sgn

(σ)

sgn

(τ ) K( w _i

1 , b _σ(i

1 ) ) . . . K( w _i

k , b _σ(i

k ) ) K( w _i

1 , b _τ(i

1 ) ) . . . K( w _i

k , b _τ(i

k ) )

,

whih isthe same asthe signof the produtofthenumerator andthedenominator. Now,

( ⋄ 1 ) : =

^sgn

(σ)

^sgn

(τ ) =

^sgn

(σ ◦ τ ) =

^sgn

(σ ◦ σ ◦ c)

^{byEquation(2.1)}

= ( − 1) ^k+1 . ( ⋄ 2 ) : = Sign ( [K( w _i

1 , b _σ(i

1 ) ) . . . K( w _i

k , b _σ(i

k ) )][K ( w _i

1 , b _τ(i

1 ) ) . . . K( w _i

k , b _{τ (i}

k ) )]

= Sign [K( w _i

1 , b _j

1 ) . . . K( w _i

k , b _j

k )][K( w _i

1 , b _j

k ) . . . K( w _i

k , b _j

k−1 )]

= Sign K( w _i

1 , b _j

1 )K( w _i

1 , b _j

k ) . . . K( w _i

k , b _j

k )K( w _i

k , b _j

k−1 )

Let

p

^{be the parity of the number of edges of the yle}

C

^{oriented lokwise. The yle}

C

^is

saidto belokwise odd if

p = 1

^{,lokwise even if}

p = 0

^{. Letus showthat}

Sign(M ₁ /M ₂ ) = +1

ifand onlyif

p = 1

^{. To this purpose, werst relate}

( ⋄ 2 )

^and

( − 1) ^p

^.

Partition

W _C := { w _i

1 , . . . , w _i

k }

^as

W ^e

C ∪ W ^o

C

^{, where}

W ^e

C

^{onsistsof white verties with 0 or 2}

inomingedges(equivalently2or 0outgoingedges),and

W ^o _C

onsistsofwhitevertieswithone inoming and one outgoing edge, then

| W ^e

C | + | W ^o

C | = k

^{. If a white vertex belongs to}

W ^e

C

^{, it}

ontributes

1

^to

( ⋄ 2 )

^and

1

^to

p

^{;ifa whitevertexbelongs to}

W ^o

C

^{, itontributes}

− 1

^to

( ⋄ 2 )

^and

0

^to

p

^{. We thus have:}

( ( ⋄ 2 ) = ( − 1) ^{| W o C |}

( − 1) ^p = ( − 1) ^{| W e C |} ⇒ ( ⋄ 2 ) = ( − 1) ^k ( − 1) ^p .

Asa onsequene

Sign(M ₁ /M ₂ ) = ( ⋄ 1 )( ⋄ 2 ) = ( − 1) ^2k+1 ( − 1) ^p

^{,so that}

Sign(M ₁ /M ₂ )

^{is positive}

ifand onlyif

p = 1

^{,i.e. ifand onlyiftheyle}

C

^{islokwiseodd.}

Following Kasteleyn [Kas67℄, an orientation of the edges of

G

suh that all yles obtained as superimpositionofdimerongurationsarelokwiseodd,isalledadmissible. Deneaontour

yle tobeayleboundinganinnerfaeofthegraph

G

. Kasteleynprovesthatiftheorientation issuhthatallontourylesarelokwiseodd,thentheorientationisadmissible. Theproofis

byindutiononthe numberoffaesinludedintheyle, refertothepaper[Kas67℄ fordetails.

Anorientation ofthe edgesof

G

suhthatallontouryles arelokwiseoddisonstrutedin thefollowingway,seeforexample[CR07 ℄. Consideraspanningtreeofthedualgraph

G ^∗

,witha vertexorrespondingto theouterfae,takento be theroot ofthetree. Chooseanyorientation

for edges of

G

not rossed by the spanning tree. Then, start from a leafof thetree, and orient

orresponding faeis lokwiseodd. Remove the leafand theedge from thetree. Iterate until

onlytherootremains. Sinethetreeisspanning,allfaesarereahed bythealgorithm, andby

onstrution all orrespondingontour ylesarelokwiseodd.

