We consider the Erdős-Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3 , for some fixed λ ∈ R. We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3 , converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.

Assign i.i.d. standard exponential edge weights to the edges of the complete graph Kn, and let Mn be the resulting minimum spanning tree. We show that Mn converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M . The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order fluctuations for which we provide explicit bounds. Our volume growth estimates confirm recent predictions from the physics literature [18], and contrast with the behaviour of invasion percolation on the PWIT [2] and on regular trees [6], which exhibit quadratic volume growth. 10 4. Properties of the forward maximal process ((X n , Z n ), n ≥ 1) 13 4.

We consider the Erdős-Rényi random graph G(n, p) inside the critical window, where p = 1/n + λn −4/3 for some λ ∈ R. We proved in [1] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n −1/3 and letting n → ∞ yields a non-trivial sequence of limit metric spaces C = (C1, C2, . . . ). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [29] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component. * MSC 2000 subject classifications: primary 05C80; secondary 60C05.

We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n → ∞, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-reversal of this Markov chain have limits as n → ∞. These results can be interpreted as describing a "post-gelation" phase of the Bolthausen-Sznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed.