An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λd(p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤd, where p∈[0,1] is the dimer density. We give upper and lower bounds for λd(p) in terms of expressions involving λd−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤd is bounded above by λd(p). We compute the first three terms in the formal asymptotic expansion of λd(p) in powers of $\frac{1}{d}$. We prove that the lower asymptotic matching conjecture is satisfied for λd(p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.

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