Aztec diamond. An Aztec diamond of order n is the union of the unit squares with lattice point coordinates in the region given by...

Transcription

1 Aztec diamond An Aztec diamond of order n is the union of the unit squares with lattice point coordinates in the region given by x + y n + 1

2 Aztec diamond An Aztec diamond of order n is the union of the unit squares with lattice point coordinates in the region given by x + y n + 1

3 Domino Tilings The number of domino tilings of aztec diamond of order n is 2 n(n+1) 2

4 Domino Tilings The number of domino tilings of aztec diamond of order n is 2 n(n+1) 2

5 Arctic circle phenomenon

6 Tilings as matchings

7 Tilings as matchings

8 Aztec rectangle (Propp s problem) diamond of order 5

9 Aztec rectangle (Propp s problem) 5 7 rectangle

10 Aztec rectangle (Propp s problem) 5 7 rectangle No matchings

11 Aztec rectangle (Propp s problem) 5 7 rectangle Add diagonal edges connecting nearest black vertices

12 Aztec rectangle (Propp s problem) 5 7 rectangle Add diagonal edges connecting nearest black vertices A matching of m n rectangle contains m n 2 diagonal edges

13 Aztec rectangle (Propp s problem) 5 7 rectangle Add diagonal edges connecting nearest black vertices A matching of m n rectangle contains m n 2 diagonal edges

14 Aztec rectangle (Propp s problem) 5 7 rectangle Add diagonal edges connecting nearest black vertices A matching of m n rectangle contains m n 2 diagonal edges 2mn + m + n m n 2 dominos m n 2

15 Matchings and Tilings

16 Matchings and Tilings

17 Matchings and Tilings

18 Matchings and Tilings

19 Our first result: Local transformation rule

20 Our first result: Local transformation rule

21 Our first result: Local transformation rule These local moves connects all tilings We can construct an ergodic Markov chain whose state space is the set of all tilings

22 Typical domino tiling of an Aztec rectangle

23 Typical domino tiling of another rectangle

24 Our result 2 Theorem For any simply connected subgraph G of the square lattice graph with a prescribed special vertex r called root, there exists a simply connected area R G which can be tiled with dominos and one diagonal impurity The probability of finding the impurity at a given position can be explicitly represented in terms of the probability concerning the simple random walk on G

25 Our result 2 Theorem For any simply connected subgraph G of the square lattice graph with a prescribed special vertex r called root, there exists a simply connected area R G which can be tiled with dominos and one diagonal impurity The probability of finding the impurity at a given position can be explicitly represented in terms of the probability concerning the simple random walk on G Remark Always possible G R G

26 Our result 2 Theorem For any simply connected subgraph G of the square lattice graph with a prescribed special vertex r called root, there exists a simply connected area R G which can be tiled with dominos and one diagonal impurity The probability of finding the impurity at a given position can be explicitly represented in terms of the probability concerning the simple random walk on G Remark Always possible Not always possible G R G R G R

27 Example Tile the 5 5 rectangle with a corner removed with 1 2 rectangles

28 Example Tile the 5 5 rectangle with a corner removed with 1 2 rectangles 192 tilings exist

29 Example Tile the 5 5 rectangle with a corner removed with 1 2 rectangles 192 = tilings exist (i, j) {0, 1, 2} 2 (i, j) (0, 0) ( 4 2 cos jπ 3 2 cos kπ ) 3

30 Example

31 Example

32 Example

33 Example No tilings

34 Example No tilings There are two more blacks

35 Example Use a special type of tile

36 Example Use a special type of tile Where should it be put?

37 Example Use a special type of tile Where should it be put? The place with the most tilings?

38 Example

39 Example 56 tilings

40 Example 56 tilings

41 Example 56 tilings

42 Example 16 tilings

43 Example 16 tilings

44 Example 16 tilings

45 Example 16 tilings

46 Example 8 tilings

47 Example

48 Simple Random Walks and Tilings

49 Simple Random Walks and Tilings absorbing state

50 Simple Random Walks and Tilings absorbing state

51 Simple Random Walks and Tilings absorbing state 1 7 Probability of arriving at

52 Simple Random Walks and Tilings absorbing state 1 7 Probability of arriving at 2 7

53 Simple Random Walks and Tilings absorbing state Probability of arriving at 1 2 7

54 Simple Random Walks and Tilings absorbing state Probability of arriving at

55 Simple Random Walks and Tilings absorbing state Probability of arriving at Combinatorial Laplacian = det

56 Enumerations of domino tilings with fixed impurities tilings

57 Random walk and hitting matrix

58 Random walk and hitting matrix

59 absorbing Random walk and hitting matrix states

60 Random walk and hitting matrix absorbing states identified

61 starting Random walk and hitting matrix absorbing states vertex

62 starting Random walk and hitting matrix absorbing states vertex 3 2 1

63 starting Random walk and hitting matrix 2 3 absorbing states vertex

64 starting Random walk and hitting matrix H = absorbing states vertex ( h ) 11 h 12 h 13 2 h 21 h 22 h 23 3 h 31 h 32 h 33 1 h i,j is the probability of a random walk starting at i reaches the absorbing state j

65 H =

66 Combinatorial Laplacian L = (l ij ), l ij = { deg(i) where i s and j s are non-absorbing states i = j edges between i and j i j, det H det L = = Z

67 Proof Sketch

68 Proof Sketch Modification of Temperley bijection ( )

69 Proof Sketch Modification of Temperley bijection ( ) S Fomin s results on LERW