Over the last 30 years, a beautiful theory of scaling limits for random trees has been established. In my first lecture, I will discuss one of the foundational examples: the Brownian continuum random tree (BCRT). Consider a uniform random labelled tree on n vertices (Cayley's formula tells us that there are n^{n-2} such labelled trees; pick one uniformly at random). It turns out that the typical distance between two vertices in such a tree scales as \sqrt{n}, and that on rescaling we get convergence in distribution to the BCRT, which is a random fractal with fascinating properties. This result is not specific only to a uniform random labelled tree, but holds (for example) for conditioned critical branching process trees with finite offspring variance. I aim to particularly emphasise the "line-breaking" approach to these results, which goes right back to Aldous' original work on the BCRT, but has recently been given new impetus.

In my second lecture, I will focus on a branching random walk whose genealogy is given by a conditioned branching process tree (in the domain of attraction of the BCRT). The associated discrete snake gives the collection of random walk paths from the root to each of the vertices in turn. I will discuss work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell in which we show the convergence of globally centred discrete snakes to the so-called "Brownian snake driven by a Brownian excursion". In this work, we make essential use of the line-breaking approach outlined in the first lecture.

Abstract

The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a very close relative of the discrete Gaussian free field (DGFF), for which instead the most likely interfaces are those preserving the mean height. However, working with the two modelspresents some key differences, in that in the MM the shape is driven by the biharmonic operator, while the DGFF is essentially a Gaussian perturbation of harmonic functions. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will review some basic properties of the MM, and we will investigate a random walk representation for the covariances of the MM and what it can bring forth in terms of its scaling limits.

This talk is based on joint works with Alessandra Cipriani (University College, London), Biltu Dan (Indian Institute of Science) and Rounak Ray (TU, Eindhoven).