AKasteleyn-Perus matrix,or simplyKasteleynmatrix,denotedby

K

^{,assoiated tothegraph}

G

is the oriented, weighted adjaeny matrix orresponding to an admissible orientation. We havethus proved thefollowing:

Theorem 1. [Kas67℄ Let

G

be a nite, planarbipartite graph withan admissible orientation of its edges,let

ν

^{be a positive weight funtionon the edges, and}

K

^{be the} orresponding Kasteleyn matrix. Then,the partition funtion of the graph

G

is:

Z ( G ) = | det(K) | .

When the graph

G

is not bipartite, lines and olumns of the adjaeny matrix are indexed by allvertiesof

G

(vertiesannotbenaturallysplitinto twosubsets). Byhoosingan admissible orientation of the edges, the partition funtion an be expressed as the square root of the

determinant of the orresponding Kasteleyn matrix or,sine this matrix isskew-symmetri, as

thePfaanof this samematrix. For more details,referto [Kas67 ℄.

2.3.2 Boltzmann measure formula

Whenthegraph

G

isbipartite,Kenyon[Ken97 ℄givesanexpliitexpressionfortheloalstatistis of the Boltzmann measure. Let

K

^{be a Kasteleyn matrix assoiated to}

G

, and let

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

^{be asubset of edgesof}

G

.

Theorem 2. [Ken97℄ The probability

µ( e ₁ , . . . , e _k )

^{of edges}

{ e ₁ , . . . , e _k }

^{ourring in a dimer}

onguration of

G

hosen withrespet to the Boltzmannmeasure

µ

^is:

µ( e ₁ , . . . , e _k ) =

Y k i=1

K ( w _i , b _i )

!

1 ≤ det i,j ≤ k K ^{− 1} ( b _i , w _j )

.

^(2.2)

Proof. The weighted sum of dimer ongurations ontaining the edges

{ e ₁ , . . . , e _k }

^{is (up to}

sign) the sum of all terms ontaining

K( w ₁ , b ₁ ) . . . K( w _k , b _k )

^{in the expansion of}

det(K)

^{. By}

expanding this determinant along lines(or olumns), itis easy to seebyindution that this is

equal to:

Y k i=1

K( w _i , b _i )

!

det(K E ) ,

where

K E

^{isthematrix obtainedfrom}

K

^{byremovingthelines}orrespondingto

w ₁ , . . . , w _k

and theolumnsorrespondingto

b ₁ , . . . , b _k

. Now byJaobi's identity,see for example[HJ90℄:

det(K E ) = det(K) det (K ^{− 1} ) E ^∗ ,

where

E ^∗

^{isthe setof edgesnot in}

E

^{. Otherwise stated,}

(K ^{− 1} ) E ^∗

^isthe

k × k

^{matrixobtained}

from

K ^{− 1}

^{by keeping the lines} orresponding to

b ₁ , . . . , b _k

and the olumns orresponding to

w ₁ , . . . , w _k

. Thus,

µ( e ₁ , . . . , e _k ) = Q k

i=1 K( w _i , b _i )

det(K E )

| det(K ) | =

Y k i=1

K( w _i , b _i )

!

det (K ^{− 1} ) E ^∗ .

thatis apoint proesssuh thatthejointprobabilities are oftheform

µ( e ₁ , . . . , e _k ) = det(M( e _i , e _j ) _{1 ≤} i,j ≤ k ),

forsomekernel

M

^{. Intheaseofbipartitedimers,}

M ( e _i , e _j ) = K( w _i , b _j )K ^{− 1} ( b _i , w _j )

^{,see[Sos07 ℄}

for an overview.

2.3.3 Expliit example

Figure2.6givesanexampleofaplanar,bipartitegraphwhoseedgesareassignedpositiveweights

and anadmissible orientation.

a a

a a

a

b b b

b b

a, b > 0

PSfrag replaements

w ₁ b ₁ w ₂

b ₂ w ₃ b ₃

w ₄ b ₄

Figure 2.6: A planar,bipartite graphwith a positive weight funtion and an admissible orien-

tation.

TheorrespondingKasteleynmatrix is:

K =



 



a 0 − b 0 a − b 0 0

b a a b

0 0 b − a



 

 ,

and thedeterminant is equal to

det(K) = 2a ³ b + 2b ³ a.

Setting

a = b = 1

^{yields that the number of perfet mathings of this graph is}

4

^{. In this ase,}

theBoltzmann measureis theuniform measureon tilings ofthis graph.

