conformal invariance : Two - dimensional clusters grafted to wedges , cones , and branch points of Riemann surfaces
نویسنده
چکیده
Lattice 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 (ZN ∼ μ N). 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 ∼ 1/α, only for small angles (α ≪ 2π), while θ ≈ const−α/2π for α ≫ 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.
منابع مشابه
Violating conformal invariance: two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces.
Lattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of two-dimensional site animals on square and triangular lattices in nontrivial geometries. The simulations are done with the pruned-enriched Rosenbluth method (PERM) algorithm, which gives very precise estimates of the partition sum, yielding precise values fo...
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