Thermodynamics of a Tiling Model

A specific two-dimensional tiling model, composed of the so-called Wang tiles has been studied at finite temperature using Monte Carlo numerical simulations. In the absence of any thermal bath the Wang tiles provide the opportunity of building a very large number of non-periodic tilings. We can construct a local Hamiltonian such that only perfectly matched tilings are ground states with zero energy. This Hamiltonian has a very large degeneracy. The thermodynamic behaviour of such a system seems to show a continuous phase transition at non-zero temperature. An order parameter with non-trivial features is proposed. Under the critical temperature the model exhibits ageing properties. The fluctuation-dissipation theorem is violated.