Stochastic Flips on Dimer Tilings

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Thomas Fernique & Damien Regnault

Moscow, June 21, 2013

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Context

Quasicrystal: non-periodic ordered material modeled by tilings.

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Context

How to model the quasicrystalgrowth?

However, non-periodicity yields “frequent”deadlocks.

Alternative: first allow mismatches to facilitate self-assembly, then perform random locally-defined corrections. Convergence?.

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Context

Natural idea: add tiles one at a time (self-assembly).

then perform random locally-defined corrections. Convergence?.

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Context

Natural idea: add tiles one at a time (self-assembly).

Alternative: first allow mismatches to facilitate self-assembly, then perform random locally-defined corrections. Convergence?.

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1 Tilings and flips

2 Cooling process

3 An upper bound

4 A second upper bound

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Dimer tilings

Bounded (simply) connected subset of the triangular grid.

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Dimer tilings

Dimer tiling: perfect matching of adjacent triangles.

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Dimer tilings

Shading dimers 3D-viewpoint.

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Dimer tilings

3D-viewpoint distance to the plane x+y+z = 0 (height).

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Errors

Error: edge between two dimers equal up to a translation.

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Errors

Errors form contours lines between dimers of different heights.

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Errors

Error-free tiling plays the quasicrystal role.

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Islands and holes

An island with height 1, area 5 and perimeter 16.

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Islands and holes

A larger island of height 1.

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Islands and holes

Another one, of height 2 this time.

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Islands and holes

A hole in the largest island.

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Flips

Flip: exchange of thee dimers'add/remove a cube.

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Flips

Flip: exchange of thee dimers'add/remove a cube.

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Flips

Flip change the area of an island or its perimeter (error number).

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Flips

Topologically, flip can merge or split islands. . .

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Flips

. . . and destroy or create holes.

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1 Tilings and flips

2 Cooling process

3 An upper bound

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Cooling

Cooling: Markov chain (ω_t)t≥0 defined by an initial tiling ω0;

ω_t →ω_t+1: perform unif. at random a flip s.t. ∆E ≤0;

stop if no flip s.t. ∆E ≤0.

The cooling stops only on error-free tilings (our “quasicrystals”).

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Example

Flips such that ∆E ≤0: around blue points.

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Example

Cooling: random flips such that ∆E ≤0 (one at a time).

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

Cooling: random flips such that ∆E ≤0 (10 at a time).

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

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Example

Until no more flips are allowed (169 flips perfomed).

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Convergence time

random variableT: number of performed flips.

Worst average convergence timeon a given region D:

Tb = max

ω0∈P(D)E(T |ω=ω₀).

Asymptotic behavior ofTb when n :=|D| → ∞?

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Numerical simulations

Worst case = maximal volume tiling? (colors: height modulo 3).

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Numerical simulations

Θ(√

n) stacked islands of area Θ(n) to clear (100 flips at a time).

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Numerical simulations

Θ(√

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Numerical simulations

Θ(√

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Numerical simulations

Θ(√

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Numerical simulations

Θ(√

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Numerical simulations

Θ(√

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Numerical simulations

Θ(√

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Numerical simulations

Numerical simulations conjecture: Tb = Θ(n²).

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1 Tilings and flips

2 Cooling process

3 An upper bound

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Tool

To boundTb: Proposition

Let (ω_t)t≥0 be a Markov chain over Ω.

If there areε >0 and a “potential function”φ: Ω→R+ s. t.

φ(ω_t)>0 ⇒ E[φ(ω_t+1)−φ(ω_t)|ω_t]≤ −εφ(ω_t), then

E(min{t |φ(ωt) = 0})≤ logφ(ω0) ε .

How to findφwhich satisfies such a “differential equation”?

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One island

Consider the border of a hole-free islande withF possible flips.

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One island

There is always 6 more salientthanreflex angles (induction).

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One island

i ≥2 salientangles in a row flip s.t. ∆(4V +E) =−2i.

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One island

i ≥2 reflex angles in a row flip s.t. ∆(4V +E) =+2i.

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One island

Let φ= 4V +E. We thus have E(∆φ)≤ −6×2/F.

