State matrix recursion method and monomer--dimer problem

TL;DR

This paper introduces the state matrix recursion method to compute monomer-dimer coverings on lattices, addressing an unsolved problem by providing exact enumeration, partition functions, and asymptotic behaviors for complex regions.

Contribution

The paper develops a novel recursive method for counting monomer-dimer coverings, extending enumeration to complex lattice regions and providing new analytical tools.

Findings

Exact enumeration of monomer-dimer coverings achieved.

Partition functions for various lattice regions derived.

Asymptotic growth rates analyzed.

Abstract

The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer-dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer--dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.