Flutuations of the height funtion

In this setion we address the question of the utuations of the height funtion around its

mean. We restritourselvesto the aseof the uniform dimermodel on theinnitehoneyomb

lattie

H

,withnomagnetield. Thegoalistoprovethatthelimitingutuationsoftheheight funtion are desribed by a Gaussian free eldof the plane, a behavior whih is harateristi

of theliquidphase.

Resultspresentedhere, and extensionsan be found in[Ken00 ,Ken01 ,Ken08 , dT07℄.

3.5.1 Uniform dimermodel on the honeyomb lattie

Letusspeifytheresultsobtainedintheprevioussetionstotheasewhere

H

isthehoneyomb lattie,edges areassignedweights

1

^,ând ^thereîs^no ^magneti êld. ^The^hoie ôffundamental domain isgiven inFigure3.10.

1/w

γ _x γ y

z 1 1

1 1

1

Figure3.10: Choieof fundamentaldomain of thehoneyomb lattie

Theharateristi polynomialis:

P (z, w) = 1 + z + 1 w .

The Gibbs measure

µ

ôbtained âs^weak ^limit ôf ^the ^Boltzmann ^measures

µ n

^on

M ( G _n )

^(with

no magnetield) hasthefollowing expressionon ylindersets:

µ( e ₁ , . . . , e _k ) = det K ⁻ ¹ ( b _i , w _j ) ₁ _≤ i,j ≤ k

,

^(3.10)

where

K ⁻ ¹ ( b _0,0 , w _x,y ) = 1 (2πi) ²

Z

T ²

1 1 + z + _w ¹ z ^y w ⁻ ^x dz z

dw w ,

and

w _x,y

representsthewhitevertexofthe

(x, y)

^opy^of^thefundamentaldomain,andsimilarly for

b _x,y

on the unit torus

T

(z 0 , w 0 ) = (e ⁱ ^2π ³ , e ⁱ ^2π ³ ),

^and

(z 0 , w 0 ).

^The ^uniform ^dimer ^model ^on ^the

honeyomb lattie isthus intheliquid phase.

InSetion 2.4, we dened Thurston'sheight funtionon lozenge tilingsof thetriangular lattie

T

,whih wasa funtion onvertiesof

T

. Equivalently, itis afuntion on dimerongurations of the honeyomb lattie

H

,dened on faesof

H

. For onveniene, from now on, we onsider

1

3

^of ^Thurston's ^height ^funtion.

The following lemma gives an expression of the height funtion using indiator funtions of

edges. We will use this inthe proof of the onvergene of the height funtion. Let

u , v

be two faes of

H

,and let

γ

^be ^an ^edge-path ⁱⁿ ^the^dual ^graph ^from

u

v

. Denote by

e ₁ , . . . , e _m

the dualedgesofedgesof

γ

^whih^haveâ^blak^vertexôn^the^left,ând^by

f ₁ , . . . , f _n

thosewhihhave awhite vertexon theleft. Then,

Lemma 21.

γ

^. ^Then, ^returning ^to ^the orientation of the edges of the triangular lattie introdued in the

onstrutionofThurston'sheightfuntion,seeSetion2.4,weknowthattheedge

u _i v _i

isoriented from

u _i

v _i

. Thus bydenition, we have:

h( v _i ) − h( u _i ) = ( ₁

3

^if^the^edge

e _i

isnot inthedimeronguration

− ² ₃

^if^the^edge

e _i

isinthedimeronguration

.

Thisan be summarized as:

h( v _i ) − h( u _i ) = − I _e

i + ¹ ₃

A similar argument holds for edges whih have a white vertex on the left. Summing over all

edges inthe path

γ

^yields ^the ^result.

3.5.2 Gaussian free field of the plane

We nowdene the Gaussian freeeld ofthe plane, whih is, aswe will see, thelimiting objet

of theheight funtion.

