Dimer Coverings on Random Polyomino Chains

Abstract

Abstract A polyomino chain is a planar square lattice that can be constructed by successively attaching squares to the previous one in two possible ways. A random polyomino chain is then generated by incorporating the Bernoulli distribution to the two types of attachment, which describes a zeroth-order Markov process. Let (ℜ n , p) be the ensemble of random polyomino chains with n squares, where p∈[0,1] is a constant. Then, in this paper, we determine the explicit expression for the expectation of the number of dimer coverings over (ℜ n , p). Our result shows that, with only one exception, i.e., p = 0, the average of the logarithm of this expectation is asymptotically nonzero when n → ∞.