Variational principles for discrete maps Martin Tassy, joint work - PowerPoint PPT Presentation

Variational principles for discrete maps Martin Tassy, joint work with Georg Menz October 12, 2017 UCLA

Limit shape “Aztec Diamond” for domino tilings Colors represent the parity of the dominos 1

Overview • We study random discrete maps. • Limit shapes are a universal phenomenon. • A variational principle explains: • the occurrence of limit shapes • the shape of the limit. 2

Overview • Methods for deducing variational principles: • so far: based on integrability/ lattice structure of the model. • now: based on Kirszbraun theorem and concentration inequality. • Main result: Variational principle for graph homomorphisms to a tree • First variational principle for a model of random discrete maps that is not integrable. 3

Object we study Graph homomorphism h : G 1 → G 2 : x ∼ y ⇒ h ( x ) ∼ h ( y ) . Example 1: Graph homomorphism to Z : h : Z 2 ⊃ Λ → Z ”height function” Example 2: Graph homomorphism to 3 regular tree T : h : Z 2 ⊃ Λ → T ”Tree-valued height function” 4

Objects we study Random graph homomorphism: 1. Fix Λ ⊂ Z d 2. Fix boundary data g : ∂ Λ → Z 3. Pick uniformly at random one element of � � h : Λ → Z | h is height function and h | ∂ Λ = g For large Λ and under proper rescaling you will see ”limit shapes”. 5

Limit shape for graph homomorphism into Z Colors represent the parity of the height in Z 6

Limit shape for graph homomorphism into 3 regular tree T 7

Appearance of limit shapes is a universal phenomenon

Limit shape for lozenge tilings Each color represents a type of lozenge 8

Limit shape “Aztec Diamond” for domino tilings Colors represent the parity of the dominos 9

Limit shape for tilings by 3-1 bars 10

Limit shape for ribbon tilings 11

Variational principle for domino tilings and height functions h : Λ → Z

Preparations Two questions: • How many height functions h : Λ → Z with given boundary values exist? • What do they asymptotically look like? 12

Preparations Microscopic entropy Ent(Λ , g ∂ Λ ) := 1 � �� �� | Λ | ln h : Λ → Z | h is height function and h | ∂ Λ = g � Microscopic surface tension ent n ( s , t ) := 1 n 2 ln |{ h : B n → Z | h is height function and for all ( x 1 , x 2 ) ∈ ∂ B n : h ( x 1 , x 2 ) ≈ s · x 1 + t · x 2 }| B n = { 1 , . . . , n } 2 with and − 1 ≤ s , t ≤ 1 13

Preparations Local surface tension ent( s , t ) := lim n →∞ ent n ( s , t ) Macroscopic entropy � Ent( R , f ) := ent( ∇ f ( x )) dx R R ⊂ R 2 with and f : R → R 1-Lipschitz 14

The variational principle for domino tilings Theorem 1 (Cohn, Kenyon, Propp ’00) 1 Assume: n Λ n → R and g ∂ Λ n → g ∂ R Then: Ent(Λ n , g ∂ Λ n ) → sup Ent( R , f ) . f : f ∂ R = g ∂ g Asymptotically characterizes the number of height-functions. 15

The variational principle for domino tilings Additionally ent( s , t ) is strict concave and therefore � sup Ent( R , f ) = f : f ∂ R = g ∂ g Ent( R , f ) = max ent( ∇ f max ( x )) dx . f : f ∂ R = g ∂ g R WHY DO LIMIT SHAPES APPEAR? 16

The variational principle for domino tilings Given f : R → R and ε > 0 define B ( ε, f , Λ n ) := { height functions h : Λ n → Z that are ε -close to f } Theorem 2 (Cohn, Kenyon, Propp ’00) 1 | Λ n | ln | B ( ε, f , Λ n ) | → Ent( R , f ) + Error( ε ) 17

Why limit shapes appear Combination of both theorems yields 1 | Λ n | ln | B ( ε, f max , Λ n ) | ≈ Ent( f max ) 1 ≈ Ent n (Λ n , g ∂ Λ n ) = | Λ n | ln | Z (Λ n , g ∂ Λ n ) | This means: Uniform measure on Z (Λ n , g ∂ Λ n ) concentrates around B ( ε, f max , Λ n ) 18

