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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

  1. 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 − β N ( f ) . 0 0 0 0 1 0 1 0 1 0 (Stationary distribution of Glauber 0 1 0 1 0 1 1 1 0 dynamics) 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0

  2. 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 − β N ( f ) . 0 0 0 0 1 0 1 0 1 0 (Stationary distribution of Glauber 0 1 0 1 0 1 1 1 0 dynamics) 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 • Often taken under external field 0 1 0 1 0 1 0 1 0 (giving a bias for seeing + vs. − ). 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0

  3. 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 − β N ( f ) . 0 0 0 0 1 0 1 0 1 0 (Stationary distribution of Glauber 0 1 0 1 0 1 1 1 0 dynamics) 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 • Often taken under external field 0 1 0 1 0 1 0 1 0 (giving a bias for seeing + vs. − ). 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 • Ferromagnet ( β < 0 ) and 0 1 0 1 0 1 0 anti-ferromagnet ( β > 0 ) are 0 0 0 equivalent.

  4. 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 ).

  5. 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

  6. 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.

  7. 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. 0 0 0 • Λ large domain. 0 0 0 0 • Condition on f ( v ) = τ for all v on 0 0 the boundary. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Even zero boundary conditions

  8. 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. 0 0 0 • Λ large domain. 0 1 0 1 0 1 0 • Condition on f ( v ) = τ for all v on 0 1 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 Sample with 0-boundary conditions on even domain

  9. 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. 0 0 0 • Λ large domain. 0 1 0 1 0 1 0 • Condition on f ( v ) = τ for all v on 0 1 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 • Does the distribution in the center 0 1 0 1 0 1 0 1 0 depend on τ ? 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 sample with 0-boundary conditions on even domain

  10. 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. 0 0 0 • Λ large domain. 0 1 0 1 0 1 0 • Condition on f ( v ) = τ for all v on 0 1 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 • Does the distribution in the center 0 1 0 1 0 1 0 1 0 depend on τ ? 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 • Ordered phase: Yes. 0 1 0 1 0 1 0 1 0 Disordered phase: No. 0 1 0 1 0 1 0 0 0 0 sample with 0-boundary conditions on even domain

  11. 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. 0 0 0 • Λ large domain. 0 1 0 1 0 1 0 • Condition on f ( v ) = τ for all v on 0 1 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 • Does the distribution in the center 0 1 0 1 0 1 0 1 0 depend on τ ? 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 • Ordered phase: Yes. 0 1 0 1 0 1 0 1 0 Disordered phase: No. 0 1 0 1 0 1 0 • Mature notions: 0 0 0 Gibbs measures & pure phases. sample with 0-boundary conditions on even domain

  12. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Questions about the model Basic Questions:

  13. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Questions about the model Basic Questions: • In which d does a phase transition occur? 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  14. 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? 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1

  15. 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? 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  16. 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? 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1

  17. 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? 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  18. 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.

  19. 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. 0 0 0 • Condition on f ( v ) = 0 for all v on 0 0 0 0 the boundary. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Even zero boundary conditions

  20. 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. 0 0 0 • Condition on f ( v ) = 0 for all v on 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 Sample with 0-boundary conditions on even domain

  21. 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. 0 0 0 • Condition on f ( v ) = 0 for all v on 0 1 0 1 0 1 0 the boundary. 0 1 0 1 0 1 0 1 0 • Show that the frequencies on even 0 1 0 1 0 1 0 1 0 and odd sublattice are unbalanced. 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 sample with 0-boundary conditions on even domain

  22. 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 /β c = Θ( d ) (error terms are known). • β < β c implies a unique pure state. • β > β c implies two pure states. • In β > β c one sublattice is biased towards + and the other towards − . Ising 2 d ferromagnets and anti-ferromagnets: β = ≪ 0 F − Crit < 0 > 0 AF − Crit ≫ 0 −∞ ∞

  23. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Beyond Ising Clock and Potts models. Cyril Domb Renfrey Po�s

  24. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Kotecky Conjecture Baxter (1982): d = 2 , q = 3 Potts AF - critical at β = ∞ . Rodney Baxter

