[PPT] - Long-range order in random 3 -colorings in high dimensions Ohad N. PowerPoint Presentation

SLIDE 1 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Long-range order in random 3-colorings in high dimensions

Ohad N. Feldheim Joint work with Yinon Spinka

IMA, University of Minnesota

June 15, 2015

SLIDE 2 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Setup and terminology

Consider a finite set Λ ⊂ Zd.

SLIDE 3 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Setup and terminology

Consider a finite set Λ ⊂ Zd.

A q-coloring of Λ is a function

f : Λ → {0, 1, . . . , q − 1}.

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

SLIDE 4 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Setup and terminology

Consider a finite set Λ ⊂ Zd.

A q-coloring of Λ is a function

f : Λ → {0, 1, . . . , q − 1}.

f is proper if

u ∼ v ⇒ f (u) = f (v) .

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

SLIDE 5 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Setup and terminology

Consider a finite set Λ ⊂ Zd.

A q-coloring of Λ is a function

f : Λ → {0, 1, . . . , q − 1}.

f is proper if

u ∼ v ⇒ f (u) = f (v) .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 6 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Setup and terminology

Consider a finite set Λ ⊂ Zd.

A q-coloring of Λ is a function

f : Λ → {0, 1, . . . , q − 1}.

f is proper if

u ∼ v ⇒ f (u) = f (v) .

Denote # of singularities in f :

N(f ) := |{u ∼ v : f (u) = f (v)}|.

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

SLIDE 7 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

SLIDE 8 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

SLIDE 9 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

β < 0 : Ferromagnetic regime.

2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1

β ≪ 0

SLIDE 10 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

β < 0 : Ferromagnetic regime.

β = −∞

SLIDE 11 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

β < 0 : Ferromagnetic regime.
β > 0 : Anti-ferromagnetic regime.

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

β ≫ 0

SLIDE 12 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

β < 0 : Ferromagnetic regime.
β > 0 : Anti-ferromagnetic regime.
β = ∞ : Uniform proper coloring.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

β = ∞

SLIDE 13 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Potts Model

We consider a random 3-coloring f of Λ ⊂ Zd with one parameter β called inverse temperature.

P(f ) proportional to e−βN(f ).

(Boltzmann distribution.)

β < 0 : Ferromagnetic regime.
β > 0 : Anti-ferromagnetic regime.
β = ∞ : Uniform proper coloring.
q-states Potts is defined similarly

5 5 1 1 2 5 1 1 1 1 5 1 1 5 1 1 5 5 1 5 5 5 1 4 4 2 2 2 5 2 4 2 2 4 4 2 4 2 4 2 2 4 4 2 2 1 5 1 1 1 5 5 5 2 2 2 2 4 1 5 5 1 5

β ≫ 0, q = 5

SLIDE 14 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation

SLIDE 15 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation from statistical mechanics

Equilibrium statistical mechanics = research of phases

SLIDE 16 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation from statistical mechanics

Phases of matter. Gas Liquid Solid Phase depends on temperature and pressure.

SLIDE 17 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation from statistical mechanics

Magnetic phases. Magnet Paramagnet Anti-Ferromagnet Phase depends on temperature and external field.

SLIDE 18 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation from statistical mechanics

Magnetic phases. Diamagnet Ferrimagnet Phase depends on temperature and external field.

SLIDE 19 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Motivation from statistical mechanics

Magnetic phases. Magnet Paramagnet Anti-Ferromagnet Phase depends on temperature and external field. Goal: explain phases through microscopic mechanics

SLIDE 20 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Wilhelm Lenz Wolfgang Paulli

SLIDE 21 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

SLIDE 22 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

Values represent spin +/− direction.
+
+

+ + + + + + + + + + + + + + + + + + + + + + + +

+
+
+

+

+

SLIDE 23 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

Values represent spin +/− direction.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 24 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

Values represent spin +/− direction.
P(f ) proportional to e−βN(f ).

