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

^,ând ôers^more însight

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

T

whih is tileable by lozenges, and a lozenge tiling

T

^of

X

. Then theheight funtion

h ^T

îs ân înteger

valuedfuntion onverties of

X

,dened indutively asfollows:

•

^Fix^a ^vertex

v ₀

X

,and set

h ^T ( v ₀ ) = 0

•

^Fôr êvery ^boundary êdge

uv

of a lozenge,

h ^T ( v ) − h ^T ( u ) = +1

^if^the ^edge

uv

is oriented from

u

v

,implyingthat

h ^T ( v ) − h ^T ( u ) = − 1

^when^the ^edge

uv

is orientedfrom

v

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

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 ân înteger^valued ^funtion ôn ^the ^verties ôf

X

,satisfying:

• h( v ₀ ) = 0

^,^where

v ₀

isa xed vertex of

X

• h( v ) − h( u ) = 1

^for ^any ^boundary ^edge

uv

X

oriented from

u

v

• h( v ) − h( u ) = 1

^or

− 2

^for âny înterior êdge

uv

X

oriented from

u

v

Then,there isa bijetion between funtions

h

^satisfying^these ^two ônditions, ând^tilings ôf

X

Proof. Let

T

^be ^a ^lozenge ^tiling ^of

X

and let

uv

be an edgeof

X

, oriented from

u

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

ⁱⁿ^the^rst ^ase, ^and

− 2

ⁱⁿ^the^seond.

Conversely,let

h

^beânînteger^funtionâsⁱⁿ^the^lemma. ^Letûsônstrutâ^tiling

T

^whose^height

funtion is

h

^. ^Consider ^a ^blak^fae^of

X

,thenthere isexatly one edge

uv

ontheboundary of this faewhose height hange is

− 2

^. ^To^this ^fae, ^we ^assoiate ^the ^lozenge ^whih ^is^rossed ^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 )

îs ^simple, ^planar, înnite, ^bipartite, ând

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 }

^, âllowing ^to ^reord ôpies ôf â

vertex

v

G

{ v + (k, l) : (k, l) ∈ Z ² }

^. ^Refer^to^Figure^3.1^for^an^example ^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 )

^be^the^quotient^of

G

bythe ationof

n Z ²

. Thenthesequeneofgraphs

{ G _n } n ≥ 1

îsânêxhaustionôf^theînnite^graph

G

bytoroidalgraphs. Thegraph

G ₁ = G / Z ²

isalledthe fundamental domain,see Figure3.2. Assumethatedgesof

G ₁

areassigned a positive weight funtion

ν

^thus ^deningâ ^periodi^weight ^funtionôn ^theêdgesôf

G

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 ôf ^this ^hapter îs ^to ûnderstand ^the ^dimer ^model ôn

G

, using the exhaustion

{ G _n } ⁿ ≥ 1

^,^by^taking^limits^as

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

^are^assumed^to^be

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

^. ^Sine^we^are^working^on^the^dimer^model

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 ^real^valued ^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

^,îsâ ^real ^valued ^funtion ^dened ôn

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

^as^follows. ^The^ow

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

ω ^M

^has^divergene ¹ât êvery^white ^vertex,ând ^-1ât êvery ^blakône, ^thatîs:

∀ w ∈ W , div ω ^M ( w ) = X

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

^, ^the^dierene

ω ^M − ω ^M ⁰

^is ^adivergene-free ow,that is:

We arenow readyto dene theheight funtion. Let

M

^be â^dimerôngurationôf

G

,thenthe height funtion

h ^M

îs ân ⁱⁿ^teger ^valued^funtion ôn ^faesôf

G

dened asfollows.

•

^Fix^a ^fae

f ₀

G

,andset

h ^M ( f ₀ ) = 0

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

^. Ân ânalogôf

Lemma 4 givesa bijetion between height funtionsanddimerongurations of

G

. Remark 5.

•

^There îs âtually ân êasy ^wayôf ômputing ^the^height ^funtion. ^Reall ^from ^Setion ^2.3

thatthesuperimpositionoftwodimerongurations

M

^and

M ₀

ônsistsôf^doubledêdges

andalternating ylesoflength

≥ 4

^(yles^may^extend^to ^innity^when

G

isinnite). Let usdenoteby

M − M ₀

^the^orientedsuperimpositionof

M

^and

M ₀

^,^with^edges^of

M

^oriented

fromwhitevertiesto blakones,and thoseof

M ₀

^from^blak^verties^to ^white^ones,^then

M − M ₀

ônsists ôf ^doubled êdges ôriented ⁱⁿ ^both ^diretions ând ôriented alternating ylesof length

≥ 4

^,^see^for ^example ^Figure^3.3. ^Returning ^to^the^denition ^of^the^height

funtion,onenotiesthattheheighthangesby

± 1

êxatly^when^rossingâ^yle,ând^the

sign onlydependson theorientation of the yle.

