An Asymptotic Expansion for the Dimer $\Lambda_d$

An Asymptotic Expansion for the Dimer $\Lambda_d$

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Paul Federbush, University of Michigan
Jadwin Hall 343

The dimer problem is to count the number of ways a $d$-dimensional "chessboard" can be completely covered by non-overlapping dimers (dominoes), each dimer covering two nearest neighbor boxes. The number is approximately $exp^{\Lambda_d*V}$ as the volume $V$ goes to infinity. It has been long known $\Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)-1)$.  We derive an asymptotic expansion whose first few terms are $\Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)-1) +(1/8)(1/d) + (5/96)(1/d2) + (5/64)(1/d3)$. The last term here was calculated by computer, and we conjecture the next term will never be explicitly computed ( just by reason of required computer time ). The expansion is not yet rigorously established.