Compact polymers on decorated square lattices

TL;DR

This paper investigates the enumeration and entropy of Hamiltonian cycles, modeling compact polymers on decorated square lattices with sublattice structures, using field theory and transfer matrix methods.

Contribution

It introduces two novel methods for estimating the number of Hamiltonian cycles on non-homogeneous lattices with sublattice structures.

Findings

Estimated the number of Hamiltonian cycles using saddle point approximation.

Numerically diagonalized transfer matrices to obtain scaling exponents.

Provided entropy estimates for compact polymers on decorated lattices.

Abstract

A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It serves as a model of a compact polymer on a lattice. I study the number of Hamiltonian cycles, or equivalently the entropy of a compact polymer, on various lattices that are not homogeneous but with a sublattice structure. Estimates for the number are obtained by two methods. One is the saddle point approximation for a field theoretic representation. The other is the numerical diagonalization of the transfer matrix of a fully packed loop model in the zero fugacity limit. In the latter method, several scaling exponents are also obtained.