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

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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.

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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.

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Object we study

Graph homomorphism h : G1 → G2: x ∼ y ⇒ h(x) ∼ h(y). Example 1: Graph homomorphism to Z: h : Z2 ⊃ Λ → Z ”height function” Example 2: Graph homomorphism to 3 regular tree T : h : Z2 ⊃ Λ → T ”Tree-valued height function”

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Objects we study

Random graph homomorphism:

• 1. Fix Λ ⊂ Zd
• 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”.

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Limit shape for graph homomorphism into Z

Colors represent the parity of the height in Z

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Limit shape for graph homomorphism into 3 regular tree T

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Appearance of limit shapes is a universal phenomenon

Limit shape for lozenge tilings

Each color represents a type of lozenge

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Limit shape “Aztec Diamond” for domino tilings

Colors represent the parity of the dominos

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Limit shape for tilings by 3-1 bars

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Limit shape for ribbon tilings

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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?

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Preparations

Microscopic entropy Ent(Λ, g∂Λ) := 1 |Λ| ln

• h : Λ → Z | h is height function and h|∂Λ = g
• Microscopic surface tension

entn(s, t) := 1 n2 ln |{h : Bn → Z | h is height function and for all (x1, x2) ∈ ∂Bn : h(x1, x2) ≈ s · x1 + t · x2}| with Bn = {1, . . . , n}2 and − 1 ≤ s, t ≤ 1

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Preparations

Local surface tension ent(s, t) := lim

n→∞ entn(s, t)

Macroscopic entropy Ent(R, f ) :=

• R

ent(∇f (x)) dx with R ⊂ R2 and f : R → R 1-Lipschitz

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The variational principle for domino tilings

Theorem 1 (Cohn, Kenyon, Propp ’00) Assume: 1 nΛn → R and g∂Λn → g∂R Then: Ent(Λn, g∂Λn) → sup

f : f∂R=g∂g

Ent(R, f ). Asymptotically characterizes the number of height-functions.

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The variational principle for domino tilings

Additionally ent(s, t) is strict concave and therefore sup

f :f∂R=g∂g

Ent(R, f ) = max

f :f∂R=g∂g Ent(R, f ) =

• R

ent(∇fmax(x)) dx. WHY DO LIMIT SHAPES APPEAR?

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The variational principle for domino tilings

Given f : R → R and ε > 0 define B(ε, f , Λn) := {height functions h : Λn → Z that are ε-close to f } Theorem 2 (Cohn, Kenyon, Propp ’00) 1 |Λn| ln |B(ε, f , Λn)| → Ent(R, f ) + Error(ε)

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Why limit shapes appear

Combination of both theorems yields 1 |Λn| ln |B(ε, fmax, Λn)| ≈ Ent(fmax) ≈ Entn(Λn, g∂Λn) = 1 |Λn| ln |Z(Λn, g∂Λn)| This means: Uniform measure on Z(Λn, g∂Λn) concentrates around B(ε, fmax, Λn)

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Main ingredients of the proof

First ingredient: limn→∞ entn(s, t) exists Second ingredient: limn→∞ entn(s, t) = limn→∞ entn,free(s, t) entn,free(s, t) = 1 n2 ln |{h : Bn → Z | h is height function and for all (x1, x2) ∈ ∂Bn : |h(x1, x2) − s · x1 + t · x2| √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 : Λn → T .

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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 fmax can be analyzed
• fluctuations can be analyzed

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Comparing both methods

The new method: more robust but not precise

• existence of limit limn→∞ entn(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.

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How is integrability substituted?

Two ingredients:

• A Kirszbraun theorem for graphs: Describe when one can glue two

graph homomorphisms together.

• Concentration inequality

P (|h(x1, x2) − (sx1 + tx2)| ≥ εn) exp

• −Cε2n
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How are those ingredients used?

Showing limn→∞ entn(t, s) = limn→∞ entn,free(t, s). In other words microscopic fluctuation on the boundary do not modify the entropy.

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Kirszbraun theorem for graph

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

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Which graphs are Kirszbraun

Which graphs have the Kirszbraun property:

• All trees
• Z2
• Modified Z3 with extra points in the center of unit cubes

However for d ≥ 3, the lattice Zd 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).

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How is the concentration inequality deduced

For graph homomorphisms h : Bn → Z several methods:

• monotonicity
• Random surfaces: cluster swapping

For graph homomorphisms h : Bn → 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

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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

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The variational principle for tree homomorphisms

Theorem 3 (Menz, T.) Assume: 1 nΛn → R and g∂Λn → g∂R Then: Ent(Λn, g∂Λn) → sup

f admissible

Ent(R, f ). Here the sup is taken over a different set of functions due to the constraints imposed by the geodesics.

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Differences with one-dimensional models

A limit shape for boundary conditions climbing on several geodesics

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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 Zd
• Consider weighted graphs instead of T ....do homogenization.

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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

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Thank you!

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