We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R 2. Specifically, for each ǫ> 0 let Hǫ be the subgraph of ǫZ 2 whose vertices lie in a fixed rectilinear polygon U. Let N(Hǫ) denote the number of vertices of Hǫ and B(Hǫ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on Hǫ has the following asymptotic expansion in ǫ:
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Richard Kenyon (2000) studied this question.
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