Tiling expression of minors

Transcription

1 Institute for Mathematics and its Applications Minneapolis, MN Colloquium University of British Columbia, Vancourver February, 0

2 Outline Introduction to the Enumeration of Tilings. Electrical networks Future work.

3 Tilings A lattice divides the plane into disjoint pieces, called fundamental regions.

4 Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge.

5 Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps.

6 Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps. The number of tilings of a chessboard is,,.

7 Tilings We would like to find the number of tilings of certain regions.

8 Tilings We would like to find the number of tilings of certain regions. Tilings can carry weights, we also care about the weighted sum of tilings: T wt(t ), called the tiling generating function.

9 Kasteleyn Temperley Fisher Theorem Theorem (Kasteleyn, Temperley and Fisher ) The number of tilings of a m n chessboard equals m n j= k= ( ( cos jπ ) ( + cos kπ ) ). m + n +

10 Connections and Applications to Other Fields Statistical Mechanics: Dimer model, double-dimer model, square ice model, -vertex model, fully packed loops configuration... Probability: Limit shapes and Arctic Curves. Graph Theory: Bijection between tilings and perfect matchings. Cluster Algebras: Combinatorial interpretation of cluster variables. Electrical networks:... Other topics in combinatorics: Alternating-sign matrices, monotone triangles, plane partitions, lattice paths, symmetric functions...

11 Aztec Diamond Theorem (Elkies, Kuperberg, Larsen and Propp ) The Aztec diamond of order n has n(n+)/ domino tilings. Figure: The Aztec diamond of order and one of its tilings.

12 An Aztec Temple

13 MacMahon s Formula Theorem (MacMahon 00) The number of (lozenge) tilings of a semi-regular hexagon of sides a, b, c, a, b, c on the triangular lattice is a b c i= j= t= i + j + t H(a) H(b) H(c) H(a + b + c) = i + j + t H(a + b) H(b + c) H(c + a), where the hyperfactorial H(n) = 0!!!... (n )!. a= b= c= c= b= a=

14 Connection to Plane Partitions a=0 O j k i b=0 c=0 c=0 b=0 a=0

15 MacMahon s q-formula Theorem (MacMahon) π q vol(π) = H q(a) H q (b) H q (c) H q (a + b + c) H q (a + b) H q (b + c) H q (c + a), where the sum is taken over all stacks π fitting in an a b c box. q-integer [n] q := + q + q q n q-factorial [n] q! := [] q [] q [] q... [n] q q-hyperfactorial H q (n) := [0] q! [] q! [] q!... [n ] q!.

16 Generalizing MacMahon s Formula a z+b+c x m y O j k i a z+b+c x m y t m x c c m a y+b b t+c t m a+b x c a z m q volume of the stack =? stacks c m y+b b m z t+c a+b

17 Generalizing MacMahon s Formula Theorem (L. 0+) For non-negative integers x, y, z, t, m, a, b, c q vol(π) H q( + x + y + z + t) = H q( + x + y + t) H q( + x + y + z) π Hq( + x + t) Hq( + x + y) Hq( + y + z) Hq( ) H q( + z + t) H q( + x) H q( + y) Hq(m + b + c + z + t) Hq(m + a + c + x) Hq(m + a + b + y) H q(m + b + y + z) H q(m + c + x + t) H q(c + x + t) H q(b + y + z) H q(a + c + x) H q(a + b + y) H q(b + c + z + t) Hq(m) H q(a) H q(b) H q(c) H q(x) H q(y) H q(z) H q(t) H q(m + a) H q(m + b) H q(m + c) H q(x + t) H q(y + z), where = m + a + b + c.

18 Quasi-hexagon a= b= c= c= a= b= In, James Propp collected open problems in enumeration of tilings.

19 Quasi-hexagon a= b= c= c= a= b= In, James Propp collected open problems in enumeration of tilings. Problem on the list asks for the number of tilings of a quasi-hexagon.

20 Quasi-hexagon a= b= c= c= a= b= Theorem (L. 0) The number of tilings of a quasi-hexagon is a power of times the number of tilings of a semi-regular hexagon.

21 Blum s Conjecture and Hexagonal Dungeon a= b= a= a= b= a= Theorem (Blum s (ex-)conjecture) The number of tilings of the hexagonal dungeon of side-lengths a, a, b, a, a, b (b a) is a a. The conjecture was proven by Ciucu and L. (0).