•

^By ^takingâ ^dierent ^hoie ôf ^referene ôw, ône ân ^reover ^Thurston's ^height ^funtion

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

1 3

ⁱⁿ^the

rst ase, and

1

4

ⁱⁿ ^the^seond ^- ^the înterested ^reader ân ^try ând ^work ôut ^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

^indued^by^a^periodi^referene

mathingof

G

,andoftheheighthangeomputationofadimeronguration

M

^of^the^toroidal

graph

G ₂

of the square-otagongraph. Theperfetmathing

M

^has^height ^hange

(0, 1)

0 f ₀

M M

0 0 0

−1

−1 0 −1

0 0

0 0 0

−1 0 −1 0 −1

−1

Figure3.3: Aperfet mathing

M

^of^the ^toroidal ^graph

G ₂

,having height hange

(0, 1)

Remark 6. Let

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

^denote^thetwo-dimensional unit torus,and let

H ₁ ( T ² , Z ) ∼ = Z ²

be the rst homology group of

T ²

Z

. The graph

G _n

being embedded in

T ²

,we takeasrepresentative of abasisof

H ₁ ( T ² , Z )

^the^vetors

ne x

^and

ne y

^embedded ^on^the

torus,where reall

{ e x , e y }

^were ^our ^hoie ^of^basis^vetor ^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

ⁱⁿ^this^basis

(1, 0)

^, ^and ^the ^height ^hange ^is

(0, 1)

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

M − M ₀

^is

(a, b)

^, ^then^the^height ^hange

(h ^M _x , h ^M _y )

^is

( − b, a)

^. ^This^is ^beause,^as^mentioned ⁱⁿ^Remark^5,

theheight funtion hanges by

± 1

êxatly^when ît^rossesôriented^yles ôf

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 â^hoie ôf^basisôf ^theûnderlying ^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 â ^Kasteleyn ^matrix ôf ^this ^graph, ^that îs

K ₁

îs ^the ôriented, ^weighted, âdjaeny ^matrix ôf

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 âny ôther ^mathing ôf

G ₁

. Then, bythe results of [Kas67 ,DZM

+

96,GL99 ,Tes00 ,CR07℄,the signofthemathing

M

^only^depends^on ^the^parity

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

havethesamesignin

det(K ₁ )

ândône^hasôpposite^sign. ^Byânappropriatehoieofadmissible orientation,one an makethe

(0, 0)

^lass^have ^positive ^sign.

Let

γ x

γ y

^be^the ^basis^vetor

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

âreôriented^ylesôf^the^dual

graph

G ∗

1

^,^see^Figure ^3.4. ^W^e ^refer ^to

γ _x

âsâ ^horizontal^yle, ând ^to

γ _y

âsâ ^vertialône.

G ₁

γ γ

x y

Figure3.4: Fundamental domain

G ₁

of the squareotagongraphwiththeorientedpaths

γ x , γ y

inthedualgraph

G ∗

1

^. ^The^two ^opies ^of

γ _x

^,respetively

γ _y

^,^are^glued^together.

For

σ, τ ∈ { 0, 1 }

^, ^let

K ₁ ^θτ

^be ^the ^Kasteleyn ^matrix ⁱⁿ ^whih ^the ^weights ^of ^the^edges ^rossing

the horizontal yle

γ x

^are ^multiplied ^by

( − 1) ^θ

^, ^and ^those ^rossing ^the ^vertial ^yle

γ y

^are

multiplied by

( − 1) ^τ

^. Ôbserving ^that ^hanging ^the ^signsâlong â ^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

^be^the^oriented^yles ⁱⁿ^the^dual^graph

G ^∗ _n

,obtainedbytaking

n

^times^the^basis^vetor

e x

n

^times ^the ^basis^vetor

e y

respetively,embedded on the torus. For

σ, τ ∈ { 0, 1 }

^, ^let

K _n ^στ

^be

thematrix

K _n

ⁱⁿ^whih^the^weights^of ^edges^rossing^the^horizontal ^yle

γ _x,n

^are^multiplied^by

( − 1) ^θ

^,^and^those ^rossing^the ^vertial^yle

γ y,n

^are^multiplied ^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)

îs ônstruted âs ^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 ^the^left.