(Stationary distribution of Glauber dynamics)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 25 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

Values represent spin +/− direction.
P(f ) proportional to e−βN(f ).

(Stationary distribution of Glauber dynamics)

Often taken under external field

(giving a bias for seeing + vs. −).

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 26 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The Ising model

The Ising model (2-states Potts).

Values represent spin +/− direction.
P(f ) proportional to e−βN(f ).

(Stationary distribution of Glauber dynamics)

Often taken under external field

(giving a bias for seeing + vs. −).

Ferromagnet (β < 0) and

anti-ferromagnet (β > 0) are equivalent.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 27 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit).

SLIDE 28 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder

SLIDE 29 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

SLIDE 30 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

Λ large domain.
Condition on f (v) = τ for all v on

the boundary. Even zero boundary conditions

SLIDE 31 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

Λ large domain.
Condition on f (v) = τ for all v on

the boundary.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Sample with 0-boundary conditions on even domain

SLIDE 32 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

Λ large domain.
Condition on f (v) = τ for all v on

the boundary.

Does the distribution in the center

depend on τ?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

sample with 0-boundary conditions on even domain

SLIDE 33 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

Λ large domain.
Condition on f (v) = τ for all v on

the boundary.

Does the distribution in the center

depend on τ?

Ordered phase: Yes.

Disordered phase: No.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

sample with 0-boundary conditions on even domain

SLIDE 34 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thermodynamical limit

Thermodynamical questions deal with large volume systems. That is fixed d, with n → ∞ (thermodynamical limit). Order vs. Disorder: dependence on boundary conditions.

Λ large domain.
Condition on f (v) = τ for all v on

the boundary.

Does the distribution in the center

depend on τ?

Ordered phase: Yes.

Disordered phase: No.

Mature notions:

Gibbs measures & pure phases.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

sample with 0-boundary conditions on even domain

SLIDE 35 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

SLIDE 36 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

In which d does a phase transition occur?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 37 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

In which d does a phase transition occur?
What does a typical β ≫ 0 sample look like?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 38 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

In which d does a phase transition occur?
What does a typical β ≫ 0 sample look like?

Advanced questions:

Behavior at/near criticality?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 39 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

In which d does a phase transition occur?
What does a typical β ≫ 0 sample look like?

Advanced questions:

Behavior at/near criticality?
Rapid/Torpid mixing?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 40 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Questions about the model

Basic Questions:

In which d does a phase transition occur?
What does a typical β ≫ 0 sample look like?

Advanced questions:

Behavior at/near criticality?
Rapid/Torpid mixing?
How fast do correlations decay?

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SLIDE 41 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Using zero-boundary conditions

How to demonstrate multiple pure phases? More specific strategy for β ≫ 0.

Λ large even domain.

SLIDE 42 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Using zero-boundary conditions

How to demonstrate multiple pure phases? More specific strategy for β ≫ 0.

Λ large even domain.
Condition on f (v) = 0 for all v on

the boundary. Even zero boundary conditions

SLIDE 43 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Using zero-boundary conditions

How to demonstrate multiple pure phases? More specific strategy for β ≫ 0.

Λ large even domain.
Condition on f (v) = 0 for all v on

the boundary.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Sample with 0-boundary conditions on even domain

SLIDE 44 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Using zero-boundary conditions

How to demonstrate multiple pure phases? More specific strategy for β ≫ 0.

Λ large even domain.
Condition on f (v) = 0 for all v on

the boundary.

Show that the frequencies on even

and odd sublattice are unbalanced.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

sample with 0-boundary conditions on even domain

SLIDE 45 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Properties of the Ising model

Answers to these questions are now known for the Ising model (q = 2):

In all d ≥ 2 there is a critical temperature 1/βc = Θ(d)

(error terms are known).

