Smirnov's work on the two-dimensional Ising model

Smirnov’s work on the two-dimensional Ising model

Hugo Duminil-Copin, Universite de Geneve

Hugo Duminil-Copin, Universite de Geneve Smirnov’s work on the two-dimensional Ising model

Recently much of the progress in understanding two-dimensionalphenomena resulted from

Conformal Field Theory (last 25 years)

Schramm-Loewner Evolution (last 10 years)

There were very fruitful interactions between mathematics andphysics. We will try to describe parts of the relations betweenthese three subjects.

Plan:(1) Brief historic(2) Study of discrete models(3) Schramm-Loewner Evolution


The Square lattice Ising model

Each cell of a square lattice iseither red or blue (correspondingrespectively to + or −). Theprobability of a configuration σ is

Pβ(σ) =1

Zβexp

(−β∑

x∼y

σ(x)σ(y)

)

where Zβ is the partition functionof the model.


Brief historic (1): Onsager’s exact solution

1944 (Onsager) : computation of the partition function(unrigorous)

1949 (Kaufman, Onsager): computation made rigorous

Later (Yang, Wu, McCoy): use the computation to derivequantities of the model (such as certain critical exponents)

Other approaches developped:(1) Kac, Ward, Potts, Dolbilin, Cimasoni : combinatorialapproach(2) Kasteleyn, Fisher : dimer approach(3) Lieb, Baxter : transfert matrices approach

All these approaches deal with simple geometries (whole plane,torus, etc...). Derive analytic properties. Harder to get geometricones (in particular dependence into boundary conditions).






























Brief historic (2): Renormalization Group

1951, Petermann-Stueckelberg

1963-1966, Kadanoff, Fisher, Widom, Wilson

Perform a block-spin renormalization which corresponds to arescaling: define an operator from the set of hamiltonians in itself.

Conclusion: At criticality, the scaling limit is described by amass-less field theory. The critical point is universal and hencetranslational, scaling and rotational invariant.

We must believe in this picture (for instance for the problem ofmarginal ’operators’).























Brief historic (3): Birth of 2D conformal field theory

Definition: Conformaltransformations are preserving theangles, or in other words arelocallytranslations+rotation+scaling.

Since the fields are local, onecan logically postulate theconformal invariance of themodel.

We must assume the RG istrue. It does not adress anyboundary conditions issue.


Brief historic (3): Birth of 2D conformal field theory

Definition: Conformaltransformations are preserving theangles, or in other words arelocallytranslations+rotation+scaling.

Since the fields are local, onecan logically postulate theconformal invariance of themodel.

We must assume the RG istrue. It does not adress anyboundary conditions issue.


Brief historic (4): 2D conformal field theory

1966, (Patashinkii-Pokrovskii, Kadanoff) scale, rotation andtranslation invariance allow to calculate two-point correlations

1970, (Polyakov): Mobius invariance allows to calculatethree-point correlations

1984, (Belavin, Polyakov, Zamolodchikov) postulate fullconformal invariance allows to compute much more things

1984, (Cardy) work with boundary fields which leads toapplications to lattice models.

highest weight of Virasoro’s algebra, Quantum gravity, etc...






























Brief historic (5): summary

We can study the discrete Ising model directly (exactlysolvable model)

we can study its continuum limit at criticality (RG and CFT)

Brief historic (6): Geometric and analytic approach

1999 (Schramm) Schramm-Loewner Evolution, a geometricdescription of the scaling limits at criticality

Recent years (Smirnov) Discrete analyticity: a way torigorously establish existence and conformal invariance ofscaling limits.























DONE

Brief historic

TO DO

The Schramm-Loewner Evolution

Discrete observables and lattice models

What is next


DONE

Brief historic

TO DO



What is next


Schramm-Loewner Evolution (pre-history)

Event 1 (1994): (Langlands, Pouilot, Saint-Aubin) check theexistence of the limit, the universality and the conformal invarianceof crossing probabilities for percolation.

Very widely read

Event 2: (Cardy) crossing formula for percolation:

limδ→0

Pδ

(C(R)

)=

Γ(

23

)

Γ(

43

)Γ(

13

)m13 2F1

(1

3,

2

3,

4

3,m

),

where m is the conformal radius of the rectangle.

Easier version by Carleson, proved by Smirnov (2001)



(Schramm) Look at interfaces in models of statistical physics.

Model these interfaces at the scaling limit by a random continuouscurve, called SLE.

It allows him to deduce (among other things) Cardy’s formula,conditionally to the fact that the discrete interface indeedconverges to SLE.