Similarly,multiply edge-weightsof edgesrossing thevertial path

γ _y

^by

w ^± ¹

^,^see^Figure ^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

^beâ^referene^dimerôngurationôf

G ₁

,andsupposetheadmissibleorientation oftheedgesishosensuhthatperfetmathingshaving

(0, 0)

^mod

(2, 2)

^height^hange^have⁺

signintheexpansionof

det(K ₁ )

^. ^We^onsider^the^altered^Kasteleyn^matrix

K ₁ (z, w)

^onstruted

from

K ₁

^. ^Let

ω ^M ⁰

^be ^the^referene^oworrespondingto thereferenedimeronguration

M ₀

andlet

x ₀

^denote ^the^total ^ux^of

ω ^M ⁰

^aross

γ _x

^,^similarly

y ₀

îs^the^totalûxôf

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

Proof. Let

M

^be ^a ^perfet ^mathing ^of

G ₁

. Then, by the hoie of Kasteleyn orientation, the signofthetermorrespondingto

M

ⁱⁿ^the^expansion^of

det(K 1 (z, w))

^is:

+

^if

(h ^M _x , h ^M _y ) = (0, 0) mod (2, 2)

^,^and

−

êlse. ^Thisân ^be^summarized âs:

( − 1) ^h ^M ^x ^h ^M ^y ^+h ^M ^x ^+h ^M ^y .

Let us denote by

ν z,w

^, ^the ^weight ^funtion ôn êdges ôf

G ₁

obtained from

G ₁

by multiplying edge-weights of edgesrossing

γ _x

^by

z ^± ¹

^,^and ^those ^rossing

γ _y

^by

w ^± ¹

^as^above. ^Then,

ν z,w (M) = ν (M )z ⁿ ^wb ^x ^(M ⁾ ⁻ ⁿ ^bw ^x ^(M) w ⁿ ^wb ^y ^(M ⁾ ⁻ ⁿ ^bw ^y ^(M ⁾ ,

where

n ^wb _x (M)

îs ^the ^numberôf êdges ôf

M

^rossing

γ x

^, ^whih ^have ^a ^white ^vertex^on ^the ^left

γ x

^, ^and

n ^bw _x (M )

îs ^the^number ôf êdges ôf

M

^rossing

γ x

^, ^whih ^have ^a ^blak ^vertex^on ^the

leftof

γ _x

^. ^The ^denition^of

n ^wb _y (M )

n ^bw _y (M)

^is^similar ^with

γ _x

^replaed^by

γ _y

Returning to the denition of the height hange for dimer ongurations of thetoroidal graph

G _n

,wehave:

Example 3.1. Letus ompute theharateristi polynomial ofthe fundamentaldomain

G ₁

of thesquare-otagongraph,withweightsoneontheedges. Figure3.5(left)desribesthelabeling

of the verties, and weights of the altered Kasteleyn matrix. The orientation of the edges is

admissible and is suh that perfet mathings having

(0, 0)

^height ^hange ^with ^respet ^to ^the

referenemathing

M ₀

^given ôn ^the ^right, ^have â ⁺^signⁱⁿ^theêxpansion ôf

det(K ₁ )

γ

Figure 3.5: Left: labeling of the verties of

G ₁

, edge-weights of the altered Kasteleyn matrix, hoieof admissible Kasteleynorientation. Right: hoie of refereneperfet mathing

M ₀

Thealtered Kasteleynmatrix

K 1 (z, w)

^is:

K 1 (z, w) =

mathing

M 0

ôf ^Figure ^3.5 ^(right), ^the ^total ûx ôf

ω ^M ⁰

^through

γ x

^and

γ y

^is

(0, 0)

^, ^so ^that

z ^x ⁰ w ^y ⁰ = 1

^. ^Sine ^all ^edges ^have ^weight ^1,

ν(M) ≡ 1

^. ^Figure ^3.6 ^desribes ^the ⁹ ^perfet

mathingsof

G ₁

,withtheir respetive height hange. Asa onsequene,theontribution ofthe perfet mathings of

G ₁

to the right hand side of (3.3) is

1

^for êah ôf ^the ⁵ ^rst ônes,

− w

respetively,

− w ¹

− z

− ¹ z

^for ^the^last ^four ^ones. ^Combining ^the^dierent ontributions yields, asexpeted, the harateristi polynomialasomputed inEquation (3.4) .

