The asymptotic determinant of the discrete Laplacian
نویسنده
چکیده
We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R2. Specifically, for each > 0 let H be the subgraph of Z2 whose vertices lie in a fixed rectilinear polygon U . Let N (H ) denote the number of vertices of H and B(H ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on H has the following asymptotic expansion in : 4G π N (H )+ log( √ 2 − 1) 2 B(H )− π 48 r2( ,U )+ o(1) where G is Catalan’s constant and r2( ,U ), which is O(log 1 ), is the Dirichlet energy of a certain canonical harmonic function h on U . As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z2 is 5/4.
منابع مشابه
The Asymptotic Determinant of the Discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a simplyconnected rectilinear region in R2. Specifically, for each > 0 let H be the subgraph of Z2 whose vertices lie in a fixed rectilinear polygon U . Let N (H ) denote the number of vertices of H and B(H ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on H has the fo...
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