In 2009 the pseudononymous Satoshi Nakamoto published a short paper on the Internet, together with accompanying software, that proposed an `electronic equivalent of cash’ called Bitcoin. At its most basic level, Bitcoin is a payment system where transactions are verified and stored in a distributed data structure called the blockchain. The Bitcoin system allows electronic transfer of funds without the presence of a trusted third party. It achieves this by making it `very hard work’ to create the payment record, so that it is not computationally-feasible for a malicious player to repudiate a transaction and create a forward history with the transaction deleted.
As its name suggests, the blockchain is comprised of discrete blocks. Blocks are added to the blockchain by `miners’ working across a distributed peer-to-peer network to solve a computationally difficult problem.
One deficiency of the Bitcoin blockchain as a payment system is that it is not fast enough: blocks are mined every ten minutes on average and they can contain information about only 1,500 transactions. In this talk, I shall discuss a couple of different models that can be used to describe a `fast mining blockchain’ in which the mining rate is much higher. It turns out that the data structure that is produced can be described as a `blocktree’ rather than a `blockchain’.
(Joint work with Rhys Bowden, Jan de Gier and Cengiz Gazi.)

The Wells-Riley model has been widely used to estimate airborne infection risk in indoor environments. In this work, we revisit the topic of uncertainty in its parameters, building upon recent probabilistic extensions, see reference [1]. We consider a framework that treats key model parameters as uncertain or random variables. We particularly focus on the duration of exposure, quanta generation rate, ventilation rate and number of infectious individuals. This approach allows us to quantify infection risk not only as the per-capita probability of infection for each susceptible individual but also as the full probability distribution of the number of infections during an indoor interaction.
Our results show that infection risk can vary significantly depending on the distribution and variability of model parameters. In particular, using mean parameter values in the classical Wells‐Riley model can lead to systematic inaccuracies: population-related uncertainties (those describing characteristics of the population) tend to cause overestimations, while environmental uncertainties (i.e. ventilation) can lead to underestimations. We find that these inaccuracies can be exacerbated by high-variance, highly-skewed random parameters.
We also investigated the stochastic dominance dynamics between pairs of simultaneously random, Gamma distributed parameters. We found that when one parameter had notably higher variance and skew over the other (such that the shape value is reduced, but the mean value is kept constant), it would typically dominate the dynamics.
These results highlight the importance of accounting for heterogeneity in population characteristics and varying environmental conditions when assessing airborne infection risk. They show that relying on the classical Wells-Riley approach with average parameter values may not adequately capture infection risk dynamics, particularly in extreme cases (e.g. where ventilation might be considerably lower). By capturing the full variability in key parameters, our probabilistic framework provides a more realistic representation of infection risk and can inform more effective public health interventions.
[1] Edwards, A.J., King, M.-F., Peckham, D., López-García, M., Noakes, C.J. (2023). The Wells–Riley model revisited: Randomness, heterogeneity, and transient behaviours. Risk Analysis 43(9): 1748--1767.

Suppose that red and blue points form independent homogeneous Poisson processes of unit intensity on $R^d$. For a positive (respectively, negative) parameter $\gamma$ we consider red-blue matchings that locally minimise (respectively, maximise) the sum of the $\gamma^th$ powers of edge lengths, subject to locally minimising the number of unmatched points. These matchings were first introduced and studied by Holroyd, Janson and Wastlund, where the $\gamma$-minimal matchings in dimension one were almost completely categorised for all $\gamma \in [\infty, \infty]$. A complete classification in dimensions d > 1 is still unknown. We will present a similiar classification to dimension 1 for $\gamma$-minimal matchings on the strip R x [0,1] where the picture is almost completely the same. We further provide a tight tail bound on the typical matching distance when $\gamma>1$ in the one-colour case.