long range order in random 3 colorings in high dimensions
play

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

Nov 04, 2022 •46 likes •1.87k views

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

Recommend

Random first-order transition of a spin glass model in three dimensions Toward a mean-field

Random first-order transition of a spin glass model in three dimensions Toward a mean-field

Random first-order transition of a spin glass model in three dimensions Toward a mean-field theory for glass transitions Koji Hukushima University of Tokyo, Dep. of Basic Sciences 14 August 2015 In collaboration with Takashi Takahasi.

534 views • 37 slides
Steganalysis in high dimensions: Fusing classifiers built on random subspaces Jan Kodovsk,

Steganalysis in high dimensions: Fusing classifiers built on random subspaces Jan Kodovsk,

Steganalysis in high dimensions: Fusing classifiers built on random subspaces Jan Kodovsk, Jessica Fridrich January 25, 2011 / SPIE Steganalysis in high dimensions:, Fusing classifiers built on random subspaces 1 / 14 Motivation Modern

536 views • 19 slides
Bootstrapping Spectral Statistics in High Dimensions Miles Lopes UC Davis Random Matrices and

Bootstrapping Spectral Statistics in High Dimensions Miles Lopes UC Davis Random Matrices and

Bootstrapping Spectral Statistics in High Dimensions Miles Lopes UC Davis Random Matrices and Complex Data Analysis Workshop Shanghai 2019 1 / 48 Bootstrap for sample covariance matrices Let X 1 , . . . , X n R p be i.i.d. observations,

1.61k views • 113 slides
Long range dependence for heavy Joint work with R. Kulik (U Ottawa), tailed random functions V.

Long range dependence for heavy Joint work with R. Kulik (U Ottawa), tailed random functions V.

Long range dependence for heavy Joint work with R. Kulik (U Ottawa), tailed random functions V. Makogin, M. Oesting (U Siegen), A. Rapp Evgeny Spodarev | Institute of Stochastics | 10.10.2019 Risk and Statistics, 2nd ISMUUlm Workshop Seite

975 views • 45 slides
Random Walks in Two Dimensions Leena Salmela January 31st, 2006 January 31st, 2006 Leena

Random Walks in Two Dimensions Leena Salmela January 31st, 2006 January 31st, 2006 Leena

Random Walks in Two Dimensions Random Walks in Two Dimensions Leena Salmela January 31st, 2006 January 31st, 2006 Leena Salmela Slide 1 Random Walks in Two Dimensions Examples Random walk in two dimensions. Escape routes and police.

626 views • 14 slides
Spatial Mixing of Coloring Random Graphs Yitong Yin Nanjing University Colorings undirected

Spatial Mixing of Coloring Random Graphs Yitong Yin Nanjing University Colorings undirected

Spatial Mixing of Coloring Random Graphs Yitong Yin Nanjing University Colorings undirected G(V,E) q colors: max-degree: d temporal mixing of Glauber dynamics approximately counting : 2 11/6 or sampling almost uniform proper q

687 views • 21 slides
Interpolation in high dimensions: Non-intrusive reduced order modeling Akil Narayan 1 1 Department

Interpolation in high dimensions: Non-intrusive reduced order modeling Akil Narayan 1 1 Department

Introduction Interpolation Non-adaptive Adaptive Conclusion Interpolation in high dimensions: Non-intrusive reduced order modeling Akil Narayan 1 1 Department of Mathematics University of Massachusetts Dartmouth June 7, 2013 GR-ROM @

658 views • 43 slides
Zeroth-order Optimization Yining Wang in High Dimensions Carnegie Mellon University Joint work

Zeroth-order Optimization Yining Wang in High Dimensions Carnegie Mellon University Joint work

AISTATS 2018, Playa Blanca, Lanzarote, Spain Zeroth-order Optimization Yining Wang in High Dimensions Carnegie Mellon University Joint work with Simon Du, Sivaraman Balakrishnan and Aarti Singh B ACKGROUND Optimization: min x X f ( x )

253 views • 23 slides
Discrete Fractal Dimensions of the Ranges of Random Walks Associate with Random Conductances

Discrete Fractal Dimensions of the Ranges of Random Walks Associate with Random Conductances

Introduction Main Results Proof Sketch and Main Ingredients Summary Discrete Fractal Dimensions of the Ranges of Random Walks Associate with Random Conductances Xinghua Zheng Department of ISOM, HKUST http://ihome.ust.hk/ xhzheng/

377 views • 16 slides
Nonlinear response theory in long-range Hamiltonian systems Yoshiyuki Y. YAMAGUCHI (Kyoto

Nonlinear response theory in long-range Hamiltonian systems Yoshiyuki Y. YAMAGUCHI (Kyoto

2014/05/27@The Galileo Galilei Institute for Theoretical Physics Advances in Nonequilibrium Statistical Mechanics: large deviations and long-range correlations, extreme value statistics, anomalous transport and long-range interactions Nonlinear

766 views • 48 slides
Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the

Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the

ST 380 Probability and Statistics for the Physical Sciences Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the union of one or more real number intervals; 1 P ( X = c ) = 0 for every c in the range of

410 views • 12 slides
Subcritical random hypergraphs, high-order components, and hypertrees Wenjie Fang, Institute of

Subcritical random hypergraphs, high-order components, and hypertrees Wenjie Fang, Institute of

Graph Hypergraph Strategy Remarks Subcritical random hypergraphs, high-order components, and hypertrees Wenjie Fang, Institute of Discrete Mathematics, TU Graz With Oliver Cooley, Nicola del Giudice and Mihyun Kang ANR-FWF-MOST meeting, TU

598 views • 25 slides
Hessian-based sampling in high dimensions for goal-oriented model order reduction Peng Chen Omar