Schramm-Loewner Evolution (definition Loewner chain)

Consider a simply connected domain D with two points on theboundary (for instance think of (H, 0,∞)) and a growing curvefrom 0 to ∞.

(Loewner 1920) Growing curves can be coded by real functions.

∞ ∞z 7→ z + C

z +O(1/z2)

g : H \ γ → H

γ

0

Moreover for every z ∈ H, up to the first time at which z isswallowed by the curve, we have:

∂tgt(z) =2

gt(z)−Wt.

The process Wt is called the driving process.





∞ ∞z 7→ z + C

z +O(1/z2)

g : H \ γ → H

γ

0


∂tgt(z) =2

gt(z)−Wt.






∞ ∞z 7→ z + 2t

z +O(1/z2)

gt : H \ γ[0, t] → H

γt

gt(γt) = Wt

0


∂tgt(z) =2

gt(z)−Wt.






∞ ∞z 7→ z + 2t

z +O(1/z2)

gt : H \ γ[0, t] → H

γt

gt(γt) = Wt

0


∂tgt(z) =2

gt(z)−Wt.

The process Wt is called the driving process.Hugo Duminil-Copin, Universite de Geneve Smirnov’s work on the two-dimensional Ising model

Schramm-Loewner Evolution (definition Schramm-LoewnerEvolution)

Observation 2: (domain Markov property) Conditionally on thestart of the curve, the remaining curve is an SLE from the tip to∞ in the slit domain.

Observation 2: curves must be conformally invariant:

a

b

D

γ

Φ conformal

Φ(a)

Φ(b)

Φ(D)

Φ(γ)

γ′

With these two properties, the driving process must haveindependent increments. Scale invariance implies that it is aBrownian motion. The SLE(κ) is the (random) Loewner chaingenerated by the

√κBt (Bt is a standard Brownian motion).





a

b

D

γ

Φ conformal

Φ(a)

Φ(b)

Φ(D)

Φ(γ)

γ′







a

b

D

γ

Φ conformal

Φ(a)

Φ(b)

Φ(D)

Φ(γ)

γ′

With these two properties, the driving process must haveindependent increments. Scale invariance implies that it is aBrownian motion.

The SLE(κ) is the (random) Loewner chaingenerated by the






a

b

D

γ

Φ conformal

Φ(a)

Φ(b)

Φ(D)

Φ(γ)

γ′




Schramm-Loewner Evolution (properties of SLE itself)

fractal curve: simple for κ ≤ 4, self-touching for κ ∈ (4, 8)and space filling for κ ≥ 8 (Rohde, Schramm)

dimHausdorff (SLE (κ)) =(1 + κ

8

)∧ 2 (Beffara)

computation of critical exponents (like intersectionexponents).

duality properties between κ and 16/κ (Zhan, Dubedat).

allows to construct more general processes, such as CLEs(Schramm, Sheffield, Werner).





8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)

computation of critical exponents (like intersectionexponents).duality properties between κ and 16/κ (Zhan, Dubedat).allows to construct more general processes, such as CLEs(Schramm, Sheffield, Werner).





8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)








8

)∧ 2 (Beffara)





Schramm-Loewner Evolution (connection with CFT)

Connection with discrete models: computation of criticalexponents (Lawler, Schramm, Werner)

κ = 2 κ = 3 κ = 4 κ = 6 κ = 8κ = 8/3

loop erased Ising DimersLevel lines of GFF

Percolation USTSAW

Construction of a highest weight representation of Virasoro’salgebra (Friedrich, Werner) for boundary CFT

Link between SLE martingales and CFTs (Bauer, Bernard)

Gaussian free field




κ = 2 κ = 3 κ = 4 κ = 6 κ = 8κ = 8/3


Percolation USTSAW



Gaussian free field




κ = 2 κ = 3 κ = 4 κ = 6 κ = 8κ = 8/3


Percolation USTSAW



Gaussian free field




κ = 2 κ = 3 κ = 4 κ = 6 κ = 8κ = 8/3


Percolation USTSAW



Gaussian free field




κ = 2 κ = 3 κ = 4 κ = 6 κ = 8κ = 8/3


Percolation USTSAW



Gaussian free field


DONE

Brief historic


TO DO


What is next


DONE

Brief historic


TO DO


What is next


Discrete observables (General philosophy)

Consider a family of interfaces between two points a and b in(discrete approximations with meshsize δ of) a fixed domainΩ.

Assume we know that the family of curves is precompact. Inorder to prove that the family of curves actually convergeswhen δ → 0, it is sufficient to identify a unique possible limit.

To identify the possible curve, one needs explicit martingales...

