DIMER MONOMER Zac Hall Dr. Bryan Shader  Planar monomer-dimer tilings  Monomer – 1 by 1 tile  Dimer – 2 by 1 tile or 1 by 2 Monomer-Dimer Tilings Monomer-Dimer Tilings  What properties does a “random” monomer-dimer tiling of a region have? ◦ Expected number of Dimers? ◦ Is a given cell more likely to be a monomer?  How can we generate a uniform, random monomer-dimer tiling of a region? Fundamental Questions:  Methods ◦ Computer-based simulations  Kastelyn’s Theorem ◦ Probabilistic Models (matrices) ◦ Analytical Models  Sage Math (Python) was used throughout the year to aid us Monomer-Dimer Tilings  N by N grids w/ 2 monomers, rest dimers  4 by 4 ◦ 552 possible tilings ◦ Monomer’s probable locations: corners, sides, center 1 6 2 2 6 1  6 by 6 ◦ 363,536 possible tilings 6 5 4 4 5 6 ◦ Probable Locations: 2 4 3 3 4 2  8 by 8 2 4 3 3 4 2 ◦ 1,760,337,760 tilings 6 5 4 4 5 6 Beginning Work 1 6 2 2 6 1  Recursive Formulas: 2𝑁−2+1 1 𝑁−2  FN = 1 + 𝑁−1 EN + FN-3 + 𝑁−1(F0 + E2 8 2 N-2 ) + 𝑁−4 𝑁−𝑘−2σ𝑘=1 𝑁−𝑘 (Fk + E2 N-k-2 ) E = σ𝑁−3 1 1  N 𝑘=0 2𝑘+1 (FN-k-2 + Ek) + 2𝑁−2 (F0 + EN-2)  EN = E(Dimers)  E20000 /20000 = 0.43344653343386  Bounded … Must Converge 1 by N : Mathematical Discoveries  Matrix A8: ◦ “Characteristic matrix” for monomer dimer tilings of 8 X 1 region ◦ A8^N represents “characteristic matrix” for 8 X N region ◦ Top-left entry represents the number of tilings possible ◦ 8 X N plane: 256 X 256 matrix ◦ Created using numerical methods/coding Enumerating 8 X N Tilings  Formula for total number of tilings  This is a 136th degree recursive formula: TN =133* TN-1 +9461* TN-2 - 444475* TN-3 +… - 23797 TN-133 + 1027 TN-134 + 21 TN-15 - TN-136  lim TN/ TN-1≈170.8771576 N->∞ Recursive Formula for 8XN  Cayley-Hamilton theorem for a square matrix  Satisfies a linear recurrence  Could do this for any kXN - first to do 8XN and find the recurrence for it  Counting problem with computational difficulties for high N Implications  Use the same “characteristic matrix” for discerning vertical and horizontal dimers  Able to characterize our tilings with more detail Extension  Original 3X3  Updated 3X3 3 2 1 1 2 1 1 1 2𝑣 + 1 ℎ𝑣 + ℎ ℎ ℎ2 ℎ𝑣 + ℎ ℎ2 ℎ2 ℎ3 2 0 1 0 1 0 1 0 𝑣 + 1 0 ℎ 0 ℎ 0 ℎ2 0 1 1 0 0 1 1 0 0 1 ℎ 0 0 ℎ ℎ2 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 2 1 1 1 0 0 0 0 𝑣 + 1 ℎ ℎ ℎ2 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 ℎ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Extension  Found a relationship for AN : 𝑣 ∗ 𝐴𝑘−2 0 0 0 𝐴 ℎ ∗ 𝐴 0 0 0 0 𝐴𝑘 = 𝑘−1 𝑘−1 + 𝐴𝑘−1 0 0 0 0 0 0 0 0 0  Can extend this to our simple numerical example  Create more characteristic equations for k X N regions Extension  Static -> Dynamic  Explored various regions: N by N, 2 by N, 1 by N  Began as all monomers, “paired” to become dimers.  Pair until “frozen” Generating tilings  Creating probability distributions for any MXN area (E(dimers)) ◦ Created every possible tiling by coding it  Represented by matrix of 0’s, 1’s, 2’s ◦ Use these tilings to create probability distibutions  Computational issues: ◦ 6 X 6 possibilities aren’t able to be calculated in less than 2 hours ◦ Probabilities require ‘permanent’ calculation – even more difficult Final work  Finding tiling distribution – need to find probabilities for ending with each  Finding a formula for E(dimers) as a recursion? Possibly the same recurrence with different initial conditions?  Applications in diatomic nuclear bonding and ice-formations Future Potential Research Thank you.