3-Coloring the Discrete Torus or Rigidity of zero temperature - PowerPoint PPT Presentation

Introduction Proof overview Ideas and Method Open problems 3-Coloring the Discrete Torus or Rigidity of zero temperature 3-states anti-ferromagnetic Potts model Ohad N. Feldheim Joint work with Ron Peled Institute for Mathematics and its Application IMA PostDoc Seminar, October, 2014

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems 3-Colorings of the Grid/Torus 0 1 0 1 0 2 0 1 0 1 0 0 0 0 1 0 1 2 1 0 1 0 1 0 1 0 1 0 2 0 1 0 2 1 0 1 0 1 2 1 0 1 2 0 1 0 1 0 1 2 1 0 0 2 1 0 1 2 0 2 1 2 0 0 1 2 1 2 1 2 1 0 2 1 2 1 2 1 2 1 0 1 2 0 1 2 0 2 1 0 2 0 1 2 1 0 1 0 1 0 1 2 1 0 1 2 1 2 1 0 1 0 0 1 0 1 2 1 0 1 0 1 0 0 1 2 1 0 1 2 1 0 2 0 1 2 1 0 1 2 1 0 2 0 1 0 1 0 2 0 1 0 0 1 2 1 0 2 0 1 0 1 0 0 2 0 1 0 2 0 1 0 1 0 2 0 1 2 1 2 1 0 0 0 0 1 0 1 0 2 0 1 0 1 0 Zero boundary conditions Periodic boundary conditions

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Random 3-Colorings 0 1 0 1 0 2 0 1 0 1 0 0 0 0 1 0 1 2 1 0 1 0 1 0 1 0 1 0 2 0 1 0 2 1 0 1 0 1 2 1 0 1 2 0 1 0 1 0 1 2 1 0 0 2 1 0 1 2 0 2 1 2 0 0 1 2 1 2 1 2 1 0 2 1 2 1 2 1 2 1 0 1 2 0 1 2 0 2 1 0 2 0 1 2 1 0 1 0 1 0 1 2 1 0 1 2 1 2 1 0 1 0 0 1 0 1 2 1 0 1 0 1 0 0 1 2 1 0 1 2 1 0 2 0 1 2 1 0 1 2 1 0 2 0 1 0 1 0 2 0 1 0 0 1 2 1 0 2 0 1 0 1 0 0 2 0 1 0 2 0 1 0 1 0 2 0 1 2 1 2 1 0 0 0 0 1 0 1 0 2 0 1 0 1 0 Zero boundary Periodic boundary conditions conditions Uniformly chosen proper 3-coloring (Given boundary conditions) High dimension Z d , and T d n .

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Additional Motivation Mathematical Physics Combinatorics Physics 2 2 0 2 0 2 0 2 1 0 2 0 1 0 1 0 2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 2 1 0 1 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 2 1 0 1 0 1 2 1 0 1 2 1 0 1 0 1 0 1 2 1 0 1 0 2 1 0 1 2 0 2 1 2 0 0 1 0 1 2 1 2 1 0 1 0 2 1 2 1 2 1 2 1 0 1 2 2 0 1 2 2 2 1 0 2 0 2 1 2 1 0 1 0 1 0 1 2 1 0 1 2 1 2 1 0 1 0 2 0 0 1 0 1 2 1 0 1 0 1 0 2 0 1 2 1 0 1 2 1 0 2 2 0 1 2 1 0 1 1 2 0 2 0 1 2 1 0 2 0 1 0 1 0 0 1 0 1 0 2 0 1 0 1 0 1 0 1 0 2 0 1 2 1 2 1 1 0 2 0 1 0 2 0 1 2 1 0 1 0 1 0 2 0 1 0 1 0 2 2 0 2 0 2 0 2 1 0 2 q-states antiferromagnetic q-colorings of the Antiferromagnetism discrete torus Potts model Generalizes the celebrated Ising model. Each point takes one of q values. Neighbors dislike getting the same color. 3-coloring is the “zero temperature” version.

