The Dimer Problem
Consider the 2nby2n 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 (nonoriented) 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 4by4
lattice (n=2):
Kasteleyn [14], Fisher [15] and Temperley & Fisher [16] proved that, if f(n)
denotes the number of distinct dimer coverings of L, then
They additionally proved that the socalled molecular freedom per
dimer is
a fascinating and unexpected occurrence of Catalan's constant [1429,45].
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
2nby2n chessboard with 2by1 or 1by2 dominoes? (See our essay on
KneserMahler polynomial constants for
another appearance of the dimer constant.)
The threedimensional analog of this problem remains one of the classical
unresolved issues of solidstate 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 monomerdimer coverings of L. An exact expression for
the asymptotic result
remains unknown.
Copyright © 19952001 by Steven Finch.
All rights reserved.