22 Circular Planar Electrical Networks Study of electrical networks comes from classical physics with the work of Ohm and Kirchhoff more than 00 years ago. The circular planar electrical networks were studied systematically by Colin de Verdière and Curtis, Ingerman, Mooers, and Morrow. A number of new properties have been discovered recently.

23 Circular Planar Electrical Networks Definition A circular planar electrical network is a finite graph G = (E, V ) embedded in a disk with a set of distinguished vertices N of V on the circle, called nodes, and a conductance function wt : E R +

24 Well-connected networks A = {a, a,..., a k } and B = {b, b,..., b l } are non-interlaced on the circle if we do not have a b j a k or b a i b l.

25 Well-connected networks A = {a, a,..., a k } and B = {b, b,..., b l } are non-interlaced on the circle if we do not have a b j a k or b a i b l. A network G is well-connected if for any pair (A, B) of non-interlaced sets with k nodes ( k n ) we can find k pairwise vertex-disjoint paths connecting nodes in A to nodes in B.

26 Well-connected networks

27 Well-connected networks

28 Well-connected networks

29 Motivational Problem We would like to test the well-connectivity of given networks.

30 Response Matrix Associated with a network is a response matrix Λ = (λ i,j ) i,j n, that measures the response of the network to potential applied at the nodes.

31 Response Matrix Associated with a network is a response matrix Λ = (λ i,j ) i,j n, that measures the response of the network to potential applied at the nodes. λ i,j is the current that would flow into node j if node i is set to one volt and the remaining nodes are set to zero volts.

32 Response Matrix Associated with a network is a response matrix Λ = (λ i,j ) i,j n, that measures the response of the network to potential applied at the nodes. λ i,j is the current that would flow into node j if node i is set to one volt and the remaining nodes are set to zero volts. Two networks is electrically equivalent of they have the same response matrix.

33 Electrical Moves a a b a ab/(a+b) a b a+b c a b ac/(a+b+c) ab/(a+b+c) bc/(a+b+c) Two networks are electrically equivalent if and only if they can be obtained from each other by using the electrical moves.

34 Circular Minors Arrange the indices,,..., n of a general matrix M = ( m i,j ) i,j n in counter-clockwise order around the circle. A = {a, a,..., a k } in counter-clockwise order, and B = {b, b,..., b k } in clockwise order around the circle. Circular minor M B A = det b b... b k b k a m a,b m a,b... m a,b k m a,b k a m a,b m a,b... m a,b k m a,b k a k m ak,b m ak,b... m ak,b k m ak,b k a k m ak,b m ak,b... m ak,b k m ak,b k

35 Circular Minors: Examples M = a b a,, M,, = a b b

36 Circular Minors: Examples M = b b b a a a M,,,, = det b b b a a a 0

37 Circular Minors: Examples a b a,, M,, = a b b = det b b b a a a 0

38 Test the well-connectivity (Colin de Verdière) A network is well-connected if and only if all non-interlaced circular minors Λ B A > 0.

39 Test the well-connectivity (Colin de Verdière) A network is well-connected if and only if all non-interlaced circular minors Λ B A > 0. Kenyon and Wilson showed a test of the well-connectivity of a network by checking the positivity of (only) ( n ) special circular minors of the response matrix.

40 Contiguous Minors The contiguous minor CON a,b,y (M) := M B A where A = {a, a +,..., a + y } and B = {b + y,..., b +, b} 0 (a) 0 (b) 0 (c) Figure: M,0,,,, CON,,(M), CON,0,(M).

41 Central Minors The central minor CM x,y (M) is the contiguous minor CON a,b,y (M), where x y a = and b = a and b are opposite or almost opposite. x y + n (n mod ). 0 Figure: CON,0,(M) = CM,(M).

42 Small Central Minors If x n, y < n/ or y = n/ and x + y odd, then we call CM x,y (M) a small central minor of M. There are total ( n ) small center minors. x= x= x= x= x= y= y=

43 Small Central Minors If x n, y < n/ or y = n/ and x + y odd, then we call CM x,y (M) a small central minor of M. There are total ( n ) small center minors. x= x= x= x= x= x= y= y= y=

44 Kenyon Wilson Test Theorem (Kenyon Wilson Test) If ( n ) small central minors of the response matrix are all positive, then the network is well-connected.

