The Laplacian on planar graphs and graphs on surfaces
Probability
2012-03-07 v1 Combinatorics
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.
Cite
@article{arxiv.1203.1256,
title = {The Laplacian on planar graphs and graphs on surfaces},
author = {Richard Kenyon},
journal= {arXiv preprint arXiv:1203.1256},
year = {2012}
}