Random Tilings: Exercises
Béatrice de Tillière
Cédric Boutillier
August 25, 2015
Exercise 1 Determine a Kasteleyn orientation for the following graph.
Exercise 2 How many rhombus tilings are there of an equilateral triangle
of size n?
Exercise 3 How many domino tilings are there of a 2 × n rectangle?
1
Exercise 4 Consider the following matrix L of size n × n:


0 1
 1 0


1

L=




1
0
..
.
0
0
1
..
1
.
..
.
0
1








1 
0
1. Find the eigenvectors and eigenvalues of the matrix L.
2. Convince yourself that, by exchanging lines and columns, the matrix
L can be transformed into
0 K
K⊥ 0
L̃ =
!
where K is the Kasteleyn matrix of the bipartite graph Gn made of n
vertices on a line.
3. Deduce the number of perfect matchings of Gn .
4. Is there an easier way of finding the answer?
2