Heat Kernel Estimates for Random Walks on Some Kinds of One-dimensional Continuum Percolation Clusters

Abstract

We consider random walks on random graphs determined by a some kind of continuum percolation on $\mathbf{R}$. The vertex set of the random graph is given by the Poisson points conditioned that all points of $\mathbf{Z}$ are contained. The edge set of the random graph is determined by the random radii of the spheres centered at each points. We give heat kernel estimates for the random walks under the condition on the moment of the random radii. We will also discuss random walks on continuum percolation clusters in $\mathbf{R}^d$, $d\ge 2$.