Effective parameters of random walks on critical random graphs

在这个项目中，我们研究了一个在图上运动的随机过程，这个图本身是由随机机制产生的。主要感兴趣的对象是过程的有效参数，如扩散系数，它揭示了底层图形的特征。特别是，我们想要研究当图形接近临界状态时这些有效参数的行为，对于该临界状态，它显示出复杂的、类分形的结构。这类图的主要例子是渗流团簇，作为非均匀、多孔介质的模型。对于接近临界值的渗流参数，这个图可以用一个初始的无限团簇来描述。这个项目的主要目标之一是估计渗流团簇接近临界值时扩散系数的量级。作为一个更易处理的模型和初始无限团簇的近似，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.