β < βc implies a unique pure state.
β > βc implies two pure states.
In β > βc one sublattice is biased towards + and the other

towards −. Ising 2d ferromagnets and anti-ferromagnets:

∞ ≫0 AF−Crit >0 <0 F−Crit ≪0 −∞

β =

SLIDE 46 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond Ising

Clock and Potts models. Cyril Domb Renfrey Pos

SLIDE 47 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Kotecky Conjecture

Baxter (1982): d = 2, q = 3 Potts AF - critical at β = ∞.

Rodney Baxter

SLIDE 48 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Kotecky Conjecture

Baxter (1982): d = 2, q = 3 Potts AF - critical at β = ∞. Roman Kotecky (1985): Conjecture - for AF 3-states Potts model on Zd, there exists a minimal d0 (probably d0 = 3) such that for d ≥ d0 there is a positive critical temperature 1/βc.

Roman Kotecký

SLIDE 49 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Kotecky Conjecture

Baxter (1982): d = 2, q = 3 Potts AF - critical at β = ∞. Roman Kotecky (1985): Conjecture - for AF 3-states Potts model on Zd, there exists a minimal d0 (probably d0 = 3) such that for d ≥ d0 there is a positive critical temperature 1/βc.

Roman Kotecký

For β > βc: six pure states (phase

co-existence).

Each state corresponds to one color

dominant on one sublattice and nearly absent from the other.

2 2 1 1 1 1 2 1 2 2 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 2 2 1 2

SLIDE 50 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Kotecky Conjecture

Baxter (1982): d = 2, q = 3 Potts AF - critical at β = ∞. Roman Kotecky (1985): Conjecture - for AF 3-states Potts model on Zd, there exists a minimal d0 (probably d0 = 3) such that for d ≥ d0 there is a positive critical temperature 1/βc.

Roman Kotecký

For β > βc: six pure states (phase

co-existence).

Each state corresponds to one color

dominant on one sublattice and nearly absent from the other.

For β < βc: one disordered pure phase,

correlations decay exponentially fast.

2 2 1 1 1 1 2 1 2 2 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 2 2 1 2

SLIDE 51 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

AF 3-states Potts

q ≥ 3 AF is more challenging because the model “defies” the third law of thermodynamics.

SLIDE 52 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

AF 3-states Potts

q ≥ 3 AF is more challenging because the model “defies” the third law of thermodynamics. 3rd law: the entropy of a perfect crystal at absolute zero is zero.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 53 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

AF 3-states Potts

q ≥ 3 AF is more challenging because the model “defies” the third law of thermodynamics. 3rd law: the entropy of a perfect crystal at absolute zero is zero. The remaining entropy is called residual entropy.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 54 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature - highly connected

Benjamini, Haggstrom and Mossel (1999): What about the case n fixed, β = ∞, d → ∞?

SLIDE 55 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature - highly connected

Benjamini, Haggstrom and Mossel (1999): What about the case n fixed, β = ∞, d → ∞? Kahn (2001) and Galvin (2003): q = 3, n = 2, β = ∞, d → ∞ has six pure states.

SLIDE 56 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature - highly connected

Benjamini, Haggstrom and Mossel (1999): What about the case n fixed, β = ∞, d → ∞? Kahn (2001) and Galvin (2003): q = 3, n = 2, β = ∞, d → ∞ has six pure states. Galvin & Engbers (2012): Any q, n fixed, β = ∞, d → ∞ has many pure states.

SLIDE 57 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature - highly connected

Benjamini, Haggstrom and Mossel (1999): What about the case n fixed, β = ∞, d → ∞? Kahn (2001) and Galvin (2003): q = 3, n = 2, β = ∞, d → ∞ has six pure states. Galvin & Engbers (2012): Any q, n fixed, β = ∞, d → ∞ has many pure states. This is very encouraging, but fixed n is irrelevant for thermodynamical limits.

SLIDE 58 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature through other model

Galvin and Kahn(2004): d ≫ 0 hard-core (independent set) model has a phase transition.

