We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.
Yong LinDepartment of Mathematics, Information School, Renmin University of China, Beijing, People’s Republic of ChinaGábor LippnerDepartment of Mathematics, Harvard University, Cambridge, MA, 02138, USAShing-Tung YauDepartment of Mathematics, Harvard University, Cambridge, MA, 02138, USA
We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.
Yong LinRenmin University, Department of Mathematics, Beijing 100872, ChinaLinyuan LuUniversity of South Carolina, Department of Mathematics, Columbia, SC 29208, USAShing-Tung YauHarvard University, Department of Mathematics, Cambridge, MA 02138, USA
We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.