Violating conformal invariance: Two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces

Abstract

We present simulations of 2-d site animals on square and triangular lattices in non-trivial geomeLattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of 2-d site animals on square and triangular lattices in non-trivial geometries. The simulations are done with the newly developed PERM algorithm which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent $θ$ ($Z_N \sim μ^N N^{-θ}$). In particular, we studied animals grafted to the tips of wedges with a wide range of angles $α$, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have $k$ sheets and no boundary, generalizing in this way cones to angles $α> 360$ degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, $θ\sim 1/α$, only for small angles ($α\ll 2π$), while $θ\approx const -α/2π$ for $α\gg 2π$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $α$ is given, and comparison is made with critical percolation.