Critical random forests

Let F(N, m) denote a random forest on a set of N vertices, chosen uniformly from all forests with m edges. Let F(N,p) denote the forest obtained by conditioning the Erdos-Renyi graph G(N,p) to be acyclic. We describe scaling limits for the largest components of F(N,p) and F(N, m), in the critical window p = N-1 + O(N-4/3) or m = N/2 + O(N-2/3). Aldous (1997) described a scaling limit for the largest components of G(N, p) within the critical window in terms of the excursion lengths of a reflected Brownian motion with time-dependent drift. Our scaling limit for critical random forests is of a similar nature, but now based on a reflected diffusion whose drift depends on space as well as on time.