Flips in Tilings

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Flips in Tilings

Thomas Fernique CNRS & Univ. Paris 13

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1 Examples

2 Questions

3 Results

4 Context-sensitive flips

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Examples Questions Results Context-sensitive flips

1 Examples

2 Questions

3 Results

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Dimers

Consider the square or triangular grid.

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Dimers

Adimer (domino or lozenge) is the union of two neighboor cells.

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Dimers

Tilings by dimers.

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Dimers

Aflipis a local exchange of tiles.

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Dimers

Aflipis a local exchange of tiles.

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Rectangles

Two dominoes two rectangles.

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Rectangles

Tiling by two rectangles.

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Rectangles

Flip: exchange rectangles which tile alcm(a,c)×lcm(b,d) box.

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Rectangles

Flip: exchange rectangles which tile alcm(a,c)×lcm(b,d) box.

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Rectangles

Flip bis: swap alcm(a,c)×(b+d) or (a+c)×lcm(b,d) box.

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Rectangles

Flip bis: swap alcm(a,c)×(b+d) or (a+c)×lcm(b,d) box.

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Parallelograms

n pairwise non-colinear vectors ⁿ₂

parallelograms.

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Parallelograms

Tiling by parallelograms (no more underlying lattice).

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Parallelograms

Flip: exchange parallelograms which tile a hexagon.

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Parallelograms

Flip: exchange parallelograms which tile a hexagon.

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Kagome

Kagome or trihexagonal lattice.

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Kagome

Tile: hexagon with two neighboor triangles (three up to rotation).

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Kagome

Tile: hexagon with two neighboor triangles (three up to rotation).

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Kagome

Flip: exchange of two triangles between two neighboor hexagons.

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Kagome

Flip: exchange of two triangles between two neighboor hexagons.

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Kagome

Different types of flips (six up to isometry).

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Kagome

Different types of flips (six up to isometry).

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T -tetraminoes

Two types of flips.

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T -tetraminoes

Two types of flips.

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3-d

Flips/trits on 3d dominoes and flips on parallelotopes.

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3-d

Flips/trits on 3d dominoes and flips on parallelotopes.

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1 Examples

2 Questions

3 Results

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Tiling graph

Graph(region, tiles, flip):

vertices: tilings of the region;

edges: connect tilings which differ by a flip.

Properties? (connexity, diameter, eccentricity, treewidth. . . )

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Limit shape

Does a tiling chosen u.a.r. satisfy a.s. some macroscopic property?

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Mixing time

Ergodic random walkP on a graph stationnary distribution π.

Convergence measured by: d(t) := max

x∈V

1 2

X

y∈V

|P^t(x,y)−π(y)|.

Mixing time(half-life):

τ_mix:= min{t |d(t)≤1/4}.

The distance toπ is divided by (at least) 2 eachτ_mix steps. Rapid mixing: τ_mix logarithmic in the size of the graph.

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Mixing time

Convergence measured by:

d(t) := max

x∈V

1 2

X

y∈V

|P^t(x,y)−π(y)|.

The distance toπ is divided by (at least) 2 eachτ_mix steps. Rapid mixing: τ_mix logarithmic in the size of the graph.

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Mixing time

d(t) := max

x∈V

1 2

X

y∈V

|P^t(x,y)−π(y)|.

The distance toπ is divided by (at least) 2 eachτ_mix steps.

Rapid mixing: τ_mix logarithmic in the size of the graph.

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Mixing time

d(t) := max

x∈V

1 2

X

y∈V

|P^t(x,y)−π(y)|.

The distance toπ is divided by (at least) 2 eachτ_mix steps.

Rapid mixing: τ_mix logarithmic in the size of the graph.

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Motivation

Flips on tilings: model local rearrangements in materials.

Limit shape: macroscopic properties of materials.

Mixing time: practicability of the model.

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1 Examples

2 Questions

3 Results

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Connexity

For a simply connected region, the following graphs are connected:

Dimers (Thurston 1990)

Parallelograms (Kenyon 1994)

Two rectangles(R´emila 2004)

Kagome (Bodini, 2006)

T-tetraminoes(Korn-Pak 2007)

Additional conditions are necessary:

Parallelotopes(Desoutter-Destainville 2005)

3-d. dominoes (Milet-Saldanha 2017)

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Height function

Height function: real map defined on the vertices of the tiling.

each tiling determined by its height function; the height is monotone w.r.t. flips.

