Matthew I. Roberts

The many-to-few lemma and multiple spines

We develop a simple and intuitive identity for calculating expectations of weighted k-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

A simple path to asymptotics for the frontier of a branching Brownian motion

We give short proofs of two classical results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour.

Catalytic Branching Processes via Spine Techniques and Renewal Theory

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.

One-point localization for branching random walk in Pareto environment

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very fine estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.

Increasing paths in regular trees

We consider a regular

The unscaled paths of branching Brownian motion

n

Intermittency for branching random walk in Pareto environment

-ary tree of height

Branching Brownian Motion: Almost Sure Growth Along Scaled Paths

h

Mixing Time Bounds via Bottleneck Sequences

, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of simple paths from the root to a leaf along vertices with increasing labels. We show that if

Intermittency for branching random walk in heavy tailed environment

\alpha = n/h