PhaseP Plenary Talks

Early in the study of random graphs it was observed that natural graph parameters can be concentrated on a small list of values. For example, it has long been known that the independence number of a binomial random graph with constant edge probability is concentrated on two values, where we say that a random variable X defined on graphs exhibits two-point concentration on the random graph G if there exists a number k such that X(G) is either k or k+1 with high probability. In this talk we discuss recent results and open questions regarding tight concentration in the random graphs with an emphasis on the independence number, domination number, chromatic number and related parameters.

Consider the family tree of a branching process with offspring distribution (p_k)_{k ≥ 0} of mean 1 and with a heavy tail such that p_k ∼ c k^{−α − 1} as k → ∞, for some constant c > 0 and α ∈ (1,2). (Then the offspring distribution is in the domain of attraction of an α-stable distribution.) Now condition the tree to have exactly n vertices. Distances in the tree vary as n^{1/α} and, on rescaling them by this factor, we obtain a limit in distribution as n → ∞ called the stable tree. In this talk I’ll discuss a new construction of the limit object, and indicate how to give a proof of the scaling limit theorem using it. This is joint work with Liam Hill (arXiv:2512.17533).