David Galvin Jeff Kahn

SLIDE 59 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero Temperature through other model

Galvin and Kahn(2004): d ≫ 0 hard-core (independent set) model has a phase transition. Peled(2010): d ≫ 0 hom(Zd, Z) with zero boundary conditions fluctuate mainly between ±1.

Ron Peled

3

1

1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

2 2 2 2

1

2 2 2

1
1
1
1

SLIDE 60 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Homomorphism height functions and 3-colorings

There is a natural bijection between 3-colorings and hom(Zd, Z).

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3

1 2

6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

SLIDE 61 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Homomorphism height functions and 3-colorings

There is a natural bijection between 3-colorings and hom(Zd, Z).

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3

1 2

6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

HHF values between ±1 ⇒ Coloring values of even 0, odd 1,2.

SLIDE 62 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

SLIDE 63 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0.

SLIDE 64 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < exp

−

cd log2 d

.

SLIDE 65 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < exp

−

cd log2 d

.
This verifies the existence of at least six pure states.

SLIDE 66 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < exp

−

cd log2 d

.
This verifies the existence of at least six pure states.
A preliminary result on Glauber dynamics’ mixing was developed by

Galvin & Randall in 2007.

SLIDE 67 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < exp

−

cd log2 d

.
This verifies the existence of at least six pure states.
A preliminary result on Glauber dynamics’ mixing was developed by

Galvin & Randall in 2007.

The bound here deviates by log2 d factor from predicted estimates.

SLIDE 68 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Zero-temperature case of the Kotecky conjecture.

...and hence for β = ∞ the conjecture has been verified: 0-boundary rigidity at zero-temperature (Peled 2010) (Galvin, Kahn, Randall & Sorkin 2012) In a typical uniformly chosen proper 3-coloring with 0-boundary conditions in high dimensions nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < exp

−

cd log2 d

.
This verifies the existence of at least six pure states.
A preliminary result on Glauber dynamics’ mixing was developed by

Galvin & Randall in 2007.

The bound here deviates by log2 d factor from predicted estimates.
Zero-temperature has no physical meaning.

SLIDE 69 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method for β = ∞

The main proposition in Peled’s method is that external level line of length L around a vertex are exp(−cL/d log2 d) unlikely.

Level lines from Peled’s paper

SLIDE 70 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method for β = ∞

The main proposition in Peled’s method is that external level line of length L around a vertex are exp(−cL/d log2 d) unlikely.

Level lines from Peled’s paper

The main ingredient is the shift-minus transformation:

3

1

1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

2 2 2 2

1

2 2 2

1
1

2

1

Sublevel set

2

1

? 2 ? 2 2 2

1

? ? ? ? 2 2 ? 1 1 1 1 1 1 1 1 1

1

1 1 3 1

1

1 1 1

1
1

2

1

Shift

1

1

? 1 ? 1 1 1

1

? ? ? ? 1 1 ? 1 1 1 1 1 1 1 1 1

1

2

1
1
1

2

1

Shift + Minus

SLIDE 71 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method for β = ∞

The main proposition in Peled’s method is that external level line of length L around a vertex are exp(−cL/d log2 d) unlikely.

Level lines from Peled’s paper

The main ingredient is the shift-minus transformation, whose entropy gain is

L 2d .

3

1

1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

2 2 2 2

1

2 2 2

1
1

2

1

Sublevel set

2

1

? 2 ? 2 2 2

1

? ? ? ? 2 2 ? 1 1 1 1 1 1 1 1 1

1

1 1 3 1

1

1 1 1

1
1

2

1

Shift

1

1

? 1 ? 1 1 1

1

? ? ? ? 1 1 ? 1 1 1 1 1 1 1 1 1

1

2

1
1
1

2

1

Shift + Minus

SLIDE 72 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method and the special case of 3-states

Write FL for colorings with contour of length L around v. We thus map: each f ∈ FL, to 2L/2d other colorings. However this map is not one-to-many.