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Tilings and flips Cooling process An upper bound A second upper bound

More islands

Applied to each of thek island of a tiling:

E(∆φ)≤ −12k F .

E(∆φ)≤ − 2 n²φ. Withε= ⁿ₂² and φ(ω₀) =O(n√

n), our tool yields Tb =O(n²logn).

Triple scam!

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More islands

E(∆φ)≤ −12k F .

Sinceφ= 4V +E ≤4kn+ 2n≤6kn andF ≤n, this yields E(∆φ)≤ − 2

n²φ.

Withε= ₂ and φ(ω₀) =O(n n), our tool yields Tb =O(n²logn).

Triple scam!

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More islands

E(∆φ)≤ −12k F .

Sinceφ= 4V +E ≤4kn+ 2n≤6kn andF ≤n, this yields E(∆φ)≤ − 2

n²φ.

Withε= ⁿ₂² and φ(ω₀) =O(n√

Triple scam!

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More islands

E(∆φ)≤ −12k F .

Sinceφ= 4V +E ≤4kn+ 2n≤6kn andF ≤n, this yields

E(∆φ)≤ − 2 n²φ.

Withε= ⁿ₂² and φ(ω₀) =O(n√

Triple scam!

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Triple scam

1 - Holes have an adverse effect on the volume, hence onφ.

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Triple scam

2 - Islands merging may increaseφ.

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Triple scam

3 - Obstructions between stacked island can preventφto decrease.

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1 Tilings and flips

2 Cooling process

3 An upper bound

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Second tool

To boundTb: Proposition

Let (ωt)t≥0 be a Markov chain over Ω.

If there areε >0 and a “potential function”φ: Ω→R+ s. t.

φ(ωt)>0 ⇒ E[φ(ωt+1)−φ(ωt)|ω_t]≤ −ε, then

E(min{t |φ(ω_t) = 0})≤ φ(ω₀) ε .

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Trick: triconvex hull

Triconvex hullω of ω: convexity in three directions.

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Trick: triconvex hull

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Trick: triconvex hull

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Trick: triconvex hull

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Trick: triconvex hull

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Trick: triconvex hull

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Trick: triconvex hull

Ifω has height 1, the hull brings us back to the one island case E[∆φ(ω)]≤ − 12

F(ω).

E[∆φ(ω)]≤ − 12 F(ω).

And sinceφ≤6n and F ≤n, our second tool yields E(min{t |φ(ω_t) = 0})≤6n². For heightk ≤√

n, the highest islands disappear inO(n²). Thus Tb =O(n²√

n).

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Trick: triconvex hull

F(ω).

Withφ(ω) :=φ(ω), on can linkω andω (technical):

E[∆φ(ω)]≤ − 12 F(ω).

E(min{t |φ(ω_t) = 0})≤6n². For heightk ≤√

n).

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Trick: triconvex hull

F(ω).

E[∆φ(ω)]≤ − 12 F(ω).

And sinceφ≤6n and F ≤n, our second tool yields E(min{t |φ(ωt) = 0})≤6n².

For heightk ≤ n, the highest islands disappear inO(n ). Thus Tb =O(n²√

n).

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Trick: triconvex hull

F(ω).

E[∆φ(ω)]≤ − 12 F(ω).

And sinceφ≤6n and F ≤n, our second tool yields E(min{t |φ(ωt) = 0})≤6n². For heightk ≤√

n).

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Where comes the √

n factor from?

No more scam here, but we somewhere lost a√

n factor.

Top-down argumentforgets lower flips that can decrease φ.

However, similuations suggestTb = Θ(n²) even for height 1.

Triconvex hullforgets inner flips that can decreaseφ.

However, the cooling naturally “triconvexifies” the tiling.

Can we get a tight bound?

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Where comes the √

n factor from?

No more scam here, but we somewhere lost a√

n factor.

Top-down argumentforgets lower flips that can decrease φ.

However, similuations suggestTb = Θ(n²) even for height 1.

Triconvex hullforgets inner flips that can decreaseφ.

However, the cooling naturally “triconvexifies” the tiling.

Can we get a tight bound?

And what about theaverage average convergence time?

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Th. Fernique, D. Regnault,Stochastic flips on dimer tilings, Disc. Math. Theor. Comput. Sci. (2010).