We investigate a generalized Ising action containing nearest neighbour, next to nearest neighbour and plaquette terms that has been suggested as a potential string worldsheet discretization on cubic lattices by Savvidy and Wegner. We use both mean field techniques and Monte-Carlo simulations to sketch out the phase diagram.The Gonihedric (Savvidy-Wegner) model has a symmetry that allows any plane of spins to be flipped with zero energy cost, which gives a highly degenerate vacuum state. We choose boundary conditions in the simulations that eliminate this degeneracy and allow the definition of a simple ferromagnetic order parameter. This in turn allows us to extract the magnetic critical exponents of the system.(a) Permanent Address:

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterisation of the phase structure, particularly in the case where there are two such parameters -such as the Ising model with inverse temperature β and external field h.In various two parameter calculable models the scalar curvature R of the information metric has been found to diverge at the phase transition point βc and a plausible scaling relation postulated: R ∼ |β − βc| α−2 . For spin models the necessity of calculating in non-zero field has limited analytic consideration to 1D, mean-field and Bethe lattice Ising models. In this paper we use the solution in field of the Ising model on an ensemble of planar random graphs (where α = −1, β = 1/2, γ = 2) to evaluate the scaling behaviour of the scalar curvature, and find R ∼ |β − βc| −2 . The apparent discrepancy is traced back to the effect of a negative α.

For the 3D gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour 3D Ising action. When the parameter κ weighting the self-intersections is small the model has a first order transition and when it is larger the transition is continuous. In this paper we investigate the scaling of the renormalized string tension, which is entirely generated by fluctuations, using Monte Carlo simulations for κ = 0.0, 0.1, 0.5 and 1.0. The scaling of the string tension allows us to obtain an estimate for the critical exponents α and ν using both finite-size-scaling and data collapse for the scaling function. The behaviour of the string tension when the self-avoidance parameter κ is small also clearly demonstrates the first order nature of the transition in this case, in contrast to larger values. Direct estimates of α are in good agreement with those obtained from the scaling of the string tension. We have also measured γ/ν.(a) Permanent Address:

We investigate a 3d Ising action which corresponds to a a class of models defined by Savvidy and Wegner, originally intended as discrete versions of string theories on cubic lattices. These models have vanishing bare surface tension and the couplings are tuned in such a way that the action depends only on the angles of the discrete surface, i.e. on the way the surface is embedded in Z 3 . Hence the name gonihedric by which they are known. We show that the model displays a rather clear first order phase transition in the limit where self-avoidance is neglected and the action becomes a plaquette one. This transition persists for small values of the self avoidance coupling, but it turns to second order when this latter parameter is further increased. These results exclude the use of this type of action as models of gonihedric random surfaces, at least in the limit where self avoidance is neglected.(a) Permanent Address:

We discuss a generalized Ising action containing nearest neighbour, next to nearest neighbour and plaquette terms that has been suggested as a potential string worldsheet discretization on cubic lattices by Savvidy and Wegner. This displays both first and second order transitions depending on the value of a "self-intersection" coupling as well as possessing a novel semi-global symmetry.