On random walk on growing graphs
  Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple random walk on slowly growing graphs, upon knowing the volume and Cheeger constant of each graph. For much more specialized cases, we establish matching lower bounds, and deduce sufficient (weak) recurrence criteria. We also address recurrence directly in relation to a universality conjecture of [DHS]. We answer a related question of [SZ, Problem 1.8] about "inhomogeneous merging" in the negative. 
Ruojun Huang
math.PR