The double-dimer model and skew Young diagrams

Richard W. Kenyon David B. Wilson

Brown University Microsoft Research

The double-dimer model

Kasteleyn matrix

11

2 2

3

3

4 455

6

67

7 8

8 9 9

(bipartite version)

6

1

2

54

3

The double-dimer model

6

1

2

54

3

6

1

2

54

3

The double-dimer model

6

1

2

54

3

6

1

2

54

3

6

1

2

54

3

6

1

2

54

3

6

1

2

54

3

The double-dimer model

Assume (wlog) that nodes alternate in color

6

1

2

54

3

Probability of pairing in DD model

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

Other connection topologies

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

1

4

2

36

5

For six nodes, get all 5 connection topologies

What if there are more nodes?

With 2n nodes, # crossing types is nth Catalan number.Can we compute all their probabilities?

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