We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent α + 1, where α ∈ (1, 2). The limiting components are constructed from random R-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive α-stable Lévy process. These spanning R-trees are measure-changed α-stable trees. In each such R-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős-Rényi random graph) and where the role of the α-stable Lévy process is played by a Brownian motion.

In this paper, we study the random walk on a supercritical branching process with an uncountable and unbounded set of types supported on the d-regular tree T d (d ≥ 3), namely the cluster C h• of the root in the level set of the Gaussian Free Field (GFF) above an arbitrary value h ∈ (−∞, h ⋆ ). The value h ⋆ ∈ (0, ∞) is the percolation threshold; in particular, C h• is infinite with positive probability. We show that on C h• conditioned to be infinite, the simple random walk is ballistic, and we give a law of large numbers and a Donsker theorem for its speed.To do so, we design a renewal construction that withstands the long-range dependencies in the structure of the tree. This allows us to translate underlying ergodic properties of C h • into regularity estimates for the random walk.

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We prove a cutoff for the random walk on random n-lifts of finite weighted graphs, even when the random walk on the base graph G of the lift is not reversible. The mixing time is w.h.p. t mix = h −1 log n, where h is a constant associated to G, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of G.In the particular case where the base graph is a vertex with d/2 loops, d even, we obtain a cutoff for a d-regular random graph (as did Lubetzky and Sly in [26] with a slightly different distribution on d-regular graphs, but the mixing time is the same).