CHUNG-YAU INVARIANTS AND RANDOM WALK ON GRAPHS

The Chung-Yau graph invariants were originated from Chung-Yau’s work on discrete Green’s function. They are useful to derive explicit formulas and estimates for hitting times of random walks on discrete graphs. In this thesis, we study properties of Chung-Yau invariants and apply them to study some questions: (1) The relationship of Chung-Yau invariants to classical graph invariants; (2) The change of hitting times under natural graph operations; (3) Properties of graphs with symmetric hitting times; (4) Random walks on weighted graphs with different weight schemes

Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

In this paper, we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.Supported in part by a grant from Ministerio de Economíıa y Competitividad (MTM 2012-30719), Spain. Supported in part by two grants from Ministerio de Economíıa y Competitividad (MTM 2013-46374-P and MTM 2015- 69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Interplay of Analysis and Probability in Physics

[no abstract available