A Lower Bound for the Monomer-Dimer Problem

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Abstract
This paper extends, to the more general case of the multidimensional monomer-dimer problem, some earlier work (Hammersley, 1968) on the dimer problem. The lower bound obtained here is an improvement on the Bondy-Welsh bound (Bondy & Welsh, 1966) at high dimer concentrations, though not at low concentrations. Section 3 shows how an exact solution of the monomer-dimer problem could be obtained if we possessed an appropriate asymptotic theory of random matrices. Section 8 discusses another topic on which some pure mathematical research would be useful, namely the asymptotic theory of non-Hermitian Toeplitz determinants defined by a (divergent but summable) trigonometric series with an Abelian sum which does not belong to the Lebesgue class L(−π,π).

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