(1) These martingales should be observables of the lattice model(crossing probabilities, magnetization, etc)...(2) We can compute them because they are discrete holomorphicand they satisfy some fixed boundary problem.(3) Hence, they converge to the continuum analogue of theproblem.




















Discrete observables (The case of the Ising model on thetriangular lattice)

Consider the hexagonal lattice for a second and define thehigh-temperature expansion of the Ising model.

a

δ

z

Ω

For a simply-connected domain Ω and a discrete approximation of it, letz be on the boundary. Define the fermionic operator

FΩ,δ,x (a, z) =∑

ω with a curve γ from a to z

e−i 12 Wγ(a,z)x#edges.

The integral along any discrete contour equals 0.

If x = xc , the function z 7→ FΩ,δ,xc (a, z) is a discrete Green function withRiemann-Hilbert boundary-value problem.




a

δ

z

Ω


FΩ,δ,x (a, z) =∑








a

δ

z

Ω


FΩ,δ,x (a, z) =∑








a

δ

z

Ω


FΩ,δ,x (a, z) =∑







Let Ω be a simply connected domain and a, b on the boundary,

Fact 1: For z inside the domain,

limδ→0

FΩ,δ,xc (a, z)

FΩ,δ,xc (a, b)=

√K ′Ω(a, z)

K ′Ω(a, b).

Fact 2: The quantityFΩ\γt ,δ,xc

(γt , z)

FΩ\γt ,δ,xc(γt , b)

is a martingale (conserved quantity) of the discrete curve from a to b.

When plugging that for each z ,

√K ′

Ω\γt(a,z)

K ′Ω\γt

(a,b) is a conserved quantity

of the limiting curve, we deduce that the only possible limit is SLE(3)!





limδ→0

FΩ,δ,xc (a, z)

FΩ,δ,xc (a, b)=

√K ′Ω(a, z)

K ′Ω(a, b).


(γt , z)




√K ′

Ω\γt(a,z)

K ′Ω\γt







limδ→0

FΩ,δ,xc (a, z)

FΩ,δ,xc (a, b)=

√K ′Ω(a, z)

K ′Ω(a, b).


(γt , z)




√K ′

Ω\γt(a,z)

K ′Ω\γt







limδ→0

FΩ,δ,xc (a, z)

FΩ,δ,xc (a, b)=

√K ′Ω(a, z)

K ′Ω(a, b).


(γt , z)




√K ′

Ω\γt(a,z)

K ′Ω\γt




Discrete observables (Implications)

convergence to SLE(3) (Chelkak, Smirnov) which leads to exponents

understanding of the geometric properties (D.-C., Hongler andNolin), leading to mixing estimates for the Glauber dynamics (Sly,Lubetzky)

construction of the energy density field (Hongler, Smirnov):

〈σ(x)σ(y)〉Ωε,free =

√2

2− 1

πρΩ(x)ε+ O(ε2).







√2

2− 1








√2

2− 1








√2

2− 1



Discrete observables (Other O(n)-models)

The O(n) model is a model on closed loops lying on a finite subgraph ofthe hexagonal lattice: the partition function equals

Zx,n,G =∑

x# edgesn# loops.

F (a, z , x , σ) :=∑


e−iσWγ(a,z)x#edgesn#loops

where 2 cos( 4(1/2+σ)π3 ) = −n.

z = 1√2+√

2−n

critical phase 2: SLE( 4πarccos(−n/2))

critical phase 1: SLE( 4π2π−arccos(−n/2))

sub-critical phase

z

n2

0 1/√

2 +√

2

1/√

2

1/√

3




Zx,n,G =∑

x# edgesn# loops.

F (a, z , x , σ) :=∑



where 2 cos( 4(1/2+σ)π3 ) = −n.

z = 1√2+√

2−n



sub-critical phase

z

n2

0 1/√

2 +√

2

1/√

2

1/√

3




Zx,n,G =∑

x# edgesn# loops.

F (a, z , x , σ) :=∑



where 2 cos( 4(1/2+σ)π3 ) = −n.

z = 1√2+√

2−n



sub-critical phase

z

n2

0 1/√

2 +√

2

1/√

2

1/√

3


DONE

Brief historic



TO DO

What is next


DONE

Brief historic



TO DO

What is next


Prove conformal invariance of other lattice models

Prove universality for the Ising model

Comprehend links between SLE (or CLE) and CFT deeper

Construct CFTs from branching SLE

Relate random planar graphs to Liouville Quantum Gravity viaSLE.


























Thank you


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Smirnov's work on the two-dimensional Ising model