Charateristi polynomial

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

^.

Proof. Let

M

^{be a perfet mathing of}

G ₁

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

M

^{intheexpansionof}

det(K 1 (z, w))

^is:

+

^if

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

^,and

−

^{else. Thisan besummarized as:}

( − 1) ^{h M x h M y +h M x +h M y} .

Let us denote by

ν z,w

^{, the weight funtion on edges of}

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}

^{asabove. 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)

^{is the numberof edges of}

M

^rossing

γ x

^{, whih have a white vertexon the left}

of

γ x

^{, and}

n ^bw _x (M )

^{is thenumber of edges of}

M

^rossing

γ x

^{, whih have a blak vertexon the}

leftof

γ _x

^{. The denitionof}

n ^wb _y (M )

^,

n ^bw _y (M)

^{issimilar with}

γ _x

^replaedby

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 on the right, have a +signintheexpansion of}

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

^{of Figure 3.5 (right), the total ux of}

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 eah of the 5 rst ones,}

− w

^,

respetively,

− w ¹

^,

− z

^,

− ¹ z

^{for thelast four ones. Combining thedierent} 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)

^{,onreoversthefourmatries}

(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

^{bethehorizontalandvertial}

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

^toblokdiagonalize

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}

^{. The}eigenvaluesof

T ^{B n}

^{aretheproduts}

of

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 a onsequene of}

Letus showthat

K n

representedinthis basis isblokdiagonal, witha blok ofsize

| V ( G ₁ ) |

,andthematrix

K n

^{writteninthebasis}

E

isblokdiagonal.

For all

w ∈ W ₁

,

b ∈ B ₁

,the

( w , b )

^{-oeient oftheblok}orrespondingto theeigenvalue

α j β k

Asaorollaryto Theorems 7and 9,wehave anexpliitexpressionfor thepartitionfuntion of

G _n

asafuntion of the harateristi polynomial

P

^.

Corollary 10. When the Kasteleyn orientation ishosen suh that the signtable of the

funda-mentaldomain isgiven by Table 3.1,then forevery

n ≥ 1

^,

Thepartition funtion isgrowing withthesize ofthegraph. Anatural questionto asknowis:

whatisthisgrowthrate? Inordertoformulatethequestionorretly,weneedtoknowabout

theorderofmagnitudeofthis growth. Intuitiontellsusthatitisgoingto beexponential inthe

areaof thegraph:

Z ( G _n ) ∼ e ^{cn 2}

^{. Thus theright quantity to lookat is:}

n lim →∞

1

n ² log Z ( G _n ).

Minus this quantity isknown asthe free energy.

Theorem 11. [CKP01,KOS06 ℄ Undertheassumptionthat

P (z, w)

^{hasonlya nitenumber of}

zeros on the unit torus

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

^{, we have:}

Proof. Sine

Z _n ^θτ

^{ounts some dimer} ongurations of

G _n

with the wrong sign, we have the following bound:

Z _n ^θτ ≤ Z( G _n ).

Moreover, looking at Table3.1, we dedue:

− Z _n ⁰⁰ ≤ +Z _n ¹⁰ + Z _n ⁰¹ + Z _n ¹¹ ,

exist. ByTheorem 9,we have:

1

n ² log Z _n ^θτ

^{anbewritteninasimilarway. Thesefour termslooklikeRiemann}

sums for the integral:

too lose to the zeros of

P(z, w)

^{mayexplode. By using thevery areful argument of Theorem}

7.3. of [CKP01℄, one an hek that this will not happen, and so the Riemann sum of the

maximumonverges to the integral

I

^.

Theproof isonluded byobserving thatsine

P (e ^{− iθ} , e ^{− iτ} ) = P (e ^iθ , e ^iτ )

^,

Example. The free energy of the dimer model on the square-otagon graph with uniform

weights is:

Notethat itisingeneral hard to expliitly ompute thisintegral.

3.3 Gibbs measures

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

M ( G )

^{oftheinnitegraph}

G

whihare,insomeappropriatesense,innitevolumeversionsofthe Boltzmann measureon

M ( G _n )

^{. Reallthat bydenition, the}probabilityof 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 a ylinder set onsists of all perfet mathings ontaining a}

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

^{,andtheir inverses.}

Usingtheblokdiagonalizationofthematries

K _n ^στ

^{oftheproofofTheorem9,oneanompute}

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

Riemann sums is againompliated bythe zerosof

P (z, w)

^{on thetorus}

T ²

,but an be shown toonvergeonasubsequeneof

n

^{'stotherighthandsideofEquation(3.7) . UsingaTheoremof}

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 ))

^{whihoinideswiththelimitoftheBoltzmannmeasuresonylinder sets,where}

σ( A )

^{isthesmallestsigma-eldontainingylindersets. ThislimitingmeasureisofGibbstype}

by onstrution. Wehave thus skethed theproof ofthefollowing theorem.

