Phases of the model - The die de

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

^if^the^edge

e

^is ^present ⁱⁿ^a^dimeronguration, and

0

^else. ^Then, ^using^the

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

^gets^large)^depends^on^the^zeros^of

thedenominatorontheunittorus,i.e. on thezeros of

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

^on^the^unit^torus. ^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]

ⁱⁿ^two^omplex ^variables,^denoted^by

A (P )

^,^is^dened

asthe image of the urve

P (z, w) = 0

ⁱⁿ

C ²

underthemap:

(z, w) 7→ (log | z | , log | w | ).

When

P

^is ^theharateristi polynomial of a dimer model, the urve

P(z, w) = 0

^is ^known^as

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)

^hasât ^leastône ^zeroôn^the ûnit^torus.

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.

•

Ân âmoeba ^reahes înnity ^by ^several ^tentales. Êah ^tentale âommodates â ^ray ând

narrows exponentiallyfast towardsit, sothatthere isonly one rayineahtentale.

•

Êah ^ray îs ôrthogonal ^to â ^side ôf ^the ^Newton ^polygon

N (P )

^, ^and ^direted ^towards ^an

outward normalof theside.

•

^Fôrêah^sideôf

N (P )

^,^thereîsât^leastône^tentale âssoiated^toît. ^The^maximal^number

of tentales orresponding to a side of

N (P )

îs ^the ^number ôf ^piees ît îs ^divided ⁱⁿ ^by

integer lattiepoints.

•

^The ^amoeba's ^omplement

R ² \ A (P )

ônsists ôf ômponents ^bet^ween ^the^tentales, ând

bounded omponents. Eahbounded omponent orrespondstoa dierent integer lattie

point of

N (P )

^, ând ^the^maximal ^number ôf ^bounded ômponents îs ^the^total ^number ôf

interiorinteger lattie points of

N (P)

•

Êahônneted ômponent ôf ^theâmoeba's ômplement

R ² \ A (P )

^is^onvex.

•

^Planar ^amoebas^are^not ^bounded,^but

Area

( A (P )) ≤ π ²

^Area

(N (P )).

log|z|

log|w|

tentacle ray

Figure3.8: Newton polygon (left) and amoeba (right,shaded) of theharateristi polynomial

P (z, w) = 5 − z − ¹ _z − w − _w ¹

ôf ^the ûniform ^dimer^modelôn ^thesquare-otagongraph.

Oneofthe main toolsusedtostudy theamoeba ofapolynomial

P

^is^the^Ronkin^funtion

R

^of

this polynomial dened by:

∀ (x, y) ∈ R ² , R(x, y) = 1 (2πi) ²

Z

T ²

log | P (e ^x z, e ^y w) | dz z

dw w .

It hasthefollowing properties:

•

^The^Ronkin ^funtion ^is^onvex

•

Ît îs^linear ôn êah ômponent ôf ^theâmoeba ômplement, ând ^the^gradient îs ^the

orre-spondinginteger pointof

N (P )

Complex urves

P (z, w) = 0

^whose âmoeba ârea îs ^maximal, î.e. êqual ^to

π ²

^Area

(N (P ))

^, ^are

alled maximal. Curves with this property are very speial: they have real oeients and

their amoeba hasthe maximalnumberofomponents. Suhurveswerealready introdued by

Harnak in

1876

^,ând âre^knownâs^Harnak ûrves. ^There îsân alternative haraterization of Harnakurvesgiven in[MR01℄,whihismoreuseful here: areal urve

P(z, w) = 0

^is^Harnak

ifand only ifthe mapfrom theurve to its amoeba is at most

2

^-to-

1

^. ^It ^will ^be

2

^-to-

1

^,^with ^a

nite numberof possible exeptionswhere itmay be

1

^-to-

1

^(the ^boundary ôf^the âmoeba, ând

whenbounded omponents shrinkto apoint).

Oneof themost fruitfultheoremof thepaper[KOS06 ℄ isthefollowing.

Theorem 17. [KOS06, KO06 ℄ The spetral urve of a dimer model isa Harnak urve.

Con-versely, every Harnak urve arises as the spetral urve of some periodi bipartite weighted

dimer model.

For theproof ofthis theorem, we referto [KOS06 ,KO06 ℄.

3.4.2 Surfae tension revisited

In this setion, we relate the magneti eld oordinates to the slope of the measure

µ _(B _x _,B _y ₎

Thisisdone byexpressingthe surfae tensionasa funtion oftheRonkin funtion.

Let us assume that the referene ow

ω ^M ⁰

^of ^the fundamental domain

G ₁

ows by

0

^through

thepaths

γ x

γ y

^,^i.e.

x ₀ = y ₀ = 0

Theorem 18. [KOS06 ℄ The surfae tension is the Legendre transform of the Ronkin funtion

of the harateristi polynomial, i.e.

σ(s, t) = max

x,y {− R(x, y) + sx + ty } .

Proof. (Sketh) Let us sketh the proof that the Ronkin funtion is the Legendre transform

of the surfae tension. Sine the surfae tension is stritly onvex [She05 ℄, then the Legendre

transform isinvolutive,and Theorem 18is obtained.