  25. 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 Z d , there exists a minimal d 0 (probably d 0 = 3 ) such that for d ≥ d 0 there is a positive critical temperature 1 /β c . Roman Kotecký 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  26. 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 Z d , there exists a minimal d 0 (probably d 0 = 3 ) such that for d ≥ d 0 there is a positive critical temperature 1 /β c . Roman Kotecký 0 0 0 • For β > β c : six pure states (phase 0 2 0 2 0 1 0 co-existence). 0 1 0 1 0 2 0 1 0 0 1 0 1 0 2 0 1 0 • Each state corresponds to one color 0 2 0 2 0 1 0 2 0 dominant on one sublattice and nearly 0 1 0 2 0 2 0 1 0 0 1 0 1 0 1 0 1 0 absent from the other. 0 2 0 2 0 2 0 2 0 0 2 0 1 0 1 0 0 0 0

  27. 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 Z d , there exists a minimal d 0 (probably d 0 = 3 ) such that for d ≥ d 0 there is a positive critical temperature 1 /β c . Roman Kotecký 0 0 0 • For β > β c : six pure states (phase 0 2 0 2 0 1 0 co-existence). 0 1 0 1 0 2 0 1 0 0 1 0 1 0 2 0 1 0 • Each state corresponds to one color 0 2 0 2 0 1 0 2 0 dominant on one sublattice and nearly 0 1 0 2 0 2 0 1 0 0 1 0 1 0 1 0 1 0 absent from the other. 0 2 0 2 0 2 0 2 0 0 2 0 1 0 1 0 • For β < β c : one disordered pure phase, 0 0 0 correlations decay exponentially fast.

  28. 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 .

  29. 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 . 0 1 2 0 2 1 0 2 0 1 1 0 1 2 1 0 1 0 1 2 2 1 0 1 0 2 0 2 0 1 1 0 1 0 2 0 1 0 1 0 0 2 0 2 1 2 0 1 0 2 2 0 1 0 2 0 1 0 2 1 1 2 0 2 0 1 2 1 0 2 2 0 2 1 2 0 1 0 1 0 0 1 0 2 0 1 2 1 2 1 1 2 1 0 1 2 0 2 0 2

  30. 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 . 0 1 2 0 2 1 0 2 0 1 1 0 1 2 1 0 1 0 1 2 2 1 0 1 0 2 0 2 0 1 1 0 1 0 2 0 1 0 1 0 0 2 0 2 1 2 0 1 0 2 2 0 1 0 2 0 1 0 2 1 1 2 0 2 0 1 2 1 0 2 2 0 2 1 2 0 1 0 1 0 0 1 0 2 0 1 2 1 2 1 1 2 1 0 1 2 0 2 0 2

  31. 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 → ∞ ?

  32. 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.

  33. 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.

  34. 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.

  35. 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

  36. 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( Z d , Z ) with zero boundary conditions fluctuate mainly between ± 1 . Ron Peled 0 0 0 0 -1 0 -1 0 1 0 0 1 0 1 0 -1 0 1 0 0 1 2 1 2 1 0 1 0 0 1 2 3 2 1 0 -1 0 0 1 2 1 2 1 0 1 0 0 1 2 1 0 1 -1 1 0 0 1 0 1 0 -1 0 1 0 0 -1 0 1 0 -1 0 0 0 0

  37. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Homomorphism height functions and 3 -colorings There is a natural bijection between 3 -colorings and hom( Z d , Z ) . 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 6 0 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 7 2 1 1 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 0 1 2 0 1 -1 0 1 2 1 0 1 2 3 4 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 2 0 2 0 1 -1 0 1 2 1 2 3 2 3 4 0 1 2 0 2 1 2 0 1 2 mod 3 0 1 2 3 2 1 2 3 4 5 1 0 1 2 0 2 0 1 2 0 1 0 1 2 3 2 3 4 5 6 0 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 Pointed 3-Colorings Pointed HHFs

  38. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Homomorphism height functions and 3 -colorings There is a natural bijection between 3 -colorings and hom( Z d , Z ) . 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 6 0 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 7 2 1 1 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 0 1 2 0 1 -1 0 1 2 1 0 1 2 3 4 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 2 0 2 0 1 -1 0 1 2 1 2 3 2 3 4 0 1 2 0 2 1 2 0 1 2 mod 3 0 1 2 3 2 1 2 3 4 5 1 0 1 2 0 2 0 1 2 0 1 0 1 2 3 2 3 4 5 6 0 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 Pointed 3-Colorings Pointed HHFs HHF values between ± 1 ⇒ Coloring values of even 0 , odd 1 , 2 .

