Critical phenomena on k-booklets.

Abstract

We define a "k-booklet" to be a set of k semi-infinite planes with -∞<x<∞ and y≥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, and for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k≥3. 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≈0, and the critical points are not shifted. In the 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.