Critical phenomena on k-booklets

Abstract

We define a `k-booklet' to be a set of k semi-infinite planes with $-\infty < x < \infty$ and $y \geq 0$, glued together at the edges (the `spine') y=0. On such booklets we study three critical phenomena: Self-avoiding random walks, the Ising model, and percolation. For k=2 a booklet is equivalent to a single infinite lattice, for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k>2. Self avoiding walks starting at y=0 show a first order transition at a shifted critical point, with no power-behaved scaling laws. The Ising model and percolation show hybrid transitions, i.e. the scaling laws of the standard models coexist with discontinuities of the order parameter at $y\approx 0$, and the critical points are not shifted. In case of the Ising model ergodicity is already broken at $T=T_c$, and not only for $T<T_c$ as in the standard geometry. In all three models correlations (as measured by walk and cluster shapes) are highly anisotropic for small y.