TheinverseKasteleyn matrix

K ^{− 1}

^is:

K ^{− 1} = 1 4



 



2 1 1 1

2 − 3 1 1

− 2 1 1 1

− 2 1 1 − 3



 

 .

Using the labeling of the verties of Figure 2.6 and Theorem 2, we ompute theprobability of

ourreneofsome subset of edges:

µ( w ₁ b ₁ ) = | K ^{− 1} ( b ₁ , w ₁ ) | = 1 2 µ( w ₃ b ₁ ) = | K ^{− 1} ( b ₁ , w ₃ ) | = 1 4 µ( w ₁ b ₁ , w ₃ b ₄ ) =

det

K ^{− 1} ( b ₁ , w ₁ ) K ^{− 1} ( b ₁ , w ₃ ) K ^{− 1} ( b ₄ , w ₁ ) K ^{− 1} ( b ₄ , w ₃ )

= 1 16

det

2 1

− 2 1

= 1

4 .

ndallpossibleongurations(

4

^{ofthem),andompute the}probabilities. However,thegoalof this exampleis to showhowto usethegeneral formulae.

2.4 Geometri interpretation of lozenge tilings

By means of the height funtion, Thurston interprets lozenge tilings of the triangular lattie

as disrete surfaes in a rotated version of

Z ³

projeted onto the plane. He gives a similar interpretationofdominotilingsofthesquarelattie. Thisapproahanbegeneralizedtodimer

ongurations ofbipartite graphs usingows. Thisyields an interpretationof thedimermodel

on a bipartite graphas a random interfae modelin dimensions

2 + 1

^{,and oersmore insight}

into the model. In this setion we exhibit Thurston's onstrution of the height funtion on

lozenge tilings. We postpone the denition of the height funtion on general bipartite graphs

until Setion 3.1.

Faes of the triangular lattie

T

an be olored in blak and white, so that blak faes (resp.

whiteones)areonlyadjaenttowhiteones(resp. blakones). Thisisaonsequeneofthefat

thatitsdualgraph,thehoneyomblattie,isbipartite. Orienttheblakfaesounterlokwise,

and the white ones lokwise, see Figure 2.7 (left). Consider a nite subgraph

X

of

T

whih is tileable by lozenges, and a lozenge tiling

T

^of

X

. Then theheight funtion

h ^T

^{is an integer}

valuedfuntion onverties of

X

,dened indutively asfollows:

•

^{Fixa vertex}

v ₀

of

X

,and set

h ^T ( v ₀ ) = 0

^.

•

^{For every boundary edge}

uv

of a lozenge,

h ^T ( v ) − h ^T ( u ) = +1

^{ifthe edge}

uv

is oriented from

u

to

v

,implyingthat

h ^T ( v ) − h ^T ( u ) = − 1

^{whenthe edge}

uv

is orientedfrom

v

to

u

. The height funtion is well dened, in the sense that the height hange around any oriented

yleis 0. Anexample ofomputation of theheight funtion isgiven inFigure2.7(right).

0

3

1

1 2

3 2 4

1 2 3 4 3 2 1 0

1 2

0 1

2 1

0 2

3 3

4 4

5 3 4 3 2 1 0

4

2 1 2

2 3

2 v 0

Figure2.7: Orientation offaesofthe triangularlattie(left). Heightfuntion orrespondingto

a lozenge tiling(right).

As a onsequene, lozenge tilings are interpreted as stepped surfaes in

Z f ³

projeted onto the plane, where

Z f ³

is

Z ³

rotated so that diagonals ofthe ubesare orthogonal to the plane. The height funtionisthen simplythe height ofthesurfae(i.e. thirdoordinate). Thisonstru-

tiongivesamathematial senseto theintuitive feelingofubesstiking inor out,whihstrikes

us whenwathinga piture oflozenge tilings.

Height funtions haraterize lozengetilings asstated bythefollowing lemma.

Lemma 4. Let

X

be a nite simply onneted subgraph of the triangular lattie

T

, whih is tileable by lozenges. Let

h

^{be an integervalued funtion on the verties of}

X

,satisfying:

• h( v ₀ ) = 0

^,where

v ₀

isa xed vertex of

X

.

• h( v ) − h( u ) = 1

^{for any boundary edge}

uv

of

X

oriented from

u

to

v

.

