The Dimer Problem
Consider the 2n-by-2n planar square lattice L and call two vertices
adjacent if the distance between them is 1. A dimer
(or diatomic molecule) consists of two adjacent vertices of L
and the (non-oriented) bond connecting them. A dimer covering of L
is a collection of disjoint dimers whose union contains all
the vertices of L. Here is an example of a dimer covering on the 4-by-4
Kasteleyn , Fisher  and Temperley & Fisher  proved that, if f(n)
denotes the number of distinct dimer coverings of L, then
They additionally proved that the so-called molecular freedom per
a fascinating and unexpected occurrence of Catalan's constant [14-29,45].
This is the solution, in graph theoretic terms, of the problem of counting
perfect matchings on the square lattice. It is also the answer
to the question: what is the number of ways of covering a
2n-by-2n chessboard with 2-by-1 or 1-by-2 dominoes? (See our essay on
Kneser-Mahler polynomial constants for
another appearance of the dimer constant.)
The three-dimensional analog of this problem remains one of the classical
unresolved issues of solid-state chemistry.
A dimer arrangement is a collection of disjoint dimers (not
necessarily a cover of L). Here is an example:
Let g(n) denote the number of distinct dimer arrangements, equivalently, the
number of monomer-dimer coverings of L. An exact expression for
the asymptotic result
Copyright © 1995-2001 by Steven Finch.
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