SLIDE 73 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method and the special case of 3-states

Write FL for colorings with contour of length L around v. We thus map: each f ∈ FL, to 2L/2d other colorings. However this map is not one-to-many. Roughly - the idea is to control the number of f with contour of length L, using the formula: |domain| < |image| · in-degree

ut-degree

SLIDE 74 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Peled’s method and the special case of 3-states

Write FL for colorings with contour of length L around v. We thus map: each f ∈ FL, to 2L/2d other colorings. However this map is not one-to-many. Roughly - the idea is to control the number of f with contour of length L, using the formula: |domain| < |image| · in-degree

ut-degree

Non-trivial. Hard to estimate in-degree, and requires either

(Peled) altering the map to avoid high in-degree.
(Galvin & al.) probabilistic biasing (flow method).

SLIDE 75 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond proper colorings of Zd

It is non-trivial to extend this result even to colorings of the torus:

1 1 1 2 2 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Periodic boundary conditions

SLIDE 76 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond proper colorings of Zd

The bijection does not extend to the torus.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3

1 2

6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

SLIDE 77 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond proper colorings of Zd

The bijection does not extend to the torus.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 2

Pointed 3-Colorings

mod 3

1 2

6 6 6 6 3 3 1 1 1 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

1

2 2 2 2 2 2 3 4 5 1 2 3 4 5 3 4 5 3 4 5 6 3 4 3 4 5 3 4 3 4 5 3 4 5 3 4 5 6 2 2 2 2 2 2 2 2 2 3 1 1

1

Pointed HHFs

However, algebraic topology says that it nearly does.

SLIDE 78 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond zero-temperature

Periodic boundary rigidity at zero-temperature (F. & Peled 2013) In high dimension, a typical uniformly chosen proper 3-coloring with periodic boundary conditions is nearly constant on either the even or odd sublattice.

SLIDE 79 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Beyond zero-temperature

Periodic boundary rigidity at zero-temperature (F. & Peled 2013) In high dimension, a typical uniformly chosen proper 3-coloring with periodic boundary conditions is nearly constant on either the even or odd sublattice.

This is a first step beyond the HHF structure.

SLIDE 80 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Positive temperature

SLIDE 81 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Positive temperature

Finding contours in positive temperature is quite problematic...

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1

β ≫ 0 sample

SLIDE 82 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Remark - Asymmetric case.

The 3-state AF Potts model has recently been studied on asymmetric planar lattices. Kotecky, Sokal and Swart (2013): In such lattices there is a phase transition at positive temperature, with 3 pure states.

Lattices from KSS paper

SLIDE 83 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Remark - Asymmetric case.

The 3-state AF Potts model has recently been studied on asymmetric planar lattices. Kotecky, Sokal and Swart (2013): In such lattices there is a phase transition at positive temperature, with 3 pure states. The proof uses the asymmetry to define and exploit better the phase interface.

Lattices from KSS paper

SLIDE 84 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Positive temperature on Zd

To implement the idea of Peled’s proof we require:

alternative for contours,
alternative for the transformation,
better method for using the entropy,
method to bound the in-degree of a coloring.

SLIDE 85 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components.

SLIDE 86 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices:

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 87 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 88 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0 Phase 3 := odd 0

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 89 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0 Phase 3 := odd 0 Phase 1 := odd 1, even 2

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 90 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0 Phase 3 := odd 0 Phase 1 := odd 1, even 2 Phase 2 := odd 2, even 1

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 91 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0 Phase 3 := odd 0 Phase 1 := odd 1, even 2 Phase 2 := odd 2, even 1 The improper edges are encoded by the phases.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 92 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

A key definition in approaching positive temperature is that of a Breakup (w.r.t. to a vertex v1), in lieu of Peled’s sublevel components. We start by defining four phases for vertices: Phase 0 := even 0 Phase 3 := odd 0 Phase 1 := odd 1, even 2 Phase 2 := odd 2, even 1 The improper edges are encoded by the phases.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 93 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

The first ingredient in our proof is a notion of a Breakup w.r.t. an

dd vertex v1. This - in lieu of Peled’s sublevel components.