• h( v ) − h( u ) = 1

^or

− 2

^{for any interior edge}

uv

of

X

oriented from

u

to

v

.

Then,there isa bijetion between funtions

h

^{satisfyingthese two onditions, andtilings of}

X

.

Proof. Let

T

^{be a lozenge tiling of}

X

and let

uv

be an edgeof

X

, oriented from

u

to

v

. Then, the edge

uv

is either a boundary edge or a diagonal of a lozenge. By denition of the height funtion,theheight hange is

1

^{intherst ase, and}

− 2

^intheseond.

Conversely,let

h

^{beanintegerfuntionasinthelemma. Letusonstrutatiling}

T

^whoseheight

funtion is

h

^{. Consider a blakfaeof}

X

,thenthere isexatly one edge

uv

ontheboundary of this faewhose height hange is

− 2

^{. Tothis fae, we assoiate the lozenge whih isrossed by}

theedge

uv

. Repeating thisproedure for allblakfaesyields a tilingof

X

.

Thurston [Thu90℄uses height funtions inorderto determine whethera subgraphof thetrian-

gular lattiean betiledbylozenges. Referto thepaperfor details.

Dimer model on infinite periodi bipartite graphs

ThishapterisdevotedtothepaperDimersandamoebas [KOS06 ℄byKenyon,Okounkovand

Sheeld.

Reallthatedgesofdimerongurationsrepresentdi-atomimoleules. Sineweareinterested

inthemarosopibehaviorofthesystem,ourgoalistostudythemodelonverylargegraphs. It

turnsout thatitiseasierto extratinformationfor themodeldened oninnitegraphs, rather

than very large ones. Indeed, on very large but nite graphs, Kasteleyn's omputation an be

done,but involvesomputingthedeterminant ofhuge matries, whih isofourse veryhard in

general, and won't tell us muh about the system. Computing expliitly theloal statistis of

theBoltzmann measurebeomeshardlytratablesine itrequiresinvertingverylarge matries.

Thismotivates the following roadmap.

•

^{Assume that the graph}

G = ( V , E )

^{is simple, planar, innite, bipartite, and}

Z ²

-periodi.

This means that

G

is embedded inthe plane sothat translations atbyolor-preserving isomorphism of

G

, i.e. isomorphisms whih map blak verties to blak ones and white verties to white ones. For later purposes, we onsider the underlying lattie

Z ²

to be a subgraph of the dual graph

G ∗

, and x a basis

{ e _x , e _y }

^{, allowing to reord opies of a}

vertex

v

of

G

as

{ v + (k, l) : (k, l) ∈ Z ² }

^{. RefertoFigure3.1foranexample when}

G

isthe square-otagongraph.

e _x e _y

Figure3.1: A piee ofthe square-otagon graph. The underlying lattie

Z ²

is inlight grey,the two blakvetors represent ahoieof basis

{ e _x , e _y }

^.

•

^Let

G _n = ( V _n , E _n )

^{bethequotientof}

G

bythe ationof

n Z ²

. Thenthesequeneofgraphs

{ G _n } n ≥ 1

^{isanexhaustionoftheinnitegraph}

G

bytoroidalgraphs. Thegraph

G ₁ = G / Z ²

isalledthe fundamental domain,see Figure3.2. Assumethatedgesof

G ₁

areassigned a positive weight funtion

ν

^{thus deninga periodiweight funtionon theedgesof}

G

.

1 1

1 1

1 1

1 1

1 1 1 1

PSfragreplaements

G ₁

Figure 3.2: Fundamental domain

G ₁

of the square-otagon graph, opposite sides in light grey areidentied. Edges areassignedweight 1.

•

^{The goal of this hapter is to understand the dimer model on}

G

, using the exhaustion

{ G _n } ⁿ ≥ 1

^{,bytakinglimitsas}

n → ∞

^of appropriate quantities.

Note that it is ruial to take an exhaustion of

G

by toroidal graphs. Indeed, the latter are invariant bytranslations intwo diretions, akeyfatwhihallows omputationsto go through

using Fourier tehniques. Note also that takingan exhaustion by planar graphs a priori leads

to dierent results beause of theinueneof the boundary whih annotbenegleted.

Althoughitmightnotseemsolearatthisstage,thefatthat

G

and

{ G _n } ⁿ ≥ 1

^{areassumedtobe}

bipartiteisalsoruialfortheresultsofthishapter, beauseitallowstorelatethedimermodel

to well behaved algebrai urves. Having a general theoryof thedimermodelon non-bipartite

graphs isone ofthe important openquestions ofthe eld.

