Planar Dimer Tilings

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O. Bodini T. Fernique

LIRMM (Montpellier, France)

CSR’06, June 8–12, 2006

O. Bodini, T. Fernique Planar Dimer Tilings

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Domino and lozenge tilings espacially studied (statistical physics).

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Domino and lozenge tilings espacially studied (statistical physics).

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Domino and lozenge tilings espacially studied (statistical physics).

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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Triangular cells ↔ directed graph:

↔

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Dimer (lozenge) tiling ↔ weighted edges (black: 1, red: −2):

↔

Note: directed closed paths have weight 0.

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Weighted edges (black: 1, red: −2) ↔ heights of vertices:

↔

0 1 2 3 2

3 2

3 2 1

2 1 1

0 1 2

3

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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Proposition

A vertex of maximum height locally enforces weights:

incoming edges have positive weights (black edges) outcoming edges have negative weights (red edges)

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proposition

A tileable domain admits a tiling whose vertices of max. height are on the boundary.

proof: flip:

2 1

2 1 1

2 3

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proposition

A tileable domain admits a tiling whose vertices of max. height are on the boundary.

proof: flip:

2 1

2 1 1

2 0

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0 1

2 3

2 3 2

3 2 1

2 1

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0 1

2 3

2 3 2

3 2 1

2 1

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0 1

2 3

2 3 2

3 2 1

2 1 1

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0 1

2 2

3 2

3 2 1

2 1 1

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0 1

2 2

3 2 1

2 1 1

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0 1

2 2

2 1

2 1 1

1

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0 1

2 2

2 1

1 1

1 0

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0 1

2 2

1 1

1 0

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0 1

2 1 1

1 0

0

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0 1

1 1 0

0

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0 1

1 0

0 −1

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0 1 0

−1

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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Intuitively:

lozenge tilings: constrained enough for deriving the whole from the boundary

dimer tilings: holes/irregularities can “hide” information to the boundary

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Consider a set of polygonal cells

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Suppose it is bipartite gears-like orientation of cells

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This allows to define a directed graph.

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1 1

1

1 1 1

1 1

0

0 0 0

0 0

Binary counter: 0-1 weight function δ s.t. δ(cell ) = 1

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Trival bijection with dimer tilings.

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v*

1 1

1

1 1 1

1 1

0

0 0 0

0 0

0 Height function: h _δ : v 7→ min{δ(p) | p : v ^∗ v}, for a fixed v ^∗ .

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But heights do not more locally enforce weights!

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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2 0

0

1 0 1 1 0

−1 −1

0 2

−1 1 1

0 1 0

0 1

0 Counter: real weight function ψ s.t. ψ(cell ) = 1.

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Proposition

One can compute a counter in linear time (using a spanning tree).

Theorem

If ψ is a counter, then one defines a binary counter by:

δ : (v , v ⁰ ) 7→ ψ(v, v ⁰ ) − (h _ψ (v ⁰ ) − h _ψ (v )).

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Proposition

One can compute a counter in linear time (using a spanning tree).

Theorem

If ψ is a counter, then one defines a binary counter by:

δ : (v , v ⁰ ) 7→ ψ(v, v ⁰ ) − (h _ψ (v ⁰ ) − h _ψ (v )).

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Algorithm

1

construct a counter ψ in time O(n);

2

compute h ψ in time O(n ln ³ n) (SSSP for a planar graph);

3

derive a binary counter in time O(n).

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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Lozenge tilings of simply connected domains are connected by flips.

Which notion of flip for general dimer tilings?

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Lozenge tilings of simply connected domains are connected by flips.

Which notion of flip for general dimer tilings?

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Lozenge tilings of simply connected domains are connected by flips.

Which notion of flip for general dimer tilings?

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In terms of binary counter:

1 1 1

1 1

1 1 1

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In terms of binary counter:

1 1 1

1 1

1 1 1

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In terms of binary counter:

1 1 1

1 1

1 1 1

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Definition (Flip)

Let A be vertices strongly connected by edges of weight 0 and s.t.

all its incoming edges have weight 0;

all its outcoming edges have weight 1.

Then, a flip on A exchanges these weights (0 ↔ 1).

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1 The Thurston’s algorithm (lozenge tilings) Weights, heights and flips

Thin out a simply connected domain

2 The general case

Binary counters and heights Relaxation: real counters

3 Structure of the set of tilings Generalized flips

A distributive lattice

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