Main ingredients of the proof First ingredient: lim n →∞ ent n ( s , t ) exists Second ingredient: lim n →∞ ent n ( s , t ) = lim n →∞ ent n , free ( s , t ) ent n , free ( s , t ) = 1 n 2 ln |{ h : B n → Z | h is height function and for all | h ( x 1 , x 2 ) − s · x 1 + t · x 2 | � √ n �� ( x 1 , x 2 ) ∈ ∂ B n : � 19

Main ingredients of the proof of Cohn, Kenyon, Propp Cohn, Kenyon, Propp use direct computations provided by the dimer model and monotonicity of the model. PROOF WITHOUT USING INTEGRABILITY? Content of recent work with Georg Menz on ArXiv. Derive variational principle for graph homomorphisms h : Λ n → T . 20

Comparing both methods Method based on integrability: not robust but very precise • local surface tension ent( s , t ) is explicitly known • local surface tension ent( s , t ) is strictly concave • limit shapes f max can be analyzed • fluctuations can be analyzed 21

Comparing both methods The new method: more robust but not precise • existence of limit lim n →∞ ent n ( s , t ) • local surface tension ent( s , t ) not explicitly known • local surface tension ent( s , t ) is concave, strict concavity open problem • applies to many other graphs. 22

How is integrability substituted? Two ingredients: • A Kirszbraun theorem for graphs: Describe when one can glue two graph homomorphisms together. • Concentration inequality − C ε 2 n � � P ( | h ( x 1 , x 2 ) − ( sx 1 + tx 2 ) | ≥ ε n ) � exp 23

How are those ingredients used? Showing lim n →∞ ent n ( t , s ) = lim n →∞ ent n , free ( t , s ). In other words microscopic fluctuation on the boundary do not modify the entropy. 24

Kirszbraun theorem for graph We say that a bipartite graph H satisfy the Kirszbraun property if any contracting map from a subset of Z d to H which have the right parity condition, can be extended to a graph homomorphism from the whole lattice Z d to H . This theorem assure that one can attach graph homomorphisms with similar boundary conditions without losing significant entropy. 25

Which graphs are Kirszbraun Which graphs have the Kirszbraun property: • All trees • Z 2 • Modified Z 3 with extra points in the center of unit cubes However for d ≥ 3, the lattice Z d is not Kirszbraun In fact there exist a complete geometric characterization of Kirszbraun graphs based on triangles and geodesics between pairs of point (work in progress with Igor Pak and Nishant Chandgotia). 26

How is the concentration inequality deduced For graph homomorphisms h : B n → Z several methods: • monotonicity • Random surfaces: cluster swapping For graph homomorphisms h : B n → T very difficult: • Under the usual dynamic on T not nice: configurations tends to diverge form each other New coupling technique: • based on Azuma-Hoeffding inequality • uses a very carefully adapted Glauber dynamics 27

Difference with one-dimensional models In a tree, you are not limited to climb on a single geodesic and limiting boundary conditions must be carefully redefined. Here is an illustration of the proper definition for the rescaling. (a) n=1 (b) n=2 (c) n=4 28

The variational principle for tree homomorphisms Theorem 3 (Menz, T.) 1 Assume: n Λ n → R and g ∂ Λ n → g ∂ R Then: Ent(Λ n , g ∂ Λ n ) → sup Ent( R , f ) . f admissible Here the sup is taken over a different set of functions due to the constraints imposed by the geodesics. 29

Differences with one-dimensional models A limit shape for boundary conditions climbing on several geodesics 30

Outlook Open questions: • Is the the local entropy strictly concave? • Is it possible to approximate the local entropy of a given slope in an efficient way? Other models: • Use different graphs instead of T .........connected to tilings, random graphs. • Use different lattices instead of Z d • Consider weighted graphs instead of T ....do homogenization. 31

References Sheffield, Scott. ”Random surfaces” Astrisque , 2005. Cohn, Henry and Kenyon, Richard and Propp, James. A variational principle for domino tilings. Journal of the American Mathematical Society- 297–346 (electronic). Menz, Georg and T. ”A variational principle for a non integrable model” https://arxiv.org/pdf/1610.08103.pdf 32

Thank you! 32