45 Kenyon Wilson Theorem Theorem (Kenyon-Wilson 0) Each contiguous minor can be written as a Laurent polynomial (with positive coefficients) in central minors. Moreover, the Laurent polynomial is the generating function of domino tilings of a truncated Aztec diamond.

46 Aztec Diamonds Definition Denote by AD h x 0,y 0 of the Aztec diamond of order h with the center (x 0, y 0 ). (c) (,) (0,) (,) (a) (0,0) y=0 (b) Figure: (a) AD,. (b) AD 0,. (c) AD,.

47 Truncated Aztec Diamonds Definition The truncated Aztec diamond TAD h,n x 0,y 0 is the portion of AD h x 0,y 0 between the lines y = 0 and y = n. y= (,) (0,) (,) (0,0) y=0 (a) (b) (c) Figure: (a) TAD,,. (b) TAD, 0,. (c) TAD,,.

48 Weight Assignment Definition Assign to each lattice point (x, y) a weight v x,y : v x,y := CM x,y (M) if 0 y n. v x,y = if y < 0 or y > n.

49 Weight Assignment Definition We assign to each domino a weight v x,y v x,y, where the lattice points (x, y ) and (x, y ) are the centers of the long sides of the domino. (x,y) (x,y) (x+,y) (x,y-) v x,y v x,y v x,y v x+,y

50 Weight Assignment Definition We assign to each domino a weight v x,y v x,y, where the lattice points (x, y ) and (x, y ) are the centers of the long sides of the domino. W(R) := T wt(t ), where wt(t ) is the product of weights of all dominoes in the tiling T. (x,y) (x,y) (x+,y) (x,y-) v x,y v x,y v x,y v x+,y

51 Weight of Dominoes Definition The covering monomial: F(R) := (x,y) v x,y, taken over all lattice points (x, y) inside R or on the boundary of R, except for the 0 -corners.

52 Weight of Dominoes Definition The covering monomial: F(R) := (x,y) v x,y, taken over all lattice points (x, y) inside R or on the boundary of R, except for the 0 -corners. The tiling-polynomial: P(R) := W(R) F(R).

53 Calculating P(TAD,, ) (,) (0,) (,) (,) (-,) (,) (0,) (,) (,) ( F TAD,, (0,0) (,0) (,0) ) = v 0,0 v,0 v,0 v, v 0, v, v, v, v 0, v, v, v,

54 Calculating P(TAD,, ) (,) (,) (0,) (,) (,) (,) (,) (,) (-,) (,) (0,) (,) (,) (-,) (0,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (,0) (,) (,) (,) (,) (0,) (,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (0,) (0,0) (,) (,) (,) (0,) (0,) (,) (0,0) (,0) wt(t ) = v,v 0, v,0v, v,v, v 0,v, v,v,

55 Calculating P(TAD,, ) (,) (,) (0,) (,) (,) (,) (,) (,) (-,) (,) (0,) (,) (,) (-,) (0,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (,0) (,) (,) (,) (,) (0,) (,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (0,) (0,0) (,) (,) (,) (0,) (0,) (,) (0,0) (,0) wt(t ) = v,v 0, v,0v, v,v, v 0,v, v,v, F(TAD,, )wt(t ) = v0,0v,0v, v,

56 Calculating P(TAD,, ) (,) (,) (0,) (,) (,) (,) (,) (,) (-,) (,) (0,) (,) (,) (-,) (0,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (,0) (,) (,) (,) (,) (0,) (,) (,) (,) (,) (-,) (0,) (,) (,) (,0) (0,) (0,0) (,) (,) (,) (0,) (0,) (,) (0,0) (,0) wt(t ) = v,v 0, v,0v, v,v, v 0,v, v,v, F(TAD,, )wt(t ) = v0,0v,0v, v, P(TAD,, ) = v0,0v,0v, v, + v0,0v,0v,0v0,v, v 0,v,v, + v0,0v,0v0,v, v,v 0, + v0,0v,0v0,v, v,v, + v,0v,0v,v, v 0,v, + v,0v,v,v, v 0,v,.

57 Kenyon Wilson Theorem (cont.) Theorem (Kenyon-Wilson 0) CON a,b,y (M) = P(TAD h,n x h,y ), where h is the integer closest to 0 so that CON a,b+h,y (M) = CM x,y (M). 0 =? Figure: CON,,(M) with n =.

