Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices
This provides a theoretical advance for combinatorial enumeration problems in statistical mechanics, though it is incremental as it extends known recurrence methods to general polymer lengths.
The authors prove that the number of polymer coverings on two-dimensional rectangular lattices satisfies simple recurrence relations for arbitrary polymer length and lattice width, potentially offering insights into #P-complete enumeration problems like monomer-dimer configurations.
The problem of counting polymer coverings on the rectangular lattices is investigated. In this model, a linear rigid polymer covers $k$ adjacent lattice sites such that no two polymers occupy a common site. Those unoccupied lattice sites are considered as monomers. We prove that for a given number of polymers ($k$-mers), the number of arrangements for the polymers on two-dimensional rectangular lattices satisfies simple recurrence relations. These recurrence relations are quite general and apply for arbitrary polymer length ($k$) and the width of the lattices ($n$). The well-studied monomer-dimer problem is a special case of the monomer-polymer model when $k=2$. It is known the enumeration of monomer-dimer configurations in planar lattices is #P-complete. The recurrence relations shown here have the potential for hints for the solution of long-standing problems in this class of computational complexity.