Heat Kernel and Essential Spectrum of Infinite Graphs

By Radoslaw K. Wojciechowski. 2008

120 citations recorded by the source corpus.

We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9974646d9867d916962348c48b67cb29202ed2bf

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Citations in the corpus

A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, background): “More specifically, this problem is studied for adjacency, (magnetic) Laplacian, and Schrödinger-type operators on graphs in [4, 5, 6, 16, 17, 18, 23, 27, 28, 30, 31, 34, 37, 39, 40, 42, 43, 44, 45] among others.”
Generalized Solutions and Spectrum for Dirichlet Forms on Graphs (2010, background): “Before we turn to the proof let us briefly discuss the geometric interpretation of the boundary measures introduced above: A well-established way to measure the boundary of a set appears in the context of Cheeger constants, which is used to estimate the bottom of the spectrum, see [4, 5, 6, 13, 30, ”
Dirichlet forms and stochastic completeness of graphs and subgraphs (2009, unknown): “Let us discuss these topics in more detail: There are recent investigations of essential selfadjointness of corresponding Laplacians by Jorgensen [17], of stochastic completeness by Dodziuk and Matthai [11], and of both essential selfadjointness and stochastic completeness by Dodziuk [9], Wojciechow”
Cheeger inequalities for unbounded graph Laplacians (2012, background): “Moreover, Cheeger estimates for the bottom of the essential spectrum and criteria for discreteness of spectrum are given in [19, 34, 49, 50].”
On the conservativeness and the recurrence of symmetric jump-diffusions (2012, background): “The conservativeness of the process associated with the physical Laplacian was studied in [6,7,39,40,15].”
Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation (2011, methodology): “This was done for locally finite weighted graphs by Jorgensen in [20] and for locally finite graphs by Wojciechowski in [33] (see [34] as well) and Weber in [32].”