We study dynamical aspects of three-dimensional gonihedric spins by using Monte-Carlo methods. These models have a purely geometrical motivation, deriving from string and random surface theory. Here, however, we shall analyze this family of models just from a statistical point of view. In particular, we shall be concerned with their ability to exhibit remarkably slow dynamics and seemingly glassy behavior below a certain temperature $T_g,$ without the need of introducing disorder of any kind. We consider first a Hamiltonian that takes into account only a four-spin term $(\kappa =0),$ where a first-order phase transition is well established. By studying the relaxation properties at low temperatures, we confirm that the model exhibits two distinct regimes. For $T_g<T<\mathrm{}{T}_c,$ with long lived metastability and a supercooled phase, the approach to equilibrium is well described by a stretched exponential. For $T<\mathrm{}{T}_g,$ the dynamics appears to be logarithmic. We provide an accurate determination of $T_g.$ We also determine the evolution of particularly long lived configurations. Next, we consider the case $\kappa =1,$ where the plaquette term is absent and the gonihedric action consists in a ferromagnetic Ising with fine-tuned next-to-nearest neighbor interactions. This model exhibits a second order phase transition. The consideration of the relaxation time for configurations in the cold phase reveals the presence of slow dynamics and glassy behavior for any $T<\mathrm{}{T}_c.$ Type-II aging features are exhibited by this model.

• Received 25 April 2002

DOI:https://doi.org/10.1103/PhysRevE.66.056112