Random walk and Hardy's inequalities

Consider an infinite Erdös-Renyi graph indexed by {0,1,2,...}. A random walk proceeds as follows: from i, choose a neighbor j with probability proportional to 1/2^j. This walk has a stationary distribution and, starting from i, converges in log*(i) steps (necessary and sufficient). The proof uses Hardy's inequalities on metric trees. I will try to explain Hardy's inequalities and their applications in probability. This is a joint work with Sourav Chatterjee and Laurent Millo.