We now repeatedly take co-connected closures:

complement → conn. component → complement 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 94 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1. We now repeatedly take co-connected closures:

complement → conn. component → complement 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 1 2 2 2 2

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 95 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 96 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 97 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 98 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 99 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase. 3 Co-conn. 1 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 100 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase. 3 Co-conn. 1 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 101 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase. 3 Co-conn. 1 phase. 4 Co-conn. 2 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 102 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase. 3 Co-conn. 1 phase. 4 Co-conn. 2 phase.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 103 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Breakup

Phase definition reminder 0: even 0 | 3: odd 0 | 1: odd 1, even 2 | 2: odd 2, even 1.

1 Co-conn. 0 phase. 2 Co-conn. 3 phase. 3 Co-conn. 1 phase. 4 Co-conn. 2 phase.

The result is the Breakup.

SLIDE 104 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 105 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 106 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 2 2 1 2 2 2 1 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 2 2 1 2 2 1 2 2 2 2 2 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2

SLIDE 107 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

2 2 2 1 2 2 2 1 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 2 2 1 2 2 1 2 2 2 2 2 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2

SLIDE 108 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 109 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Here we gain energy!

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 110 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 111 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 112 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1. Step 2: Shift + Minus transformation.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 113 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1. Step 2: Shift + Minus transformation.

1 Shift.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 114 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1. Step 2: Shift + Minus transformation.

1 Shift.

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 1 1 2 1 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 115 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1. Step 2: Shift + Minus transformation.

1 Shift. 2 Minus 1 (mod 3).

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 1 1 2 1 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 116 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

We now extend Peled’s transformation to breakups. Step 1: Flip.

1 Phase 2: 1 ⇐

⇒ 2.

2 Phase 3: x ⇒ x + 1.

Now treat: Phase 3 as 0, Phase 2 as 1. Step 2: Shift + Minus transformation.

1 Shift. 2 Minus 1 (mod 3).

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 117 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 118 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 119 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 120 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 121 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 122 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 1 1 2 1 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

SLIDE 123 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 124 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2 1

SLIDE 125 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 2 2 1 2 2 2 1 2 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 2 2 1 2 2 1 2 2 2 2 2 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 1 1 1 1 1 2 1 1 1 1 2 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 2

SLIDE 126 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Transformation family

The transformation is good since:

1 The boundary from one

direction gives us entropy...

2 ... except near certain

improper edges.

3 Given the breakup

everything is reversible.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 127 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

Properties of breakups:

When the coloring is proper it coincides with Peled’s contours.
We can show, using improved flow methods, that breakups

with long boundary are unlikely.

SLIDE 128 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0.

SLIDE 129 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < e−cd.

SLIDE 130 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < e−cd. This verifies the Kotecky conjecture for d ≫ 1.

SLIDE 131 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < e−cd. This verifies the Kotecky conjecture for d ≫ 1.

In particular - Implies the existence of at least 6 pure states.

SLIDE 132 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < e−cd. This verifies the Kotecky conjecture for d ≫ 1.

In particular - Implies the existence of at least 6 pure states.
Obtains the correct order of magnitude.

SLIDE 133 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Results

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a typical sample of the 3-state AF Potts with 0-boundary conditions and β > β0, nearly all the even vertices take the color 0. Formally: E |{v ∈ V even : f (v) = 0}| |V even| < e−cd. This verifies the Kotecky conjecture for d ≫ 1.

In particular - Implies the existence of at least 6 pure states.
Obtains the correct order of magnitude.
Open: show that β0 decreases with d.

SLIDE 134 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

SLIDE 135 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

SLIDE 136 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

2

P(f (u) = 0) < e−cd for all odd u,

SLIDE 137 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

2

P(f (u) = 0) < e−cd for all odd u,

3

P(f (u) = f (v)) < e−cd−β for all u ∼ v.

SLIDE 138 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

2

P(f (u) = 0) < e−cd for all odd u,

3

P(f (u) = f (v)) < e−cd−β for all u ∼ v.