Hessian-based sampling in high dimensions for goal-oriented model order reduction Peng Chen Omar

Hessian-based sampling in high dimensions for goal-oriented model order reduction Peng Chen Omar Ghattas Center for Computational Geosciences and Optimization The Institute for Computational Engineering and Sciences The University of Texas at

795 views • 60 slides
Riflescopes for Long-range Hunting March, 2020 LONG-RANGE HUNTING Rapidly gaining in

Riflescopes for Long-range Hunting March, 2020 LONG-RANGE HUNTING Rapidly gaining in

Riflescopes for Long-range Hunting March, 2020 LONG-RANGE HUNTING Rapidly gaining in popularity; Several Youtube channels (THLR.NO, EuLRH) have attracted a considerable number of followers; A riflescope is a must-have ; Often

649 views • 11 slides
Short Walks in Higher Dimensions Ghislain McKay Febuary 3, 2015 What is a Random Walk? A path

Short Walks in Higher Dimensions Ghislain McKay Febuary 3, 2015 What is a Random Walk? A path

Short Walks in Higher Dimensions Ghislain McKay Febuary 3, 2015 What is a Random Walk? A path formed by a succession of n steps (of unit length) in random directions. Figure: A 26-step random walk in the plane What is a Random Walk? A path

474 views • 44 slides
Long Range Facility Planning Long Range Facility Planning Phase 1 Facility recommendation to

Long Range Facility Planning Long Range Facility Planning Phase 1 Facility recommendation to

September 18, 2017 Richardson Independent School District Long Range Facility Planning Long Range Facility Planning Phase 1 Facility recommendation to address overcrowding at White Rock Elementary School Capacity and Utilization Analysis:

930 views • 46 slides
Long-Range Tactical Riflescopes April, 2020 WHAT IS CONSIDERED A LONG-RANGE SHOOTING DISTANCE?

Long-Range Tactical Riflescopes April, 2020 WHAT IS CONSIDERED A LONG-RANGE SHOOTING DISTANCE?

Long-Range Tactical Riflescopes April, 2020 WHAT IS CONSIDERED A LONG-RANGE SHOOTING DISTANCE? Today, shooting beyond 800 m is considered long-range; 20 years ago, hitting a target 1000 m away was considered an achievement but this is no

277 views • 10 slides
Polymer Physics Long-range interactions and chain scaling 10:10 11:45 Flory-Krigbaum theory

Polymer Physics Long-range interactions and chain scaling 10:10 11:45 Flory-Krigbaum theory

General Descriptions Overview Physical description of an isolated polymer chain Dimensionality and fractals Short-range and long-range interactions Packing length and tube diameter Polymer Physics Long-range interactions and chain scaling

1.16k views • 98 slides
Recent results around long-range Ising models: I) g versus Gibbs, II) Inhomogeneous fields, III)

Recent results around long-range Ising models: I) g versus Gibbs, II) Inhomogeneous fields, III)

Recent results around long-range Ising models: I) g versus Gibbs, II) Inhomogeneous fields, III) Metastates and random boundary conditions, IV) Metastability V) Interfaces A. C. D. v.E., with R. Bissacot, L. Coquille, E. Endo, B. Kimura, A.

904 views • 42 slides
February 13, 2017 Richardson Independent School District Long Range Facility Planning Long Range

February 13, 2017 Richardson Independent School District Long Range Facility Planning Long Range

February 13, 2017 Richardson Independent School District Long Range Facility Planning Long Range Facility Planning Phase 1 Facility recommendation to address overcrowding at White Rock Elementary School (WRE) Long Range Facility Planning

614 views • 59 slides
HAUSDORFF DIMENSIONS FOR RANDOM WALKS AND THEIR WINDING SECTORS LPTMS ORSAY CNRS/Universit e

HAUSDORFF DIMENSIONS FOR RANDOM WALKS AND THEIR WINDING SECTORS LPTMS ORSAY CNRS/Universit e

HAUSDORFF DIMENSIONS FOR RANDOM WALKS AND THEIR WINDING SECTORS LPTMS ORSAY CNRS/Universit e Paris-Sud with Jean Desbois JSTAT (2008) P08004 arXiv: 0804.1002 about some recent numerical simulations for random walks on a square lattice

317 views • 20 slides
2015 - 2019 Long Range Financial Plan 2015 - 2019 Long Range Financial Plan Tonights Goals

2015 - 2019 Long Range Financial Plan 2015 - 2019 Long Range Financial Plan Tonights Goals

2015 - 2019 Long Range Financial Plan 2015 - 2019 Long Range Financial Plan Tonights Goals Discuss the economic context over planning horizon. Review the 2015-2019 Long Range Financial Plan with a focus on FY 2014-15. Explain the

1k views • 35 slides
Rehashing Kernel Evaluation in High Dimensions Paris Siminelakis* Ph.D. Candidate Kexin Rong*,

Rehashing Kernel Evaluation in High Dimensions Paris Siminelakis* Ph.D. Candidate Kexin Rong*,

Intro Contribution Sketching Diagnostics Evaluation Conclusion Rehashing Kernel Evaluation in High Dimensions Paris Siminelakis* Ph.D. Candidate Kexin Rong*, Peter Bailis, Moses Charikar, Phillip Levis (Stanford University) ICML @ Long

1.07k views • 80 slides
Long-Range Planning and Behavioral Biases: A Computational Approach Jon Kleinberg Including

Long-Range Planning and Behavioral Biases: A Computational Approach Jon Kleinberg Including

Long-Range Planning and Behavioral Biases: A Computational Approach Jon Kleinberg Including joint work with Manish Raghavan and Sigal Oren. Cornell University Long-Range Planning Growth in on-line systems where users and groups have long

552 views • 44 slides

More recommend