3.1 Height funtion

Inthissetion,wedesribetheonstrutionoftheheightfuntionondimerongurationsofthe

innitegraph

G

andofthetoroidalgraphs

G _n

,

n ≥ 1

^{. Sineweareworkingonthedimermodel}

and noton thetiling model,asin Thurston'sonstrution, theheight funtion isa funtion on

faesof

G

or equivalently, afuntion on vertiesof thedualgraph

G ^∗

.

Thedenitionofthe heightfuntionreliesonows. Denoteby

~ E

thesetofdiretededgesofthe graph

G

,i.e. every edgeof

E

yieldstwo orientededges of

~ E

. A ow

ω

^{is a realvalued funtion}

dened on

~ E

,that isevery diretededge

( u , v )

^of

~ E

isassigned aow

ω( u , v )

^.

The divergene of a ow

ω

^{,denoted by}

div ω

^{,isa real valued funtion dened on}

V

givingthe dierene between totaloutow and totalinow at verties:

∀ u ∈ V , div ω( u ) = X

v ∼ u

ω( u , v ) − X

v ∼ u

ω( v , u ).

Sine

G

is bipartite, we split verties

V

into white and blak ones:

V = W ∪ B

. Then, every dimeronguration

M

^of

G

denesawhite-to-blakunitow

ω ^M

^{asfollows. Theow}

ω ^M

^takes

value0 onall diretededges arisingfrom edgesof

E

whihdo not belong to

M

^,and

∀ wb ∈ M, ω ^M ( w , b ) = 1, ω ^M ( b , w ) = 0.

Sineeveryvertexofthegraph

G

isinident toexatlyoneedgeoftheperfetmathing

M

^,the

ow

ω ^M

^{hasdivergene 1at everywhite vertex,and -1at every blakone, thatis:}

∀ w ∈ W , div ω ^M ( w ) = X

b ∼ w

ω ^M ( w , b ) − X

b ∼ w

ω ^M ( b , w ) = 1,

∀ b ∈ B , div ω ^M ( b ) = X

w ∼ b

ω ^M ( b , w ) − X

w ∼ b

ω ^M ( w , b ) = − 1.

Let

M ₀

^{be a xed periodi referene perfet mathing of}

G

, and

ω ^{M 0}

^{be the} orresponding ow, alled the referene ow. Then, for any other mathing

M

^{with ow}

ω ^M

^{, thedierene}

ω ^M − ω ^{M 0}

^{is a}divergene-free ow,that is:

∀ w ∈ W , div(ω ^M − ω ^{M 0} )( w ) = X

b ∼ w

(ω ^M ( w , b ) − ω ^{M 0} ( w , b )) − X

b ∼ w

(ω ^M ( b , w ) − ω ^{M 0} ( b , w ))

= div ω ^M ( w ) − div ω ^{M 0} ( w ) = 1 − 1 = 0,

and similarlyfor blak verties.

We arenow readyto dene theheight funtion. Let

M

^{be adimerongurationof}

G

,thenthe height funtion

h ^M

^{is an integer valuedfuntion on faesof}

G

dened asfollows.

•

^{Fixa fae}

f ₀

of

G

,andset

h ^M ( f ₀ ) = 0

^.

•

^{For any other fae}

f ₁

, onsider an edge-path

γ

^{of the dual graph}

G ∗

, from

f ₀

to

f ₁

. Let

( u ₁ , v ₁ ), . . . , ( u _k , v _k )

^{denoteedges of}

G

rossing the path

γ

^{,where for every}

i

^,

u _i

ison the leftofthe path

γ

^and

v _i

ontheright. Then

h ^M ( f ₁ ) − h ^M ( f ₀ )

^{isthetotalux of}

ω ^M − ω ^{M 0}

aross

γ

^{,that is:}

h ^M ( f ₁ ) − h ^M ( f ₀ ) = X k i=1

[(ω ^M ( u _i , v _i ) − ω ^M ( v _i , u _i )) − (ω ^{M 0} ( u _i , v _i ) − ω ^{M 0} ( v _i , u _i ))].