58 Kenyon Wilson Theorem (cont.) Theorem (Kenyon-Wilson 0) CON a,b,y (M) = P(TAD h,n x h,y ), h is the integer closest to 0 so that CON a,b+h,y (M) = CM x,y (M). 0 0 Figure: h = +, x =, y =

59 Kenyon Wilson Theorem (cont.) Theorem (Kenyon-Wilson 0) CON a,b,y (M) = P(TAD h,n x h,y ), where h is the integer closest to 0 so that CON a,b+h,y (M) = CM x,y (M). 0 0 = CON,, (M) = P(TAD,, )

60 Kenyon Wilson Theorem (cont.) Theorem (Kenyon-Wilson 0) CON a,b,y (M) = P(TAD h,n x h,y ), h is the integer closest to 0 so that CON a,b+h,y (M) = CM x,y (M). 0 0 = CON,, (M) = P(TAD,, )

61 Semicontiguous minors A semicontiguous minor is a circular minor MA B and B is contiguous. when exactly one of A (a) (b)

62 Structure of Semicontiguous Minors t k t k k k k k k t k SM a,b (k,..., k s ; t,..., t s )

63 Structure of Semicontiguous Minors t k t k k k k k k t k SM a,b (k,..., k s ; t,..., t s )

64 Structure of Semicontiguous Minors t k t k k k k k k t k SM a,b (k,..., k s ; t,..., t s ) The h- and x-parameters are that of the truncated Aztec diamond corresponding to the contiguous minor M B A

65 Kenyon Wilson Conjecture Conjecture (Kenyon-Wilson 0) Any semicontiguous minor can be represented as the tiling-polynomial P(R) of some region R on the square lattice.

66 Main Result Theorem (L. 0) Any semicontiguous minor can be represented as the tiling-polynomial P(R) of some region R on the square lattice. =

67 Goal Target Define a family of regions R x,h (k,..., k s ; t,..., t s ) corresponding to the minors SM a,b (k,..., k s ; t,..., t s ).

68 Aztec Rectangles n= n= (,) (,) (0,0) m= (a) m= (b) y=0 Figure: (a) AR,,. (b) AR,,.

69 Family of Regions R x,h (k,..., k s ; t,..., t s ) k k k k t t t Figure: Z(k, k,..., k s; t, t,..., t s ) The infinite extension to the right of Z is denoted by Z +. The infinite extension to the left of Z is denoted by Z.

70 Case. h > t + k AR h k+k,h+k x h,0 y=n h-k+k h+k y=0 k = k + + k s and t = t + + t s

71 Case. h 0. AR k+t h k,k+t h k x h,0 k+t-h-k k+t-h-k y=n y=0

72 Case. 0 < h t + k. (a) (b) (c) Figure: AD AD. L AD := AD h+k x h,0 AD := AD k+t h k x h+t,0

73 Case. 0 < h t + k. y=n k+t h k h+k y=0

74 Case. 0 < h t + k. y=n h+k k+t h k y=0

75 Case. 0 < h t + k. y=n h+k k+t h k y=0

76 Main Theorem Theorem (L. 0) SM a,b (k,..., k s ; t,..., t s ) = P(R x,h (k,..., k s ; t,..., t s )) Idea of the proof. Prove by induction on s + k + t: Base case s =, following from Kenyon-Wilson s Theorem. Induction step: Show that two sides satisfy the same recurrence. Apply Dodgson condensation to obtain a recurrence on semicontiguous minors. Apply Kuo condensation to obtain the same recurrence on tiling-polynomials of R-type regions.

77 Dodgson Condensation Lemma (Dodgson condensation) X C = NW SE X - NE X SW = = ( ) ( ) ( ) ( )

78 Dodgson Condensation Lemma (Dodgson condensation) X C = NW SE X - NE X SW Charles Lutwidge Dodgson ( ).

79 Dodgson Condensation Lemma (Dodgson condensation) X C = NW SE X - NE X SW Charles Lutwidge Dodgson ( ). Who is he?

80 Alice s Adventures in Wonderland

81 Future Work Open Problem When can a circular minor be expressed as the tiling-polynomial P(R) of a region R on the square lattice? We only need to consider the case of MA B non-contiguous. with A and B both y=0 Z Z

82 The end Thank you! Website: arxiv: