Enumeration Algorithm for Lattice Model

Transcription

1 Enumeration Algorithm for Lattice Model Seungsang Oh Korea University International Workshop on Spatial Graphs 2016 Waseda University, August 5, 2016

2 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer Problem (best application) 3 Multiple Self-Avoiding Polygon Enumeration 4 Further Applications in Lattice Statistics

3 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer Problem (best application) 3 Multiple Self-Avoiding Polygon Enumeration 4 Further Applications in Lattice Statistics

4 State matrix recursion algorithm State matrix recursion algorithm enumerates 2-dimensional lattice models such as Monomer-dimer coverings Multiple self-avoiding walks and polygons Independent vertex sets Quantum knot mosaics These are famous problems in Combinatorics and Statistical Mechanics studied by topologists, combinatorialists and physicists alike.

5 State matrix recursion algorithm is divided into three stages: Stage 1. Conversion to appropriate mosaics Stage 2. State matrix recursion formula Stage 3. State matrix analyzing During this talk, the algorithm will be briefly demonstrated by solving the Monomer-Dimer Problem.

6 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer Problem (best application) 3 Multiple Self-Avoiding Polygon Enumeration 4 Further Applications in Lattice Statistics

7 Monomer-dimer coverings Monomer-dimer covering in m n rectangle on the square lattice Z m n Generating function D m n (z) = k(t) z t where k(t) is the number of monomer-dimer coverings with t monomers. D m n (1) is the number of monomer-dimer coverings. D m n (0) is the number of pure dimer coverings (i.e., no monomers).

8 [Kasteleyn and Temperley-Fisher 1961] Pure dimer problem for even mn m j=1 n k=1 ( 2 cos πj m + 1 ) ( ) πk + 2i cos n + 1 Breakthrough results [Tzeng-Wu 2003] Single boundary monomer problem for odd mn (it has a fixed single monomer on the boundary) m 1 2 j=1 n 1 2 k=1 [4 cos 2 ( πj ) ( )] πk + 4 cos 2 m + 1 n + 1 Question: How about if we allow many monomers? Generating function?

9 Monomer-Dimer Theorem Theorem D m n (z) = (1, 1)-entry of (A m ) n where A m is a 2 m 2 m matrix defined by the recurrence relation [ ] Ak 2 O z A A k = k 1 + k 2 A O k 2 O k 1 k 2 starting with A 0 = [ 1 ] and A 1 = A k 1 O k 1 [ ] z 1 where O 1 0 k is the 2 k 2 k zero-matrix. Note that it is not a closed form solution, but a sparse recurrence algorithm.

10 Exact enumeration n D n n (1) (D n n (1)) 1 n

11 Stage 1. Conversion to monomer-dimer mosaics 5 mosaic tiles labeled with two letters a, b Adjacency Rule : Attaching edges of adjacent tiles have the same letter. Boundary state requirement : All boundary edges are labeled with letter a.

12 Stage 2. State matrix recursion formula State polynomial : Twelve suitably adjacent 3 3-mosaics associated with b-state aba, t-state bab and the trivial l- and r-states aaa to produce the associated state polynomial 1 + 5z 2 + 5z 4 + z 6.

13 State matrix A m n for the set of suitably adjacent m n-mosaics is a 2 m 2 m matrix (a ij ) where a ij is the state polynomial associated to i-th b-state, j-th t-state, and the trivial l- and r-states. (Trivial state condition is needed for the boundary state requirement) We arrange 2 m states of length m in the lexicographic order. For example, (3,6)-entry of A 3 3 is a 3,6 = 1 + 5z 2 + 5z 4 + z 6.

14 Recursion strategy to find the state matrix A m n. 1. Find the starting state matrices A 1 and B 1 for 1 1-mosaics. 2. Find the bar state matrices A k and B k for suitably adjacent k 1-mosaics (or bar mosaics) by attaching a mosaic tile recursively on the right side. 3. Find the state matrix A m k for suitably adjacent m k-mosaics by attaching a bar mosaic of length m on the top side.

15 Summary First, we get the recursive relation from the bar state matrix recursion lemma [ ] [ ] z Ak 1 + B A k = k 1 A k 1 Ak 1 O and B A k 1 O k = k 1 k 1 O k 1 O k 1 starting with A 0 = [ 1 ] and B 0 = [ 0 ]. Then, we have the state matrix from the state matrix multiplication lemma A m n = (A m ) n.

16 Stage 3. State matrix analyzing Monomer-dimer generating function w.r.t. the number of monomers D m n (z) = (1,1)-entry of A m n.

17 Monomer-Dimer Theorem Theorem D m n (z) = (1, 1)-entry of (A m ) n where A m is a 2 m 2 m matrix defined by the recurrence relation [ ] Ak 2 O z A A k = k 1 + k 2 A O k 2 O k 1 k 2 starting with A 0 = [ 1 ] and A 1 = A k 1 O k 1 [ ] z 1 where O 1 0 k is the 2 k 2 k zero-matrix.

18 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer Problem (best application) 3 Multiple Self-Avoiding Polygon Enumeration 4 Further Applications in Lattice Statistics

19 Self-avoiding polygon (SAP) on the square lattice Z 2 Self-avoiding polygons p n = number of SAPs of length n up to translations Finding p n is the central unsolved problem during last 70 years in Combinatorics and Statistical Mechanics. There are many numerical datas, but few mathematically proved results.

20 Breakthrough results [Hammersley 1957] The limit µ = lim (p n ) 1 n exists. µ = ± : best estimate on Z 2 during 50 years. [Duminil-Copin and Smirnov 2012, Annals of Math.] µ = on the hexagonal lattice H 2 (easier than on Z 2 ). Nobody expects that there will be a closed form of p n.

21 Multiple self-avoiding polygons Multiple self-avoiding polygon (MSAP) in Z m n p m n = number of MSAPs in Z m n (not up to translations) Theorem p m n = (1, 1)-entry of (A m ) n 1 where the 2 m 2 m matrix A m is defined by [ ] [ ] Ak B A k+1 = k Bk A and B B k A k+1 = k k A k O k starting with A 0 = [ 1 ] and B 0 = [ 0 ].

22 MSAPs in the 1-slab square lattice Multiple self-avoiding polygons (links) in the 1-slab square lattice Z m n 2 (2 layers of the planes)

23 Conversion to 1-slab MSAP mosaics by using 65 mosaic tiles

24 MSAP enumeration in Z m n 2 Theorem The number of MSAPs in the 1-slab square lattice Z m n 2 is (1, 1)-entry of (A m ) n 1 where the 4 m 4 m matrix A m is defined by A k +D k B k +C k B k +C k A k +D k B k +C k A k A k +D k C k B A k+1 = k +C k A k A k +D k C k A, B B k +C k A k +D k A k B k+1 = k O k C k O k, k A k +D k C k B k A k A k +D k C k B k A k C k O k A k O k B k +C k A k +D k A k B k A k +D k C k B k A k A C k+1 = k +D k C k B k A k C and D A k B k O k O k+1 = k O k A k O k, k B k A k O k O k B k A k O k O k A k O k O k O k starting with A 0 = [ 1 ] and B 0 = C 0 = D 0 = [ 0 ]. - The number of MSAPs in Z is

25 Links in the 3-dimensional cubic lattice Links in the 3-dimensional cubic lattice Z l m n (not up to translations and ambient isotopies)

26 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer Problem (best application) 3 Multiple Self-Avoiding Polygon Enumeration 4 Further Applications in Lattice Statistics

27 Different regular lattices Hexagonal (honeycomb) lattice H m n (MSAP model)

28 Different regular lattices Triangular lattice T m n (Monomer-dimer model)

29 Different regular lattices 1-slab square lattice Z m n 2 (Multiple self-avoiding polygon (link) model)

30 Polymer model Monomer-dimer-trimer-tetramer covering

31 Polyomino model Monomino-domino-tromino tiling

32 Independent vertex model Independent vertex sets

33 Quantum knot model Quantum knot mosaic with 11 knot mosaic tiles as follows

34 Squared rectangle model Tiling a rectangle by squares with various integer sizes

35 Tetris configuration by 7 tetrominoes Tetris model

36 Thank you!