  39. 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 0 0 0 1 0 2 0 1 0 0 1 0 1 0 1 2 1 0 0 1 2 1 2 1 2 1 0 0 1 2 0 2 1 0 2 0 0 1 2 1 2 1 0 1 0 0 1 2 1 0 1 2 1 0 0 1 0 1 0 2 0 1 0 0 2 0 1 0 2 0 0 0 0

  40. 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 .

  41. 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 }| � � cd < exp − . log 2 d | V even |

  42. 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 }| � � cd < exp − . log 2 d | V even | • This verifies the existence of at least six pure states.

  43. 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 }| � � cd < exp − . log 2 d | V even | • This verifies the existence of at least six pure states. • A preliminary result on Glauber dynamics’ mixing was developed by Galvin & Randall in 2007.

  44. 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 }| � � cd < exp − . log 2 d | V even | • 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 log 2 d factor from predicted estimates.

  45. 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 }| � � cd < exp − . log 2 d | V even | • 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 log 2 d factor from predicted estimates. • Zero-temperature has no physical meaning.

  46. 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 log 2 d ) unlikely. Level lines from Peled’s paper

  47. 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 log 2 d ) unlikely. Level lines from Peled’s paper The main ingredient is the shift-minus transformation: 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 1 0 0 -1 0 -1 0 1 0 0 -1 0 -1 0 1 0 0 ? 0 ? 0 -1 0 1 0 0 1 0 1 0 -1 0 1 0 0 ? 0 ? 0 -1 0 1 0 0 1 0 1 0 ? 0 1 0 0 1 2 1 2 1 0 1 0 0 2 1 2 1 ? 0 1 0 0 1 2 3 2 1 0 -1 0 0 2 3 2 1 ? 0 -1 0 0 1 2 1 0 ? 0 -1 0 0 1 2 1 2 1 0 1 0 0 2 1 2 1 ? 0 1 0 0 1 0 1 0 ? 0 1 0 0 1 2 1 0 1 2 1 0 0 2 1 ? 0 1 2 1 0 0 1 0 ? 0 1 2 1 0 0 1 0 1 0 -1 0 1 0 0 1 0 ? 0 -1 0 1 0 0 1 0 ? 0 -1 0 1 0 0 -1 0 1 0 -1 0 0 -1 0 1 0 -1 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 Shift + Minus Sublevel set Shift

  48. 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 log 2 d ) unlikely. Level lines from Peled’s paper The main ingredient is the shift-minus transformation, L whose entropy gain is 2 d . 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 1 0 0 -1 0 -1 0 1 0 0 -1 0 -1 0 1 0 0 ? 0 ? 0 -1 0 1 0 0 1 0 1 0 -1 0 1 0 0 ? 0 ? 0 -1 0 1 0 0 1 0 1 0 ? 0 1 0 0 1 2 1 2 1 0 1 0 0 2 1 2 1 ? 0 1 0 0 1 2 3 2 1 0 -1 0 0 2 3 2 1 ? 0 -1 0 0 1 2 1 0 ? 0 -1 0 0 1 2 1 2 1 0 1 0 0 2 1 2 1 ? 0 1 0 0 1 0 1 0 ? 0 1 0 0 1 2 1 0 1 2 1 0 0 2 1 ? 0 1 2 1 0 0 1 0 ? 0 1 2 1 0 0 1 0 1 0 -1 0 1 0 0 1 0 ? 0 -1 0 1 0 0 1 0 ? 0 -1 0 1 0 0 -1 0 1 0 -1 0 0 -1 0 1 0 -1 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 Shift + Minus Sublevel set Shift

  49. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Peled’s method and the special case of 3 -states Write F L for colorings with contour of length L around v . We thus map: each f ∈ F L , to 2 L / 2 d other colorings. However this map is not one-to-many.

  50. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Peled’s method and the special case of 3 -states Write F L for colorings with contour of length L around v . We thus map: each f ∈ F L , to 2 L / 2 d 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 out-degree

  51. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Peled’s method and the special case of 3 -states Write F L for colorings with contour of length L around v . We thus map: each f ∈ F L , to 2 L / 2 d 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 out-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).

  52. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Beyond proper colorings of Z d It is non-trivial to extend this result even to colorings of the torus: 0 1 0 1 0 2 0 1 0 1 0 1 0 1 2 1 0 1 0 1 0 1 2 1 0 1 0 1 2 1 0 1 2 0 2 1 0 1 2 0 2 1 2 0 2 1 2 1 2 1 2 1 0 1 2 1 2 1 0 1 0 1 0 1 2 1 0 1 0 1 2 1 0 1 0 1 0 2 0 1 2 1 0 1 2 1 0 2 0 1 2 1 0 2 0 1 0 1 0 1 0 1 0 2 0 1 2 1 2 1 0 1 0 1 0 2 0 1 0 1 0 Periodic boundary conditions

  53. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Beyond proper colorings of Z d The bijection does not extend to the torus. 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 6 0 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 7 2 1 1 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 0 1 2 0 1 -1 0 1 2 1 0 1 2 3 4 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 2 0 2 0 1 -1 0 1 2 1 2 3 2 3 4 0 1 2 0 2 1 2 0 1 2 mod 3 0 1 2 3 2 1 2 3 4 5 1 0 1 2 0 2 0 1 2 0 1 0 1 2 3 2 3 4 5 6 0 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 Pointed 3-Colorings Pointed HHFs

  54. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Beyond proper colorings of Z d The bijection does not extend to the torus. 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 6 0 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 7 2 1 1 2 1 0 1 2 0 1 2 0 1 2 1 0 1 2 3 4 5 6 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 0 1 2 0 1 -1 0 1 2 1 0 1 2 3 4 0 1 0 1 2 1 2 0 1 2 0 1 0 1 2 1 2 3 4 5 2 0 1 2 1 2 0 2 0 1 -1 0 1 2 1 2 3 2 3 4 0 1 2 0 2 1 2 0 1 2 mod 3 0 1 2 3 2 1 2 3 4 5 1 0 1 2 0 2 0 1 2 0 1 0 1 2 3 2 3 4 5 6 0 1 0 1 2 0 1 2 0 1 0 1 0 1 2 3 4 5 6 7 Pointed 3-Colorings Pointed HHFs However, algebraic topology says that it nearly does.

  55. 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.

  56. 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.

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

  58. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Positive temperature Finding contours in positive temperature is quite problematic... 1 2 1 0 2 0 2 0 1 2 0 2 0 1 0 2 0 1 0 2 1 0 1 0 1 0 1 2 1 0 0 1 2 2 2 1 2 1 0 2 1 0 0 1 0 2 1 0 2 0 0 1 2 1 2 1 0 1 0 1 1 0 1 2 1 0 1 2 1 0 0 1 0 1 0 2 0 1 0 2 2 0 2 0 1 0 2 0 2 1 1 1 0 2 0 1 0 1 1 2 β ≫ 0 sample

  59. 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

  60. 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

  61. Introduction Kotecky Conjecture Zero-temperature Positive-temperature Approximations Positive temperature on Z d 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.

  62. 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 v 1 ), in lieu of Peled’s sublevel components.

  63. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  64. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  65. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 Phase 3 := odd 0 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  66. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 Phase 3 := odd 0 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 Phase 1 := odd 1 , even 2 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  67. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 Phase 3 := odd 0 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 Phase 1 := odd 1 , even 2 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 Phase 2 := odd 2 , even 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  68. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 Phase 3 := odd 0 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 Phase 1 := odd 1 , even 2 2 0 2 1 2 1 0 1 0 0 1 0 1 0 2 1 0 1 Phase 2 := odd 2 , even 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 The improper edges are 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 encoded by the phases. 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  69. 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 v 1 ), in lieu of Peled’s sublevel components. We start by defining four 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 phases for vertices: 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 Phase 0 := even 0 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 Phase 3 := odd 0 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 Phase 1 := odd 1 , even 2 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 Phase 2 := odd 2 , even 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 The improper edges are 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 encoded by the phases. 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  70. 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 odd vertex v 1 . This - in lieu of Peled’s sublevel components. We now repeatedly take 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 co-connected closures : 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 complement → conn. component → complement 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 0 2 0 2 0 1 2 1 2 2 0 2 0 2 0 1 2 1 2 0 1 0 1 1 2 1 0 1 0 1 1 2 1 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 2 2 0 2 1 2 2 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 2 0 2 1 2 0 1 0 0 1 2 0 2 1 2 0 1 0 0 1 0 1 0 2 2 1 2 1 2 1 2 0 1 0 2 2 1 2 1 2 1 2 2 0 2 0 2 0 2 2 0 2 0 2 0 2 0 2 2 0 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 0 2 0 2 0 1 2 1 2 2 0 2 0 2 0 1 2 1 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 0 1 0 1 1 2 1 0 1 0 1 1 2 1 2 0 2 1 2 2 2 0 2 1 2 2 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 2 0 2 1 2 0 1 0 0 1 2 0 2 1 2 0 1 0 0 1 0 1 0 2 2 1 2 1 2 1 2 0 1 0 2 2 1 2 1 2 1 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 2 0 2 0 2 2 0 2 0 2 0 2 0 2 2 0 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 0 2 1 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  71. 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 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 co-connected closures : 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 complement → conn. component → complement 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 0 2 0 2 0 1 2 1 2 2 0 2 0 2 0 1 2 1 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 0 1 0 1 1 2 1 0 1 0 1 1 2 1 2 0 2 1 2 2 2 0 2 1 2 2 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 2 0 2 1 2 0 1 0 0 1 2 0 2 1 2 0 1 0 0 1 0 1 0 2 2 1 2 1 2 1 2 0 1 0 2 2 1 2 1 2 1 2 2 0 2 0 2 0 2 2 0 2 0 2 0 2 0 2 2 0 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 1 2 1 2 2 0 2 0 2 0 1 2 1 2 0 1 0 1 1 2 1 0 1 0 1 1 2 1 2 0 2 1 2 2 2 0 2 1 2 2 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 2 0 2 1 2 0 1 0 0 1 2 0 2 1 2 0 1 0 0 1 0 1 0 2 2 1 2 1 2 1 2 0 1 0 2 2 1 2 1 2 1 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 2 0 2 0 2 2 0 2 0 2 0 2 0 2 2 0 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 2 2 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 2 0 2 1 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  72. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  73. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  74. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 Co-conn. 3 phase. 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  75. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 Co-conn. 3 phase. 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  76. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 Co-conn. 3 phase. 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 3 Co-conn. 1 phase. 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

  77. 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 . 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Co-conn. 0 phase. 2 0 2 0 2 0 1 2 1 2 1 2 1 0 2 0 2 0 1 0 1 0 1 0 1 2 0 2 1 2 2 0 1 0 2 Co-conn. 3 phase. 2 0 2 0 1 2 1 2 1 2 0 2 0 1 2 0 2 0 1 0 1 1 1 2 0 2 0 1 0 1 0 0 1 0 3 Co-conn. 1 phase. 2 0 2 1 2 1 0 1 0 1 0 1 0 2 1 0 1 0 1 0 2 1 2 1 2 1 2 1 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 1 2 0 2 0 1 0 2 0 1 0 1 0 1 0 2 0 2 0 2 0 1 2 1 0 2 0 2 0 2 0 2 0 2 0 2 1 2 1 1 0 2 0 1 0 1 0 1 0 2 0 1 0 2 0 2 0 1 0 2 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

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