Effective parameters of random walks on critical random graphs

在这个项目中，我们研究了一个随机过程移动的图，这本身是由一个随机机制产生的。感兴趣的主要对象是过程的有效参数，如扩散率，它揭示了底层图形的特征。特别是，我们要研究这些有效参数的行为时，图形接近临界状态，它显示一个复杂的，分形结构。这种图的最好例子是渗流簇，作为非均匀多孔介质的模型。当逾渗参数接近临界值时，该图可用初始无限簇来描述。这个项目的主要目标之一是估计的顺序的扩散，当渗流集团接近临界。作为一个更易处理的模型和初始无限集群的近似，我们还考虑了分支随机游动上的随机游动。在这种情况下，底层图具有树结构，并且生成树的分支过程具有接近于零的生存概率。

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.