Research: Probability theory; probabilitistic models and stochastic processes; (random) processes on (random) graphs; probabilistic combinatorics
- Deposition, diffusion, and nucleation on an interval. Submitted for publication. Available on arXiv. Joint with Andrew Wade.
- Markov chains with heavy-tailed increments and asymptotically zero drift, Electron. J. Probab., 24 (2019), no. 62, 1–28. Also available on arXiv. Joint with: Mikhail Menshikov, Dimitri Petritis and Andrew Wade.
- Invariance principle for non-homogeneous random walks, Electron. J. Probab., 24 (2019), no. 48, 1–38. Also available on arXiv. Joint with: Aleksandar Mijatović and Andrew Wade.
- A radial invariance principle for non-homogeneous random walks, Electron. Commun. Probab., 23 (2018), no. 56, 1–11. Also available on arXiv. Joint with: Aleksandar Mijatović and Andrew Wade.
- Anomalous recurrence properties of many-dimensional zero-drift random walks, Advances in Applied Probability, 48A (2016), 99–118. Also available on arXiv. Joint with: Mikhail Menshikov, Aleksandar Mijatović and Andrew Wade.
- New constructions and bounds for Winkler’s hat game, SIAM Journal on Discrete Mathematics, 29 (2015), no. 2, 823–834. Also available on arXiv. Joint with Maximilien Gadouleau.
- Non-homogeneous random walks on a semi-infinite strip, Stochastic Processes and their Applications, 124 (2014), no. 10, 3179–3205. Also available on arXiv. Joint with Andrew Wade.
- Modular decomposition and the Reconstruction Conjecture, Journal of Combinatorics, 3 (2012), no. 1, 123–134. Also available on arXiv. Joint with: Robert Brignall and Robert Waters.
- The simple harmonic urn, Annals of Probability 39 (2011), no. 6, 2119–2177. Also available on arXiv. Joint with: Edward Crane, Stanislav Volkov, Andrew Wade and Robert Waters.
- Continuum limits for classical sequential growth models, Random Structures and Algorithms 36 (2010), no. 2, 218–250. Preprint available. Joint with Graham Brightwell.
- On a universal best choice algorithm for partially ordered sets, Random Structures and Algorithms 32 (2008), no. 3, 263–273. Preprint available. Joint with: Małgorzata Kuchta, Michał Morayne and Jarosław Niemiec.
- Embeddings and other mappings of rooted trees into complete trees, Order 22 (2005), no. 3, 257–288. Preprint available.
- The random binary growth model, Random Structures and Algorithms 27 (2005), no. 4, 520–552. Preprint available (note the change in title).
Other research interests:
- Approximation and mixing times in the ferromagnetic Potts model, research with Magnus Bordewich (originally funded by his EPSRC grant). The project was focussed on Markov chain Monte Carlo techniques, the mixing time of Glauber dynamics and the approximability of the partition function of the Potts model. One topic we investigated was the negative correlation of forests, and whether this property could be used to create an efficient sampling/counting algorithm for the associated partition function.
- Antichains in the k-dimensional random order, collaborative research project with Ed Crane. We held a related workshop in July 2011, funded by the Bristol Mathematics department’s GRASP initiative, attended by Cedric Boutillier, Graham Brightwell, Malwina Luczak and Peter Winkler.
- Subcritically indecomposable graphs and reconstruction, collaborative research with Robert Brignall. We’re investigating the structure of indecomposable graphs (where indecomposable means with respect to modular decomposition) that have the property that very few of the vertex-deleted subgraphs (subgraphs obtained by deleting a single vertex) are indecomposable.
My thesis, Random Structures for Partially Ordered Sets, comprises two parts: in the first I studied random models that produce partial orders sequentially; in the second I studied a correlation of embeddings of rooted trees. The main results were subsequently published in my first two papers, and one section was expanded on in my “Continuum limits…” paper with Graham Brightwell, my PhD supervisor.