By Corollary15, we knowthat:

R(B x , B y ) = lim

Z Z

N(P )

e ⁿ ² ^(B ^x ^s+B ^y ^t ⁻ σ(s,t)+o(1)) ds dt ≤ e ⁿ ² ^[max ^{(s,t)∈N(P ^)} ^(B ^x ^s+B ^y ^t ⁻ σ(s,t))+o(1)] | N (P ) |

A lower bound is obtained by performing Taylor expansion of

φ(s, t) := B x s + B y t − σ(s, t)

aroundits maximum andusing thefatthatthesurfae tensionis stritlyonvex.

Taking

1 n ² log

^yields ^that^the^Ronkin ^funtion^is ^the^Legendre ^transform^of ^the^surfae ^tension:

R(B x , B y ) = max

(s,t) ∈ N(P ) { B x s + B y t − σ(s, t) } .

Corollary 19. The slope of the measure

µ _(B _x _,B _y ₎

îs ^the ^gradient ôf ^the ^Ronkin ^funtion ât ^the

point

(B _x , B _y )

3.4.3 Phases of the dimer model

In the introdutory part of Setion 3.4, we mentioned that a way to haraterize phases of a

given model is to use the rate of deay of edge-edge orrelations and that, in the ase of the

dimermodel,itrequiredaharaterizationofthe zerosof

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

^on^the^unit^torus. ^This

information is nowavailableand one an givea preisedesription ofthephasediagram ofthe

dimermodel.

Atually,Kenyon,OkounkovandSheelduseheightfuntionutuationstodenethedierent

phases,but showthatthis isequivalent tolassifyingphasesusing rateofdeayoforrelations.

Hereare theirdenitions:

•

^An^EGM

µ

îsâlledâ^frozen^phaseîf^thereêxists^distint^faes

f

f ′

G

forwhih

h( f ) − h( f ′ )

isdeterministi.

•

^An ^EGM

µ

îs âlled â ^gaseous ôr ^smooth ^phase, îf^the ^height ûtuations ^have ^bounded

variane.

•

^An ^EGM

µ

îs âlledâ ^liquid ôr ^rough ^phase, îf ^the

µ

^-variane ^of ^the^height ^dierene ^is

unbounded.

Nowomes the theoremharaterizing phases.

Theorem 20. [KOS06 ℄The measure

µ _(B _x _,B _y ₎

^is:

•

^frozen^when

(B x , B y )

îsⁱⁿ^the^losureôfânûnboundedônnetedômponentôf^theâmoeba's

omplement

R ² \ A (P)

•

^liquid^when

(B _x , B _y )

îsⁱⁿ ^the înterior ôf ^the âmoeba

A (P )

•

^gaseous^when

(B x , B y )

îsⁱⁿ ^the^losure ôf â^bounded ônneted ômponent ôf^the âmoeba's

omplement

R ² \ A (P)

Figure3.9representsthe amoebaofthedimermodelonthesquare-otagongraphwithuniform

weights, and the orresponding phasesof the model.

We only give a fewideas onthe proof ofthis theorem.

B _y

B x

frozen

frozen gas liquid

frozen frozen

Figure3.9: Phasediagram ofthe uniform dimermodel onthesquare-otagongraph.

•

^Suppose ^that

(B x , B y )

îs ⁱⁿ ^the ^losure ôf ân ûnbounded ômponent ôf ^the âmoeba's

omplementandreall(Corollary19)thattheslopeofthemeasure

µ _(B _x _,B _y ₎

^is^the^gradient

of the Ronkin funtion at the point

(B x , B y )

^. ^The ^slope îs ^thus ân înteger ^point ôn ^the

boundary of

N (P )

^. Ûsing ^the ^max-ow ^min ût ^theorem ûsed ^by ^Thurston ^[Thu90℄ ^to

give a riterion of existene of dimer ongurations, one dedues that: if the slope is a

non-extremal boundary point of

N (P)

^,^then ^there îs ân êdge-path ⁱⁿ^the ^dual ^graph

G ∗ 1

whosediretion isorthogonaltothesideoftheNewtonpolygon,andsuhthatdualedges

rossing opies of this path appear with probability 1 or 0; if the slope is an extremal

point of

N (P )

^,^there ^are^twonon-parallel edge-pathsinthedual

G ∗

1

^,^whose ^diretions^are

orthogonalto thetwosideofthe Newtonpolygonmeetingattheextremalpoint,andsuh

that dualedges rossing opies of these paths appearwith probability 1 or 0. Otherwise

stated, inthe tiling representation ofthe dimermodel, there are pathsinthe dualgraph

G ∗

1

^generating ^a ^lattie ^of

G ∗

, onsisting of frozen paths for all tilings of

G ∗

hosen with

respet to the measure

µ _(B _x _,B _y ₎

^. ^Tilings ⁱⁿ ônneted ômponents ôf ^the ômplement ôf

this lattieare independent.