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )?

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern?

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern? Is there long range correlation?

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern? 1 0 1 0 1 0 2 0 1 0 Is there long range correlation? 0 1 0 1 0 1 0 1 0 1 Does it look roughly like this? 2 0 2 0 1 0 2 0 1 0 0 1 0 2 0 1 0 1 0 1 1 0 2 0 1 0 0 1 0 1 2 0 1 0 2 0 1 0 2 0 1 2 0 1 0 2 0 1 0 1 0 0 1 0 1 0 2 0 2 0 2 1 0 1 0 2 0 1 0 2 0 0 1 0 1 0 2 0 1 0 1

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern? 1 0 1 0 1 0 2 0 1 0 Is there long range correlation? 0 1 0 1 0 1 0 1 0 1 Does it look roughly like this? 2 0 2 0 1 0 2 0 1 0 0 1 0 2 0 1 0 1 0 1 1 0 2 0 1 0 0 1 0 1 2 Conjecture: 0 1 0 2 0 1 0 2 0 1 2 0 1 0 2 0 1 0 1 0 0 1 0 1 0 2 0 2 0 2 1 0 1 0 2 0 1 0 2 0 0 1 0 1 0 2 0 1 0 1

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern? 1 0 1 0 1 0 2 0 1 0 Is there long range correlation? 0 1 0 1 0 1 0 1 0 1 Does it look roughly like this? 2 0 2 0 1 0 2 0 1 0 0 1 0 2 0 1 0 1 0 1 1 0 2 0 1 0 0 1 0 1 2 Conjecture: 0 1 0 2 0 1 0 2 0 1 d = 2 No. 2 0 1 0 2 0 1 0 1 0 0 1 0 1 0 2 0 2 0 2 1 0 1 0 2 0 1 0 2 0 0 1 0 1 0 2 0 1 0 1

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Properties of Interest In a typical coloring: What is the typical relative frequency of the colors? Is it ( 1 / 3 , 1 / 3 , 1 / 3 )? Does a typical coloring follow some pattern? 1 0 1 0 1 0 2 0 1 0 Is there long range correlation? 0 1 0 1 0 1 0 1 0 1 Does it look roughly like this? 2 0 2 0 1 0 2 0 1 0 0 1 0 2 0 1 0 1 0 1 1 0 2 0 1 0 0 1 0 1 2 Conjecture: 0 1 0 2 0 1 0 2 0 1 d = 2 No. 2 0 1 0 2 0 1 0 1 0 0 1 0 1 0 2 0 2 0 2 d > 2 Yes. 1 0 1 0 2 0 1 0 2 0 0 1 0 1 0 2 0 1 0 1

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Previous Results - Rigidity for 0-boundary The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0.

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Previous Results - Rigidity for 0-boundary The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC , has: E |{ v ∈ V e ven : g ( v ) � = 0 }| � � cd − < exp . log 2 d | V even |

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Previous Results - Rigidity for 0-boundary The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC , has: E |{ v ∈ V e ven : g ( v ) � = 0 }| � � cd − < exp . log 2 d | V even | Does not work for periodic BC.

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Previous Results - Rigidity for 0-boundary The conjecture has been established for 0-boundary conditions in high dimension. 0-boundary rigidity (Peled 2010) In a typical 3-coloring with 0-boundary conditions nearly all the even vertices take the color 0. Formally: Let d be large enough, a uniformly chosen 3-coloring with 0-BC , has: E |{ v ∈ V e ven : g ( v ) � = 0 }| � � cd − < exp . log 2 d | V even | Does not work for periodic BC. Open in low dimensions.

Introduction Coloring the Grid Proof overview Previous results Ideas and Method Our results Open problems Previous Results - Rigidity for the hypercube The conjecture has also been supported on bounded tori. Periodic boundary on the even hypercube (Galvin & Engbers 2011) For every fixed n , for high enough dimension (depndeing on n ), a typical 3-coloring with periodic boundary conditions is nearly constant on either the even or the odd sublattice.