The height funtion is well dened if itis independent of thehoie of

γ

^,or equivalently ifthe height hange aroundeveryfae

f ^∗

ofthedualgraph

G ^∗

is0. Let

u

bethevertexofthegraph

G

orrespondingto the fae

f ∗

,and let

v ₁ , . . . , v _k

be itsneighbors. Then, bydenition, theheight hange aroundthe fae

f ∗

,inounterlokwiseorder, is:

X k i=1

[(ω ^M ( u , v _i ) − ω ^M ( v _i , u )) − (ω ^{M 0} ( u , v _i ) − ω ^{M 0} ( v _i , u ))] = div(ω ^M − ω ^{M 0} )( u ) = 0.

Theheightfuntion isthus well dened asa onsequeneof thefatthattheow

ω ^M − ω ^{M 0}

^is

divergenefree, uptothe hoieofa basefae

f ₀

and ofareferenemathing

M ₀

^{. An analogof}

Lemma 4 givesa bijetion between height funtionsanddimerongurations of

G

. Remark 5.

•

^{There is atually an easy wayof omputing theheight funtion. Reall from Setion 2.3}

thatthesuperimpositionoftwodimerongurations

M

^and

M ₀

^{onsistsofdoublededges}

andalternating ylesoflength

≥ 4

^{(ylesmayextendto innitywhen}

G

isinnite). Let usdenoteby

M − M ₀

^theorientedsuperimpositionof

M

^and

M ₀

^,withedgesof

M

^oriented

fromwhitevertiesto blakones,and thoseof

M ₀

^{fromblakvertiesto whiteones,then}

M − M ₀

^{onsists of doubled edges oriented in both diretions and oriented} alternating ylesof length

≥ 4

^{,seefor example Figure3.3. Returning tothedenition oftheheight}

funtion,onenotiesthattheheighthangesby

± 1

^{exatlywhenrossingayle,andthe}

sign onlydependson theorientation of the yle.

•

^{By takinga dierent hoie of referene ow, one an reover Thurston's height funtion}

inthe aseof lozenge and domino tilings, up to a global multipliative fator of

1 3

^inthe

rst ase, and

1

4

^{in theseond - the interested reader an try and work out this relation}

expliitly.

Let us now onsider the toroidal graph

G _n = G /n Z ²

. In this ase, the height funtion is not well dened sine there might besome period,or heighthange, alongyles inthedualgraph

winding around the torus horizontally or vertially, see Figure 3.3. More preisely, a perfet

mathing

M

^of

G _n

an be liftedto a perfet mathingof theinnitegraph

G

,also denoted

M

^.

Then, theperfetmathing

M

^of

G _n

issaid tohave height hange

(h ^M _x , h ^M _y )

^if:

h ^M ( f + (n, 0)) = h ^M ( f ) + h ^M _x h ^M ( f + (0, n)) = h ^M ( f ) + h ^M _y .

Note that height hange is well dened, i.e. does not depend on the hoie of fae

f

, beause theow

ω ^M − ω ^{M 0}

^is divergene-free.

Figure3.3givesanexampleoftherefereneperfetmathing

M ₀

^{induedbyaperiodireferene}

mathingof

G

,andoftheheighthangeomputationofadimeronguration

M

^{ofthetoroidal}

graph

G ₂

of the square-otagongraph. Theperfetmathing

M

^{hasheight hange}

(0, 1)

^.

0 f ₀

M M

0

0 0

0 0 0

−1

−1 0 −1

0 0

0 0 0

−1 0 −1 0 −1

−1

−1

−1

−1

−1

Figure3.3: Aperfet mathing

M

^{ofthe toroidal graph}

G ₂

,having height hange

(0, 1)

^.

Remark 6. Let

T ² = { (z, w) ∈ C ² : | z | = | w | = 1 }

^denotethetwo-dimensional unit torus,and let

H ₁ ( T ² , Z ) ∼ = Z ²

be the rst homology group of

T ²

in

Z

. The graph

G _n

being embedded in

T ²

,we takeasrepresentative of abasisof

H ₁ ( T ² , Z )

^thevetors

ne x

^and

ne y

^{embedded onthe}

torus,where reall

{ e x , e y }

^{were our hoie ofbasisvetor for}

Z ²

,seeFigure3.1. Inthease of thesquare-otagon graphand

G ₂

ofFigure3.3, the rstbasisvetoristhelightgrey horizontal yle oriented from left to right, and the seond is the light grey vertial yle oriented from

bottomto top. Then, the homology lassofthe orientedsuperimposition

M − M ₀

^inthisbasis

is

(1, 0)