1

DLRstandsforDobrushin,LanfordandRuelle

Theorem12. [CKP01 ,KOS06 ℄Let

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

^{beasubset ofedgesof}

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

^{is a Kasteleynmatrix assoiated tothe 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)

^isthe

( b , w )

^{element of the adjugatematrix (transpose of the ofatormatrix) of}

K ₁ (z, w)

^{. Itis a polynomial in}

z, w, z ^{− 1} , w ^{− 1}

^.

3.3.2 Ergodi Gibbs measures

Inthe previous setion,wehave expliitlydetermined aGibbsmeasure on

M ( G )

^{. We 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 theslope}

(s, t)

^{tobetheexpetedhorizontalandvertial}

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 )

^{. For 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 theonditional measure indued by}

µ _n

on

M s,t ( G _n )

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

M ( G )

^{isgiven bythe}

following theoremofSheeld.

Theorem 13. [She05 ℄For eah

(s, t)

^{for whih}

M s,t ( G _n )

^{is non-emptyfor}

n

^{suientlylarge,}

µ _n (s, t)

^{onverges as}

n → ∞

^{to an EGM}

µ(s, t)

^{of slope}

(s, t)

^. Furthermore

µ _n

^{itselfonverges}

to

µ(s ₀ , t ₀ )

^where

(s ₀ , t ₀ )

^{is the limit of the slopes of}

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)

^{as above; that is}

µ(s, t)

^{is the unique ergodi}

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}

^,thatis:

N (P ) =

^onvexhull

{ (i, j) ∈ Z ² | z ^{i+x 0} w ^{j+y 0}

^{is amonomial 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)

^{if and onlyif}

(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 )

^{,thenthere isa Gibbsmeasure ofslope}

(s, t)

^,or

equiva-lently

M s,t ( G _n )

^{is non-emptyfor}

n

^{large enough. For onveniene, we will assumethat theset}

of possible slopesis losed.

ByLemma8,theabsolutevalueoftheoeient

z ⁱ w ^j

ⁱⁿ

P (z, w)

^{istheweightedsumof}

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)

^{isanextremalpointof}

N (P)

^,then

M s,t ( G ₁ ) 6 = ∅

^{. Itsuestoshow}

thatif

M s 1 ,t 1 ( G _{n 1} )

^and

M s 2 ,t 2 ( G _{n 2} )

^{arenon-emptyfor 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

^{. Indeed, by indution, this allows to prove existene of a Gibbs}

measure of slope

(s, t)

^{for a dense subset of the Newtonpolygon. The proofis ended by using}

theassumption thatthe set ofpossibleslopesis losed.

Withoutlossof generality,weanassumethat

n ₁ = n ₂

^{,otherwisetake thelmoftheir 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 ⌋ )

^ofthe

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

H 1 ( T ² , Z )

^,sothat

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 )

^{isnon-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 )

^{. Denoteby}

~ E ₁

thesetofdiretededgesofthefundamentaldomain

G ₁ = ( V ₁ , E ₁ )

^.

Realling thatthe divergene

div

^{is alinearfuntionof ows,thesetof} 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 assigns to a ow}

ω

^{the total uxaross}

γ x

^and

γ y

^{isa linearmapping}

fromthepolytopeto

R ²

,implyingthattheimage ofthepolytopeunder

ψ

^{istheonvexhull of}

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 }

^{oneverydiretededgeof}

~ 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)

^{,theimage ofa dimeronguration under}

ψ

^{is its height hange. This means thatthe image of extremal pointsof the polytopeontains}

the extremal points of the Newton polygon

N (P )

^{; the image of the polytope under}

ψ

^{is thus}

N (P )

^.

TheGibbs measure

µ(s, t)

^{of slope}

(s, t)

^denesa 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)

^isaprobabilitymeasure,theow

ω ^µ(s,t)

^{hasdivergene1ateverywhitevertexand}

-1ateveryblakvertex. Itthusbelongstothepolytopeanditsimageunder

ψ

^belongsto

N (P )

^.

Theproof isonluded byobserving thatthe image of

µ(s, t)

^under

ψ

^{is theslope}

(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 )

^{bethe partition funtion of}

M s,t ( G _n )

^,thatis:

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 )

^ofslope

(s ₀ , t ₀ )

^{. Our goalnow isto obtainanexpliit expressionfor theGibbsmeasures}

µ(s, t)

^withall

possibleslopes

(s, t)

^.

