The asymptotic determinant of the discrete Laplacian

2000/01/01 by Richard Kenyon · 5 citations
Mathematics · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Markov Chains and Monte Carlo Methods #Mathematics #Exponent #Random walk #Laplace operator #Laplacian matrix #Pure mathematics #Combinatorics #Mathematical analysis #Statistics

paper · pdf · doi:10.1007/bf02392811

openalex publication_date 2000/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

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:

Citations

• The statistics of dimers on a lattice
• Choosing a Spanning Tree for the Integer Lattice Uniformly
• Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird
• Can One Hear the Shape of a Drum?
• Dimer problem in statistical mechanics-an exact result
• Extremals of determinants of Laplacians
• Conformal invariance of domino tiling
• Can One Hear the Shape of a Drum?

Cited by

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• The scaling limit of loop-erased random walk in three dimensions
• Loop-erased random walk [wikipedia]