Yields an intuitive lift inR³, used to prove connexity: Dimers (Thurston)

Rectangles (R´emila)

Kagome (Bodini)

T-tetraminoes(Korn-Pak)

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Height function

Useful when

each tiling determined by its height function;

the height is monotone w.r.t. flips.

Yields an intuitive lift inR³, used to prove connexity: Dimers (Thurston)

Kagome (Bodini)

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Height function

Useful when

each tiling determined by its height function;

the height is monotone w.r.t. flips.

Yields an intuitive lift inR³, used to prove connexity:

Dimers (Thurston)

Kagome (Bodini)

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Diameter

Easy for dimers via height function.

Logarithmic in the number of tilings in any non-degenerate case.

Not necessarily a good indicator for mixing time. . .

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Straightness

Straighttiling graph: any two tilings can be connected by a sequence of flips, with no flip being performed back and forth.

Holds for dimers (the height function orientates flips).

Holds for parallelogram iffn≤4 (Bodini-F.-Rao-R´emila 2011). Does it imply rapid mixing?

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Limit shape

Well-studied for dimers(Kenyon, Okounkov, Propp. . . ). Other cases?

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Limit shape

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Limit shape

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Mixing time

Rapid for dimers(Luby-Randall-Sinclair 1995, Randall-Tetali 1999, Wilson 2004, Caputo-Martinelli-Toninelli 2011, Laslier-Toninelli 2013).

Connexity still to be characterized in this case.

Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016). Claimed rapid for T-tetraminoes(Kayibi-Pirzada 2015).

Conjectured rapid for parallelograms whenn= 4 (Destainville 2001). Conjectured slow for parallelograms whenn= 4 (F. 2016).

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Mixing time

Rapid for Kagome without “fish”(Ugolnikova 2016). Connexity still to be characterized in this case.

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Mixing time

Conjectured to be rapid for 1×1 and 2×2 squares (Ugolnikova 2016).

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Mixing time

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Mixing time

Conjectured rapid for parallelograms whenn= 4 (Destainville 2001).

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Mixing time

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1 Examples

2 Questions

3 Results

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Principle

Consider the Markov chain

choose a vertex of the tiling uniformly at random,

perform a flip around it with probability P = exp(−∆E)/T, where

T is a temperatureparameter,

theenergy E of a tiling is a sum of local interactions.

This is the Metropolis-Hastings algorithm for the Gibbs distribution P(tiling) = 1

Z(T)exp

−E(tiling) T

.

It aims to model the cooling of ordered materials (quasicrystals).

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Principle

Consider the Markov chain

choose a vertex of the tiling uniformly at random,

perform a flip around it with probability P = exp(−∆E)/T, where

T is a temperatureparameter,

theenergy E of a tiling is a sum of local interactions.

This is the Metropolis-Hastings algorithm for the Gibbs distribution P(tiling) = 1

Z(T)exp

−E(tiling) T

.

It aims to model the cooling of ordered materials (quasicrystals).

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Example 1: dimer tilings

Ergodic atT >0, Θ(n²) mixing at T =∞,O(n^2.5) atT = 0.

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Example 1: dimer tilings

Ergodic atT >0, Θ(n²) mixing at T =∞,Θ(n²) atT = 0.

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Example 1: dimer tilings

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Example 1: dimer tilings

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Example 1: dimer tilings

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Example 1: dimer tilings

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Example 2: Beenker tilings

Ergodic atT >0,unknown mixing atT =∞,Θ(n²)at T = 0.

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Example 2: Beenker tilings

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Example 2: Beenker tilings

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Example 2: Beenker tilings

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Example 2: Beenker tilings

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Example 3: Penrose tilings

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Example 3: Penrose tilings

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Example 3: Penrose tilings

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Example 3: Penrose tilings

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Example 3: Penrose tilings

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Flips in Tilings

Thomas Fernique CNRS & Univ. Paris 13