ReallthatbyTheorem 13,theGibbsmeasure

µ(s, t)

^{isthelimit oftheonditionalBoltzmann}

measures

µ n (s, t)

^on

M s,t ( G _n )

^{. The problem is that onditional measures are hard objets}

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 arelooking 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

^{areorientedhorizontal and vertial yles inthe dualgraph}

G ∗

n

^obtained

by taking

n

^{times the basisvetor}

e x

^{of the underlying lattie}

Z ²

,

n

^{times the basisvetor}

e y

respetively,embeddedonthetorus. Then,on

G ∗

n

^thereare

n

^{horizontalopiesoftheyle}

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 thevertial yle}

γ _y,n

^{are multipliedby}

e ^{± B y}

^{. This denes amagneti}

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

^thetotaluxof

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 )}

^{astheweight funtion}

ν

^{,multiplied bya quantitywhih onlydepends}

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)

^betheharateristipolynomialofthegraph

G ₁

orrespondingtothemagneti altered weight funtion. The key fat is that

P _{(B x} ,B y ) (z, w)

^{an easily be expressed using the}

harateristipolynomial

P (z, w)

^ofthegraph

G ₁

: expressing

P _{(B x ,B y )} (z, w)

^{usingLemma8and}

replaing

ν _{(B x ,B y )} (M )

^{bythe right handside of (3.8)inthease 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 )

^{bethe partitionfuntionand}

µ ^(B n ^{x ,B y )}

^{betheBoltzmannmeasureofthegraph}

G _n

withthemagneti alteredweight funtion. Denote by

µ _{(B x ,B y )}

^{theGibbsmeasureobtained}

asweak limit of the Boltzmann measures

µ ^(B n ^{x ,B y )}

^{. Then, asa diret orollaryof Theorems 11}

and 12,we have:

So,itisquiteremarkablethattheresultsobtainedfortheweightfuntion

ν

^{alsoyieldtheresults}

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 )}

^{. Thisis postponeduntil Setion 3.4.2.}

3.4 Phases of the model

In this setion, we desribe one of the most beautiful results on the bipartite dimer model

obtained byKenyon, Okounkovand Sheeld [KOS06 ℄,namely thefulldesriptionofthephase

diagramofthe dimermodel. Itinvolvesmagnetieldoordinatesandanobjetfromalgebrai

geometry alledHarnakurves.

A way to haraterize phases is by the rate of deay of edge-edge orrelations. In the dimer

model, this amountsto studying asymptotisof

K _(B ^{− 1}

x ,B y )

^{. Indeed, let}

e ₁ = w ₁ b ₁

and

e ₂ = w ₂ b ₂

betwoedgesof

G

,whiharethoughtofasbeingfarawayfromeahother. Let

I _e

betherandom variablewhihis

1

^iftheedge

e

^{is present inadimer}onguration, and

0

^{else. Then, usingthe}

expliitexpression for theGibbsmeasure

µ _{(B x ,B y )}

^yields:

Cov( I _e

1 , I _e

2 ) = µ _{(B x ,B y )} ( e ₁ , e ₂ ) − µ _{(B x ,B y )} ( e ₁ )µ _{(B x ,B y )} ( e ₂ ),

= K _{(B x ,B y )} ( w ₁ , b ₁ )K _{(B x ,B y )} ( w ₂ , b ₂ )K _(B ^{− 1}

x ,B y ) ( b ₂ , w ₁ )K _(B ^{− 1}

x ,B y ) ( b ₁ , w ₂ ).

Theasymptotibehaviorof

K _(B ^{− 1}

x ,B y ) ( b , w + (x, y))

^(as

x ² +y ²

^{getslarge)dependsonthezerosof}

thedenominatorontheunittorus,i.e. on thezeros of

P (e ^{B x} z, e ^{B y} w)

^{ontheunittorus. Hene,}

thegoalis to studythe set:

{ (z, w) ∈ T ² : P (e ^{B x} z, e ^{B y} w) = 0 }

^,or equivalently,the set:

{ (z, w) ∈ C ² : | z | = e ^{B x} , | w | = e ^{B y} , P (z, w) = 0 } .

Thisisthe subjetof thenextsetion.

3.4.1 Amoebas, Harnak urves and Ronkin funtion

Theamoeba of apolynomial

P ∈ C [z, w]

^{intwoomplex variables,denotedby}

A (P )

^,isdened

asthe image of the urve

P (z, w) = 0

ⁱⁿ

C ²

underthemap:

(z, w) 7→ (log | z | , log | w | ).

When

P

^{is the}harateristi polynomial of a dimer model, the urve

P(z, w) = 0

^{is knownas}

thespetral urve of the dimer model. Note that a point

(x, y) ∈ R ²

isin theamoeba

A (P )

^,if

and only if

| z | = e ^x , | w | = e ^y

^,and

P (z, w) = 0

^{. Otherwise stated, a point}

(x, y) ∈ R ²

is inthe amoeba ifand onlyifthepolynomial

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

^{hasat leastone zeroonthe unittorus.}

The theory of amoebas is a fresh and beautiful eld of researh. The paper What is ... an

amoeba? in the noties of the AMS, by Oleg Viro gives a very nie overview of the results

obtained over a period of 8 years by [FPT00 , GKZ94 , Mik00, MR01℄. It provides a preise

geometri piture of the objet, whih heavily depends on the Newton polygon

N (P )

^of

Se-tion3.3.3. Loosely stated,an amoeba satisesthefollowing, seealso Figure3.8.

•

^{An amoeba reahes innity by several tentales. Eah tentale aommodates a ray and}

narrows exponentiallyfast towardsit, sothatthere isonly one rayineahtentale.

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