2 2 1 1 1 1 2 1 2 2 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 2 2 1 2

SLIDE 139 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

2

P(f (u) = 0) < e−cd for all odd u,

3

P(f (u) = f (v)) < e−cd−β for all u ∼ v.

We also have a “structural” theorem

which provides similar bounds on other deviations from the pure state.

2 2 1 1 1 1 2 1 2 2 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 2 2 1 2

SLIDE 140 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Result

0-boundary rigidity at positive temperature (F. & Spinka 2015+) For every d high enough, there exists β0 such that in a sample of the 3-state AF Potts with 0-BC and β > β0, we have

1

P(f (v) = 0) < e−cd for all even v,

2

P(f (u) = 0) < e−cd for all odd u,

3

P(f (u) = f (v)) < e−cd−β for all u ∼ v.

We also have a “structural” theorem

which provides similar bounds on other deviations from the pure state.

Our results allow us to prove

convergence to an infinite-volume measure under 0-boundary conditions.

2 2 1 1 1 1 2 1 2 2 1 2 1 2 1 2 2 1 1 1 2 1 1 1 1 2 2 2 1 2

SLIDE 141 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

SLIDE 142 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

SLIDE 143 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

SLIDE 144 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

Periodic boundary conditions.

SLIDE 145 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

Periodic boundary conditions.
Four colors (and more), i.e., q ≥ 4.

SLIDE 146 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

Periodic boundary conditions.
Four colors (and more), i.e., q ≥ 4.
Other graph homomorphisms.

SLIDE 147 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

Periodic boundary conditions.
Four colors (and more), i.e., q ≥ 4.
Other graph homomorphisms.
Showing the existence of a single critical point.

SLIDE 148 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Open problems

Decrease of critical temperature with the dimension.

It is conjectured that βc(d) = Θ(1/d).

Periodic boundary conditions.
Four colors (and more), i.e., q ≥ 4.
Other graph homomorphisms.
Showing the existence of a single critical point.
Low dimensions, e.g., d = 3.

SLIDE 149 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

We are left with the the challenge of showing that a breakup is

rare. To explain this we should understand:

SLIDE 150 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

We are left with the the challenge of showing that a breakup is

rare. To explain this we should understand:
How to use the entropy wisely.

SLIDE 151 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

We are left with the the challenge of showing that a breakup is

rare. To explain this we should understand:
How to use the entropy wisely.
how to bound the indegree of every configuration.

SLIDE 152 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges

SLIDE 153 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges ≈ 2L/2d images

SLIDE 154 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges ≈ 2L/2d images eβM more likely

SLIDE 155 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges ≈ 2L/2d images eβM more likely BL,M breakups

SLIDE 156 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges ≈ 2L/2d images eβM more likely BL,M breakups

P(∃B ∈ BL,M : f ∈ B) ≤ |BL,M| · e−βM · 2−L/2d

SLIDE 157 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Bounding the indegree

Flow one measure unit from every coloring.

Potts samples L,M Breakup samples Potts samples Colorings in a Breakup T L - Boundary M - Improper edges ≈ 2L/2d images eβM more likely BL,M breakups

P(∃B ∈ BL,M : f ∈ B) ≤ |BL,M| · e−βM · 2−L/2d

Not good enough!

SLIDE 158 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Flow

Flow principle: Let S, D be two finite sets. Given a flow ν : S × D → [0, 1], such that for every s ∈ S, we have

d∈D ν(s, d) ≥ 1 and

for every d ∈ D, we have

s∈S ν(s, d) ≤ p,

we can deduce |S| ≤ p|D|.

SLIDE 159 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The transformation with a flow

Flow one measure unit from every coloring.

Potts samples L,M Breakup Approx. Potts samples Colorings in a Breakup T L - Boundary M - Improper edges

SLIDE 160 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The transformation with a flow

Flow one measure unit from every coloring.

