Conformal invariance and simulations in curved geometries

Conformal mappings serve as useful tools for the determination of universal properties of critical models. Typical applications are subject to a major restriction, namely that the pertinent conformal mapping should lead to a geometry that can be investigated by means of numerical methods such as Monte Carlo simulations. Since conformal mappings of 3-D systems usually lead to curved geometries which are difficult to investigate numerically, most applications have thus far been restricted to 2-D systems. We present a solution of this problem for discrete spin models, by taking the anisotropic or Hamiltonian limit which renders one of the lattice directions continuous, such that the dimensionality in effect reduces by one. Applications to the 3-D Ising and percolation models confirm the predictions obtained from the assumption of conformal invariance, and lead to accurate numerical results for the scaling dimensions.