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arXiv · 1203.1256

The Laplacian on planar graphs and graphs on surfaces

Richard Kenyon
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Abstract

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdi\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.

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Richard Kenyon. 2012-03-06. The Laplacian on planar graphs and graphs on surfaces. https://arxiv.org/abs/1203.1256

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