Question

How does the growth of a graph affect its recurrence or transience? If we want our graph to be recurrent, should it grow fast or slow? Why?

Answer 1

The growth of the graph is based on the behavior of the random walk on the infinite cluster. It can be seen in two ways as follows:--

a) When we see percolation in dimension $d<s<2d$ then there is transient walk (transience).

b) When we see precolation in dimension $s\geq 2d$ then there is recurrent walk (recurrence).

So the growth of graph must affect both transience and recurrence because both these factors are the reason of its growth. But it depends on growth rate. It can be fast or slow.

If we want our graph to be recurrent then it will grow at a faster rate because it is based on general stability results for recurrence in random electrical networks which is the reason for its fast growth.

And if we want our graph to be transient then it will grow at a slower rate beacuse transience is based on a renormalization argument

Thank You...