^{, and the height hange is}

(0, 1)

^{. More generally, if the homology lass of}

M − M ₀

^is

(a, b)

^{, thentheheight hange}

(h ^M _x , h ^M _y )

^is

( − b, a)

^{. Thisis beause,asmentioned inRemark5,}

theheight funtion hanges by

± 1

^{exatlywhen itrossesorientedyles of}

M − M ₀

^,implying

that the height hange

(h ^M _x , h ^M _y )

^{an be identied through the} intersetion pairing with the homology lassoftheorientedonguration

M − M 0

ⁱⁿ

H 1 ( T ² , Z )

^.

Letusreallthesetting:

G

isasimple,planar,innite,bipartite

Z ²

-periodigraph,and

{ G _n } ⁿ ≥ 1

is the orresponding toroidal exhaustion. Edges of

G

are assigned a periodi, positive weight funtion

ν

^,and

{ e x , e y }

^{denotes ahoie ofbasisof theunderlying lattie}

Z ²

.

Inthis setionwepresent theomputation ofthepartition funtion for dimerongurations of

thetoroidal graph

G _n

,and the omputation of the free energy,whih is minus theexponential growth rate of the partition funtion of the exhaustion

G _n

. More preisely, Setion 3.2.1 is devoted to Kasteleyn theory on the torus. Then, in Setion 3.2.2 we dene the harateristi

polynomial, one of the keyobjets underlying thedimer model on bipartite graphs, yielding a

ompat losed formula for the partition funtion, see Corollary 10. With this expression at

hand,one thenderivestheexpliit formula forthe free energy,seeTheorem 11.

3.2.1 Kasteleyn matrix

InSetion 2.3, we proved thatthe partition funtionof anite, simplyonneted planargraph

is given bythedeterminant of aKasteleyn matrix. Whenthegraph isembedded on thetorus,

a single determinant is not enough, what is needed is a linear ombination of determinants

determined inthe following way.

Consider the fundamental domain

G ₁

and let

K ₁

^{be a Kasteleyn matrix of this graph, that is}

K ₁

^{is the oriented, weighted, adjaeny matrix of}

G ₁

for a hoie of admissible orientation of theedges. Notethatadmissible orientationsalsoexistforgraphsembeddedinthetorus. Letus

nowlookat the signsof the weighted mathings in the expansion of

det(K ₁ )

^{. Consider a xed}

referene mathing

M ₀

^of

G ₁

,and let

M

^{be any other mathing of}

G ₁

. Then, bythe results of [Kas67 ,DZM

+

96,GL99 ,Tes00 ,CR07℄,the signofthemathing

M

^{onlydependson theparity}

ofthe vertial andhorizontalheighthange. Moreover,of thefour possible paritylasses,three

havethesamesignin

det(K ₁ )

^{andonehasoppositesign. Byan}appropriatehoieofadmissible orientation,one an makethe

(0, 0)

^{lasshave positive sign.}

Let

γ x

^,

γ y

^{bethe basisvetor}

e x

^,

e y

^{of the underlying lattie}

Z ²

,embedded onthetorus. Sine we have hosen

Z ²

to be asubgraphof thedualgraph

G ∗

,

γ x

^,

γ y

^{areorientedylesofthedual}

graph

G ∗

1

^{,seeFigure 3.4. We refer to}

γ _x

^{asa horizontalyle, and to}

γ _y

^{asa vertialone.}

G ₁

γ γ

x y

Figure3.4: Fundamental domain

G ₁

of the squareotagongraphwiththeorientedpaths

γ x , γ y

inthedualgraph

G ∗

1

^{. Thetwo opies of}

γ _x

^,respetively

γ _y

^{,aregluedtogether.}

For

σ, τ ∈ { 0, 1 }

^{, let}

K ₁ ^θτ

^{be the Kasteleyn matrix in whih the weights of theedges rossing}

the horizontal yle

γ x

^{are multiplied by}

( − 1) ^θ

^{, and those rossing the vertial yle}

γ y

^are

multiplied by

( − 1) ^τ

^{. Observing that hanging the signsalong a horizontal dual yle has the}

eet of negating theweight of mathings withodd horizontal height hange, and similarly for

vertial;thefollowing tableindiates thesignofthemathings intheexpansionof

det(K ₁ ^θτ )

^,as

a funtionof the horizontaland vertial height hange mod

2

^.

