Compact Polymers on Fractal Lattices

Abstract

We study compact polymers, modelled by Hamiltonian walks (HWs), i.e. self‐avoiding walks that visit every site of the lattice, on various fractal lattices: Sierpinski gasket (SG), Given‐Mandelbrot family of fractals, modified SG fractals, and n‐simplex fractals. Self‐similarity of these lattices enables establishing exact recursion relations for the numbers of HWs conveniently divided into several classes. Via analytical and numerical analysis of these relations, we find the asymptotic behaviour of the number of HWs and calculate connectivity constants, as well as critical exponents corresponding to the overall number of open and closed HWs. The nonuniversality of the HW critical exponents, obtained for some homogeneous lattices is confirmed by our results, whereas the scaling relations for the number of HWs, obtained here, are in general different from the relations expected for homogeneous lattices.