For fixed r ≥ 2, we consider bootstrap percolation with threshold r on the Erdős-Rényi graph Gn,p. We identify a threshold for p above which there is with high probability a set of size r which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants.As an application of our results, we also obtain an upper bound for the threshold for K4-bootstrap percolation on Gn,p, as studied by Balogh, Bollobás and Morris. We conjecture that our bound is asymptotically sharp.These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities which are of interest in their own right.

The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure.Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure.We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints.Finally, we classify the types of geodesic networks which are dense. For each k ∈ {1, 2, 3, 4, 6, 9}, there is a dense set of pairs of points which are joined by networks of exactly k geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense.

The vertex isoperimetric number of a graph G = (V,E ) is the minimum of the ratio |∂ V U |/|U | where U ranges over all nonempty subsets of V with |U |/|V | ≤ u and ∂ V U is the set of all vertices adjacent to U but not in U . The analogously defined edge isoperimetric number-with ∂ V U replaced by ∂ E U , the set of all edges with exactly one endpoint in U -has been studied extensively. Here we study random regular graphs. For the case u = 1/2, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all d ≥ 3. Moreover, we obtain a lower bound on the asymptotics as d → ∞. We also provide asymptotically almost sure lower bounds on |∂ E U |/|U | in terms of an upper bound on the size of U and analyse the bounds as d → ∞.

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