Effective parameters of random walks on critical random graphs

点击翻译按钮获取中文摘要

In this project we investigate a random process moving on a graph which itself is generated by a random mechanism. The main objects of interest are effective parameters of the process like the diffusivity, which reveal features of the underlying graph. In particular, we want to study the behaviour of these effective parameters when the graph is close to a critical state for which it displays a complicated, fractal-like structure. The prime example of such a graph is a percolation cluster, as a model for an inhomogeneous, porous medium. For a percolation parameter approaching the critical value, this graph can be described by an incipient infinite cluster. One of the main goals of this project is to estimate the order of the diffusivity, when the percolation cluster is close to criticality. As a more tractable model and an approximation of the incipient infinite cluster, we also consider the random walk on a branching random walk. In this case, the underlying graph has a tree structure and the branching process generating the tree has a survival probability approaching zero.