Suppose I have some big rectangle made of $n \times m$ squares, and I want to place tiles on it in a manner that makes a picture of a hamiltonian loop.
I can transform this problem into a problem of putting squares with matching borders in the rectangle. Say I color the bottom and left sides of the rectangle yellow (and keep the top and right side white). Then, hamiltonian loops correspond to tilings of the rectangle with the following tiles, while keeping matching borders between tiles
If I fix one dimension of the rectangle (say $m$), one can compute the number of matching tilings by computing a particular coefficient of the $n$-th power of a square matrix of size $5^m$ $A_m$ : let $I_m$ be the set of sequences $s$ of length $m$ of vertical colors. Consider $V = \Bbb Z^{I_m}$ and let $A \in End(V)$ be the matrix such that $A_{s,t}$ is the number of ways to have a column of $m$ tiles, with yellow on the bottow, white at the top, whose left border is $s$ and whose right border is $t$. Then $N(m,n) = (A^n)_{yellow,white}$ counts the number of hamiltonian loops in the rectangle $n \times m$.
Computing the eigenvalues of $A$, we learn about the asymptotic behaviour of $N(m,n)$ for large $n$. For example if the eigenvalue with greatest modulus $\lambda_m$ is real and positive then $N(m,n) \sim C_m \lambda_m^n$.
Can we say anything about the asymptotic behaviour for large $n$ and $m$ ? Because we originally use $6$ tiles, we obviously have $N(m,n) \le 6^{nm}$, so the set $\{|\lambda_m|^{1/m}\}$ has a $\limsup$ $\lambda \le 6$. Now, I would be interested in a few questions :
Is $\lambda$ algebraic ? Can we compute it ? Do we have $\lim_{m \to \infty} |\lambda_m|^{1/m} = \lambda$ ?
Do we have $\lim_{m,n \to \infty} N(m,n)^{1/mn} = \lambda$ ?
(this kind of tiling game can answer other kinds of problems, such as how many ways are there to tile a rectangle with polyominos with specific shapes)