4 matchings with height change (0,0) matching with height change (0,0) or

matching with height change (0,1)

(0,−1), (1,0) and (−1,0) + 3 symmetries with height change

0

Figure3.6: 9 possibledimerongurations of

G ₁

withtheir height hange.

Inthespeiasewhen

(z, w) ∈ {− 1, 1 } ²

ⁱⁿ

K ₁ (z, w)

^,^on^reovers^the^four^matries

(K ₁ ^θτ ) _θ,τ _∈{ _0,1 _}

Using Equation(3.2),wereoverthatthenumberof dimer ongurations of

G ₁

is:

Z( G ₁ ) = 1

2 ( − P (1, 1) + P( − 1, 1) + P (1, − 1) + P ( − 1, − 1)) = 9.

^(3.5)

Charateristi polynomials oflargergraphs maybe omputedreursively asfollows. Let

K n

^be

aKasteleynmatrixofthegraph

G _n

asabove,andlet

γ x,n

^and

γ y,n

^be^the^horizontal^and^vertial

yles of

G ∗

Proof. Theproofisageneralization of[CKP01℄wherethesameresultisobtainedfor thegraph

G = Z ²

. We only give theargument when

z = w = 1

^. ^The ^proof ^for ^general

z, w

^'s ^follows ^the

Let

W _n

,respetively

B _n

,denotethe set ofwhite, respetively blak, verties of

G _n

. The ideais tousethetranslationinvariane ofthegraph

G _n

andofthematrix

K _n

^to^blokdiagonalize

K _n

and toompute itsdeterminant by omputingthedeterminant ofthedierent bloks.

Let

C ^W ⁿ

be the setof omplex-valued funtions onwhite verties

W _n

,and

C ^B ⁿ

those on blak deompositionof

C ^B ⁿ

,onsistingofeigenvetorsof

T ^B ⁿ

^. ^Theeigenvaluesof

T ^B ⁿ

^are^the^produts

n

^-th ^roots ^of ^unity:

(α j β k ) _j,k _∈{ _0,...,n ₋ ₁ _}

^, ^where

α j = e ⁱ ^2πj ⁿ

β k = e ⁱ ^2πk ⁿ

^(as â ônsequene ôf

n

^-th ^roots ^of ^unity:

(α j β k ) _j,k _∈{ _0,...,n ₋ ₁ _}

^, ^where

α j = e ⁱ ^2πj ⁿ

β k = e ⁱ ^2πk ⁿ

^(as â ônsequene ôf

Dans le document The die de (Page 18-0)

Geometri interpretation of lozenge tilings

Z 3

2 + 1

T

X

T

T

X

h T

X

•

v 0

X

h T ( v 0 ) = 0

•

uv

h T ( v ) − h T ( u ) = +1

uv

u

v

h T ( v ) − h T ( u ) = − 1

uv

v

u

0

Z f 3

Z f 3

Z 3

X

T

h

X

• h( v 0 ) = 0

v 0

X

• h( v ) − h( u ) = 1

uv

X

u

v

• h( v ) − h( u ) = 1

− 2

uv

X

u

v

h

X

T

X

uv

X

u

v

uv

1

− 2

h

T

h

X

uv

− 2

uv

X

•

G = ( V , E )

Z 2

G

G

Z 2

G ∗

{ e x , e y }

v

G

{ v + (k, l) : (k, l) ∈ Z 2 }

G

e x e y

Z 2

{ e x , e y }

Z ³

h ^T

v ₀

h ^T ( v ₀ ) = 0

h ^T ( v ) − h ^T ( u ) = +1

h ^T ( v ) − h ^T ( u ) = − 1

Z f ³

Z f ³

Z ³

• h( v ₀ ) = 0

v ₀

Z ²

Z ²

{ e _x , e _y }

{ v + (k, l) : (k, l) ∈ Z ² }

e _x e _y

Z ²

{ e _x , e _y }

G _n = ( V _n , E _n )

n Z ²

{ G _n } n ≥ 1

G ₁ = G / Z ²

G ₁

G ₁

G ₁

{ G _n } ⁿ ≥ 1

{ G _n } ⁿ ≥ 1

G _n

G ^∗

ω ^M

ω ^M

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

ω ^M

∀ w ∈ W , div ω ^M ( w ) = X

M ₀

ω ^M ⁰

ω ^M

ω ^M − ω ^M ⁰

h ^M