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,therstquestionaddressedisthatof

omput-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)

...

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:

whih isthe same asthe signof the produtofthenumerator andthedenominator. Now,

( ⋄ 1 ) : =

^sgn

(σ)

^sgn

(τ ) =

^sgn

(σ ◦ τ ) =

^sgn

(σ ◦ σ ◦ c)

^{byEquation(2.1)}

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

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

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 ) =

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

equal to:

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.

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

orien-tation.

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

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

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

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). This

onstru-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 the

trian-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

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

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