Numerical studies of field theories on random lattices

In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2nd order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.