We first consider irreducible critical multitype Bienaymé trees and extend the results to the case, when they possess a critical irreducible component with attached subcritical components. We study these trees under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong non-asymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.

Let T be any Galton-Watson tree. Write vol(T) for the volume of T (the number of nodes), ht(T) for the height of T (the greatest distance of any node from the root) and wid(T) for the width of T (the greatest number of nodes at any level). We study the relation between vol(T), ht(T) and wid(T). In the case when the offspring distribution p = (p_i, i >= 0) has mean one and finite variance, both ht(T) and wid(T) are typically of order vol(T)^{1/2}, and have sub-Gaussian upper tails on this scale. Heuristically, as the tail of the offspring distribution becomes heavier, the tree T becomes "shorter and bushier". I will describe a collection of work which can be viewed as justifying this heuristic in various ways In particular, I will explain how classical bounds on Lévy's concentration function for random walks may be used to show that for any offspring distribution, the random variable ht(T)/wid(T) has sub-exponential tails. I will also describe a more combinatorial approach to coupling random trees with different degree sequences which allows the heights of randomly sampled vertices to be compared.

Given a large graph G and k agents on this graph, we consider the Voronoi tessellation induced by the graph distance. Each agent gets control of the portion of the graph that is closer to itself than to any other agent. We study the limit law of the vector Vor: = (V1/n, V2/n, …, Vk/n), whose i'th coordinate records the fraction of vertices of G controlled by the i'th agent, as n tends to infinity. We show that if G is a uniform random tree, and the agents are placed uniformly at random, the limit law of Vor is uniform on the (k – 1)-dimensional simplex. In particular, when k = 2, the two agents each get a uniform random fraction of the territory. In fact, we prove the result directly on the Brownian continuum random tree (CRT), and we also prove the same result for a “higher genus” analogue of the CRT that we call the continuum random unicellular map, indexed by a genus parameter g ≥ 0. As a key step of independent interest, we study the case when G is a random planar embedded graph with a finite number of faces. The main idea of the proof is to show that Vor has the same distribution as another partition of mass Int: = (I1/n, I2/n, …, Ik/n) where Ij is the contour length separating the i-th agent from the next one in clockwise order around the graph.

“Small worlds” are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman–Watts small world, the mixing time is of order log2 n. This confirms a prediction of Richard Durrett, who proved a lower bound of order log2 n and an upper bound of order log3 n.