(0, 0) (1, 0) (0, 1) (1, 1)

det(K ₁ ⁰⁰ ) + − − −

det(K ₁ ¹⁰ ) + + − +

det(K ₁ ⁰¹ ) + − + +

det(K ₁ ¹¹ ) + + + −

(3.1)

From Table3.1, we dedue

Z( G ₁ ) = 1

2 − det(K ₁ ⁰⁰ ) + det(K ₁ ¹⁰ ) + det(K ₁ ⁰¹ ) + det(K ₁ ¹¹ )

.

^(3.2)

A similar argument holds for the toroidal graph

G _n

,

n ≥ 1

^{. The} orientation of edges of

G ₁

denes a periodi orientation of edges of

G

, and thus an orientation of edges of

G _n

. Let

γ _x,n

^,

γ y,n

^{betheorientedyles inthedualgraph}

G ^∗ _n

,obtainedbytaking

n

^{timesthebasisvetor}

e x

^,

n

^{times the basisvetor}

e y

respetively,embedded on the torus. For

σ, τ ∈ { 0, 1 }

^{, let}

K _n ^στ

^be

thematrix

K _n

^{inwhihtheweightsof edgesrossingthehorizontal yle}

γ _x,n

^{aremultipliedby}

( − 1) ^θ

^{,andthose rossingthe vertialyle}

γ y,n

^{aremultiplied by}

( − 1) ^τ

^{. Then,}

Theorem 7. [Kas67, DZM

+

96,GL99 , Tes00, CR07, CR08℄

Z ( G _n ) = 1

2 − det(K _n ⁰⁰ ) + det(K _n ¹⁰ ) + det(K _n ⁰¹ ) + det(K _n ¹¹ ) .

3.2.2 Charateristi polynomial

Let

K ₁

^{be a Kasteleyn matrix of the} fundamental domain

G ₁

. Given omplex numbers

z

^and

w

^{, an altered Kasteleyn matrix}

K ₁ (z, w)

^{is onstruted as follows. Let}

γ _x

^,

γ _y

^{be the oriented}

horizontal and vertial yles of

G ∗

1

^{. Then, multiply} edge-weights of edges rossing

γ x

^by

z

whenever the white vertex is on the left, and by

z ^{− 1}

^{whenever the blak vertex is on theleft.}

Similarly,multiply edge-weightsof edgesrossing thevertial path

γ _y

^by

w ^{± 1}

^{,seeFigure 3.5.}

The harateristi polynomial

P (z, w)

^{of the graph}

G ₁

is dened as the determinant of the alteredKasteleyn matrix:

P (z, w) = det(K ₁ (z, w)).

Aswewillsee,theharateristipolynomialontainsmostoftheinformationonthemarosopi

behaviorof the dimermodel onthegraph

G

.

The next very useful lemma expresses the harateristi polynomial using height hanges of

dimerongurations of

G ₁

. Referto Setion 3.1for denitions andnotations onerningheight hanges. Let

M ₀

^{beareferenedimerongurationof}

G ₁

,andsupposetheadmissibleorientation oftheedgesishosensuhthatperfetmathingshaving

(0, 0)

^mod

(2, 2)

^{heighthangehave+}

signintheexpansionof

det(K ₁ )

^{. WeonsiderthealteredKasteleynmatrix}

K ₁ (z, w)

^onstruted

from

K ₁

^{. Let}

ω ^{M 0}

^{be therefereneow}orrespondingto thereferenedimeronguration

M ₀

^,

andlet

x ₀

^{denote thetotal uxof}

ω ^{M 0}

^aross

γ _x

^,similarly

y ₀

^{isthetotaluxof}

ω ^{M 0}

^aross

γ _y

^.

Then, we have:

Lemma 8. [KOS06℄

P (z, w) = z ^{x 0} w ^{y 0} X

M ∈M ( G ₁ )

ν(M )z ^{h M x} w ^{h M y} ( − 1) ^{h M x h M y +h M x +h M y} ,

^(3.3)

where, for every dimeronguration

M

^of

M ( G ₁ )

^,

(h ^M _x , h ^M _y )

^{is the height hange of}

M

^.