The Green's funtion of the plane, denoted by

g

^satises

∆ x g(x, y) = δ x (y)

^, ^where

δ x

^is ^the

Diradistribution at

x

^. Ûp^to ân âdditive ônstant,

g

^is ^given ^by

g(x, y) = − 1

implying that the bilinear form

G

^is ^positive ^denite. ^The^quantity

G(ϕ ₁ , ϕ ₁ )

^is ^known ^as^the

Dirihlet energy of

f 1

A random generalized funtion

F

^assigns ^to ^every ^test ^funtion

ϕ

ⁱⁿ

C _c,0 ^∞ ( R ² )

^, ^a ^real ^random

variable

F ϕ

^. Îtîsâssumed^to^be^linearândôntinuous,^whereôntinuity^means^thatônvergene

of the funtions

ϕ n j

⁽

1 ≤ j ≤ k

⁾ ^to

ϕ j

^, împlies ônvergene ⁱⁿ ^law ôf ^the ^random ^vetor

(F ϕ n ₁ , . . . , F ϕ n _k )

^to

(F ϕ ₁ , . . . , F ϕ k )

^. ^A^randomgeneralized funtionissaidtobeGaussianiffor every linearly independent funtions

ϕ ₁ , . . . , ϕ _k ∈ C _c,0 ^∞ ( R ² )

^, ^the^random ^vetor

(F ϕ ₁ , . . . , F ϕ _k )

isGaussian.

Theorem22. [Bo55 ℄If

B : C _c,0 ^∞ ( R ² ) × C _c,0 ^∞ ( R ² ) → R

isabilinear,ontinuous,positivedenite form,then there existsaGaussian random generalized funtion

F

^,^whose ^ovariane ^funtion^is

given by:

E [F ϕ ₁ F ϕ ₂ ] = B (ϕ ₁ , ϕ ₂ ).

Bydenition aGaussian free eldof the plane GFFisa Gaussianrandom generalized funtion

whose ovariane funtionis the bilinear form

G

^dened ^above,^i.e.

E [

^GFF

(ϕ 1 )

^GFF

(ϕ 2 )] = − 1

There areother ways of deningtheGaussian freeeld, seefor example [GJ81 ℄ and[She07 ℄.

3.5.3 Convergene of the height funtion to a Gaussian free field

Let

H ^ε

be the graph

H

whose edge-lengths have been multiplied by

ε

^, ^and ^onsider ^the

un-normalized height funtion

h

^on ^faes^of

H ^ε

. Dene,

H ^ε : C _c,0 ^∞ ( R ² ) → R

ϕ 7→ H ^ε ϕ = ε ² X

v ∈ F( H ^ε )

ϕ( v )h( v ).

Theorem 23. The random generalized funtion

H ^ε

^onverges ^weakly ⁱⁿ ^law ^to

√ ¹ π

^times ^a

Gaussian free eld,i.e. for every

ϕ ₁ , . . . , ϕ k ∈ C _c,0 ^∞ ( R ² )

(H ^ε ϕ ₁ , . . . , H ^ε ϕ k )

^onverges ⁱⁿ ^law^(as

where

g( u , v , u ′ , v ′ ) = g( v , v ′ ) + g( u , u ′ ) − g( v , u ′ ) − g( u , v ′ )

g

^is^the ^Green^'s^funtion^of ^the ^plane,

and the sumisover all

(k − 1)!!

^pairings ^of

{ 1, . . . , k }

Proof. We prove this proposition ina partiular ase. The proof of thegeneral ase issimilar,

although notationally muh more umbersome. Let us assume that

k = 2

^, ^and ^suppose ^that

u ₁ , v ₁ , u ₂ , v ₂

are suh that we an hoose

u ^ε

2

respetively asinFigure 3.11.

ε

ε lim → 0

E [(h( v ^ε

1 ) − h( u ^ε

1 ))(h( v ^ε

2 ) − h( u ^ε

2 ))] = − 2 Re Z v ₁

u ₁

Z v ₂ u ₂

1 4π ² (z 2 − z 1 ) ² dz ₁ dz ₂

= − 1 2π ² log

( v ₁ − v ₂ )( u ₁ − u ₂ ) ( v ₁ − u ₂ )( u ₁ − v ₂ )

= 1

π g( u ₁ , v ₁ , u ₂ , v ₂ ).

One Proposition 24 is available, one then introdues test funtions and prove onvergene of

all moments of (3.11) . This implies handling moments of height dierenes involving verties

whiharelose. Theasymptoti formulafor theinverse Kasteleynmatrix doesnot holdinthis

ase, and onehasto do areful boundsto see thateverything still works.

