GEOMETRICAL STRING AND SPIN SYSTEMS

We formulate the geometrical string which has been proposed in earlier articles on the euclidean lattice. There are two essentially distinct cases which correspond to non-self-avoiding surfaces and to soft-self -avoiding ones. For the last case it is possible to find such spin sys tems with local interaction which reproduce the same surface dynamics. In the three-dimensional case this spin system is a usual Ising ferro magnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case t he spin system coincides with the gauge Ising system with an additiona l double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hype rsurfaces (membrane and p-branes) of high dimensionality. We compare t hese spin systems with the eight-vertex model and BNNNI models.