Diffusions on graphs, Poisson problems and spectral geometry |
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Authors: | Patrick McDonald Robert Meyers |
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Institution: | New College of Florida, 5700 N. Tamiani Trail, Sarasota, Florida 34243 ; The Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012-1110 |
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Abstract: | We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function. |
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Keywords: | Poisson problem random walk variational principles spectral graph theory Stirling numbers zeta functions |
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