Scaling limit for the random walk on the largest connected component of   the critical random graph
  A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree. 
David A. Croydon
math.PR