The height dierene of faes

f , f ′

whih belong to thelattie of frozen pathsis onstant,

i.e.

h( f ) − h( f ^′ )

^isdeterministi andthemeasure

µ _(B _x _,B _y ₎

^is ⁱⁿ^the^frozen^phase.

Inthe other twoases, the proofonsistsof expliitasymptoti expansionsof

K _(B ⁻ ¹

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

whih isthe

(x, y)

^-Fôurier ôeient ôf

^Q _P ^bw ^(e ^Bx ^z,e ^By ^w)

(e ^Bx z,e ^By w)

^seenâsâ ^funtion ôf

(z, w) ∈ T ²

•

^When

(B x , B y )

îsⁱⁿ^theînteriorôf^theâmoeba,^then

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

^has

1

^or

2

^zeros^on^the

unittorus,andKOSshowthattheonlyontributiontothe

(x, y)

^-Fôurierôeientômes

from aneighborhood of thepole(s). Theyextrat theexat asymptotis by doingTaylor

approximations and ontour integration, and show that

K _(B ⁻ ¹

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

^dereases

linearly, implying that the edge ovarianes dereases quadratially. The authors then

prove that the height dierene between two faes

f , f ′

grows universallylike

1 π

^times^the

logarithm of the distanefrom

f

f ′

. Themeasure

µ _(B _x _,B _y ₎

^is^thus ^a ^liquid ^phase.

•

^When

(B x , B y )

^belongs ^to ^the ômplement ôf ^the âmoeba, ^then

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

^has ^no

zeroontheunittorus,and

Q bw (e ^Bx z,e ^By w)

P (e ^Bx z,e ^By w)

îsânalyti,împlying^thatîts^Fôurier ôeients

derease exponentially fast. When,

(B x , B y )

îs ⁱⁿân ûnbounded ômponent, ^then ûsing

the above spei argument of the frozen phase, the authors show that some Fourier

oeientsare

0

^. ^When

(B x , B y )

îsⁱⁿâ^bounded ômponent,^then^the^Fôurier ôeient

dereases exponentially fast, and the height utuations have bounded variane. The

measure

µ _(B _x _,B _y ₎

^is^thus ^a^gaseous ^phase.

Dans le document The die de (Page 36-41)

Phases of the model

K (B − 1

x ,B y )

e 1 = w 1 b 1

e 2 = w 2 b 2

G

I e

1

e

0

µ (B x ,B y )

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 ).

K (B − 1

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

x 2 +y 2

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

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

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

P ∈ C [z, w]

A (P )

P (z, w) = 0

C 2

(z, w) 7→ (log | z | , log | w | ).

P

P(z, w) = 0

(x, y) ∈ R 2

A (P )

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

P (z, w) = 0

(x, y) ∈ R 2

P (e x z, e y w)

N (P )

•

•

N (P )

•

N (P )

N (P )

•

R 2 \ A (P )

N (P )

N (P)

•

R 2 \ A (P )

•

( A (P )) ≤ π 2

(N (P )).

log|z|

log|w|

tentacle ray

P (z, w) = 5 − z − 1 z − w − w 1

P

R

∀ (x, y) ∈ R 2 , R(x, y) = 1 (2πi) 2

Z

T 2

log | P (e x z, e y w) | dz z

dw w .

•

•

N (P )

P (z, w) = 0

π 2

(N (P ))

1876

P(z, w) = 0

2

1

2

1

1

1

µ (B x ,B y )

ω M 0

G 1

K _(B ⁻ ¹

e ₁ = w ₁ b ₁

e ₂ = w ₂ b ₂

I _e

µ _(B _x _,B _y ₎

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 ).

K _(B ⁻ ¹

x ² +y ²

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

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

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

C ²

(x, y) ∈ R ²

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

(x, y) ∈ R ²

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

R ² \ A (P )

R ² \ A (P )

( A (P )) ≤ π ²

P (z, w) = 5 − z − ¹ _z − w − _w ¹

∀ (x, y) ∈ R ² , R(x, y) = 1 (2πi) ²

T ²

log | P (e ^x z, e ^y w) | dz z

π ²

µ _(B _x _,B _y ₎

ω ^M ⁰

G ₁

x ₀ = y ₀ = 0

e ⁿ ² ^(B ^x ^s+B ^y ^t ⁻ σ(s,t)+o(1)) ds dt ≤ e ⁿ ² ^[max ^{(s,t)∈N(P ^)} ^(B ^x ^s+B ^y ^t ⁻ σ(s,t))+o(1)] | N (P ) |

n ² log

µ _(B _x _,B _y ₎

(B _x , B _y )

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

µ _(B _x _,B _y ₎

R ² \ A (P)

(B _x , B _y )

R ² \ A (P)

B _y

µ _(B _x _,B _y ₎

µ _(B _x _,B _y ₎

h( f ) − h( f ^′ )

µ _(B _x _,B _y ₎

K _(B ⁻ ¹

^Q _P ^bw ^(e ^Bx ^z,e ^By ^w)

(e ^Bx z,e ^By w)

(z, w) ∈ T ²