Potts samples L,M Breakup Approx. Potts samples Colorings in a Breakup T L - Boundary M - Improper edges uneven flow

SLIDE 161 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The transformation with a flow

Flow one measure unit from every coloring.

Potts samples L,M Breakup Approx. Potts samples Colorings in a Breakup T L - Boundary M - Improper edges uneven flow ≈ e−cL/d mass, eβM more likely

SLIDE 162 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The transformation with a flow

Flow one measure unit from every coloring.

Potts samples L,M Breakup Approx. Potts samples Colorings in a Breakup T L - Boundary M - Improper edges uneven flow ≈ e−cL/d mass, eβM more likely AL,M breakups

SLIDE 163 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

The transformation with a flow

Flow one measure unit from every coloring.

Potts samples L,M Breakup Approx. Potts samples Colorings in a Breakup T L - Boundary M - Improper edges uneven flow ≈ e−cL/d mass, eβM more likely AL,M breakups

P(∃B ∈ AL,M : f ∈ B) ≤ |AL,M| · e−βM · e−cL/d

SLIDE 164 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 165 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 166 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

SLIDE 167 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges.

SLIDE 168 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges.

SLIDE 169 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges. We carefully increase the family using properties of the breakup.

SLIDE 170 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges. We carefully increase the family using properties of the breakup.

SLIDE 171 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges. We carefully increase the family using properties of the breakup. Finally we combine the information about our breakup from all phases, forming an information scheme.

SLIDE 172 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Approximation

A key step inspired by previous methods is to obtain a small family

f approximations for the Breakup.

First we obtain a small family

f crude dist-5-connected
approx. for each phase set.

We then add all the improper edges. We carefully increase the family using properties of the breakup. Finally we combine the information about our breakup from all phases, forming an information scheme. Here much of the technical innovation is hidden.

SLIDE 173 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Flow

We have more than enough entropy to find our breakup’s approximation, but not enough to enumerate over the missing information.

SLIDE 174 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Flow

We have more than enough entropy to find our breakup’s approximation, but not enough to enumerate over the missing information. Let us show how uneven flows help (in the simplest case).

1 2 1 1 1 2 1 1 1 1 2 2 1 2 1 1 2 1 1 1

SLIDE 175 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Flow

We have more than enough entropy to find our breakup’s approximation, but not enough to enumerate over the missing information. Let us show how uneven flows help (in the simplest case). If we use our entropy uniformly we get:

1 2 1 1 1 2 1 1 1 1 2 2 1 2 1 1 2 1 1 1

1/2 1/2 1/2 1/2 1/2 1/2 1/2 2/2 2/2 1/2

SLIDE 176 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Flow

We have more than enough entropy to find our breakup’s approximation, but not enough to enumerate over the missing information. Let us show how uneven flows help (in the simplest case). However if we use it more carefully we get:

1 2 1 1 1 2 1 1 1 1 2 2 1 2 1 1 2 1 1 1

3/4 1/4 1/2 1/2 1/4 3/4 3/4 3/4 3/4 3/4

SLIDE 177 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

What was not included?

SLIDE 178 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

What was not included?

How do we obtain the approximation?

SLIDE 179 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

What was not included?

How do we obtain the approximation?
How to generalize the flow?

(beyond the simplest case)

SLIDE 180 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

What was not included?

How do we obtain the approximation?
How to generalize the flow?

(beyond the simplest case)

Is there a height function counterpart? what does it mean?

SLIDE 181 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

What was not included?

How do we obtain the approximation?
How to generalize the flow?

(beyond the simplest case)

Is there a height function counterpart? what does it mean?

2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 2 1 2 2 1 1 1 2 2 2 2 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 2 2 1 2 2 1 1 1 2 1 2 2 1 1 2 2 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 2 2 2

SLIDE 182 Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations

Thank ank you!

Room temperature AF memory resistor

(Marti et al.)

Ferromagnetic Memories

1 1

Anti-Ferromagnetic Memories

1 1