********************************

We have reahed the end of these leture notes overing the foundations of the dimer model

and thepaper Dimersand amoeba ofKenyon,Okounkovand Sheeld [KOS06 ℄, givinga full

desription of the phase diagram of the dimer model on periodi, bipartite graphs. Although

these arespetaularresults, thereare others,suh asthespei behaviorof thedimermodel

dened on isoradial graphs [Ken02 , KO06 ℄,theunderstanding of limit shapes[CKP01,KO07 ℄,

the appliation of dimertehniques to study theIsing model[BdT10b,BdT10a ℄. We do hope

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Dans le document The die de (Page 41-49)

Flutuations of the height funtion

H

H

1

1/w

γ x γ y

z 1 1

1 1

1

P (z, w) = 1 + z + 1 w .

µ

µ n

M ( G n )

µ( e 1 , . . . , e k ) = det K − 1 ( b i , w j ) 1 ≤ i,j ≤ k

,

K − 1 ( b 0,0 , w x,y ) = 1 (2πi) 2

Z

T 2

1

1 + z + w 1 z y w − x dz z

dw w ,

w x,y

(x, y)

b x,y

T

(z 0 , w 0 ) = (e i 2π 3 , e i 2π 3 ),

(z 0 , w 0 ).

T

T

H

H

1

3

u , v

H

γ

u

v

e 1 , . . . , e m

γ

f 1 , . . . , f n

γ

u i v i

u i

v i

h( v i ) − h( u i ) = ( 1

3

e i

− 2 3

e i

.

h( v i ) − h( u i ) = − I e

i + 1 3

γ

g

∆ x g(x, y) = δ x (y)

δ x

x

g

g(x, y) = − 1

G

G(ϕ 1 , ϕ 1 )

f 1

F

ϕ

C c,0 ∞ ( R 2 )

F ϕ

ϕ n j

1 ≤ j ≤ k

ϕ j

(F ϕ n 1 , . . . , F ϕ n k )

(F ϕ 1 , . . . , F ϕ k )

ϕ 1 , . . . , ϕ k ∈ C c,0 ∞ ( R 2 )

(F ϕ 1 , . . . , F ϕ k )

B : C c,0 ∞ ( R 2 ) × C c,0 ∞ ( R 2 ) → R

F

E [F ϕ 1 F ϕ 2 ] = B (ϕ 1 , ϕ 2 ).

G

E [

(ϕ 1 )

γ _x γ y

M ( G _n )

µ( e ₁ , . . . , e _k ) = det K ⁻ ¹ ( b _i , w _j ) ₁ _≤ i,j ≤ k

K ⁻ ¹ ( b _0,0 , w _x,y ) = 1 (2πi) ²

T ²

1 + z + _w ¹ z ^y w ⁻ ^x dz z

w _x,y

b _x,y

(z 0 , w 0 ) = (e ⁱ ^2π ³ , e ⁱ ^2π ³ ),

e ₁ , . . . , e _m

f ₁ , . . . , f _n

u _i v _i

u _i

v _i

h( v _i ) − h( u _i ) = ( ₁

e _i

− ² ₃

e _i

h( v _i ) − h( u _i ) = − I _e

i + ¹ ₃

G(ϕ ₁ , ϕ ₁ )

C _c,0 ^∞ ( R ² )

(F ϕ n ₁ , . . . , F ϕ n _k )

(F ϕ ₁ , . . . , F ϕ k )

ϕ ₁ , . . . , ϕ _k ∈ C _c,0 ^∞ ( R ² )

(F ϕ ₁ , . . . , F ϕ _k )

B : C _c,0 ^∞ ( R ² ) × C _c,0 ^∞ ( R ² ) → R

E [F ϕ ₁ F ϕ ₂ ] = B (ϕ ₁ , ϕ ₂ ).

H ^ε

H ^ε

H ^ε : C _c,0 ^∞ ( R ² ) → R

ϕ 7→ H ^ε ϕ = ε ² X

v ∈ F( H ^ε )

H ^ε

√ ¹ π

ϕ ₁ , . . . , ϕ k ∈ C _c,0 ^∞ ( R ² )

(H ^ε ϕ ₁ , . . . , H ^ε ϕ k )

u ₁ , v ₁ , u ₂ , v ₂

u ^ε

E [(h( v ^ε

1 ) − h( u ^ε

1 ))(h( v ^ε

2 ) − h( u ^ε

2 ))] = − 2 Re Z v ₁

u ₁

Z v ₂ u ₂

4π ² (z 2 − z 1 ) ² dz ₁ dz ₂

= − 1 2π ² log

( v ₁ − v ₂ )( u ₁ − u ₂ ) ( v ₁ − u ₂ )( u ₁ − v ₂ )

π g( u ₁ , v ₁ , u ₂ , v ₂ ).