My research is based in probability theory. This year, I have studied various aspects of two- and three-dimensional uniform spanning trees, and discrete integrable systems started from random initial conditions.
Two- and three-dimensional uniform spanning trees
The uniform spanning tree (UST) is a natural model in probability theory, with links to problems in electrical potential theory, statistical mechanics and combinatorics, amongst other areas. In the two-dimensional case, I have an ongoing joint project with Martin Barlow (UBC) and Takashi Kumagai (Kyoto) concerning the associated random walk . Moreover, building on the two-dimensional results, in recent work with Omer Angel (UBC), Sarai Hernandez-Torres (UBC) and Daisuke Shiraishi (Kyoto), it was established that the three-dimensional version of the UST has a scaling limit as a metric measure space, and various properties of the associated random walk were obtained [1,2]. I highlight that, in contrast to the two-dimensional case, where remarkable progress has been made over the last decade, statistical mechanics in three dimensions (which is physically the most relevant) is still particularly challenging, as there is no general theory about what kind of scaling limits might arise. In the coming year, I plan to further study (together with my collaborators) properties of the limiting object in three dimensions.
Discrete integrable systems started from random initial conditions
Previously, I studied invariant measures of the box-ball system (BBS), which is a discrete model that is connected to the Korteweg-de Vries (KdV) equation, and also exhibits solitons (i.e.\ solitary wave solutions). In recent work with Makiko Sasada (Tokyo) and Satoshi Tsujimoto (Kyoto), I have extended the framework of the aforementioned research to cover more general discrete integrable systems, including the ultradiscrete and discrete KdV and Toda equations . In each case, we are able to describe the dynamics in terms of a variation of Pitman's path transformation of reflection in the past maximum, which is well-known in the probability community. Moreover, together with Sasada, I have shown how a certain soliton decomposition of BBS configurations can be used to derive generalized hydrodynamic limits of the system . For suitably smooth initial conditions, the limiting behaviour can be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the `effective speeds' of solitons locally. We henceforth plan to explore this result in a more general setting.
- O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraishi, Scaling limit of the three-dimensional uniform spanning tree and the associated random walk, arXiv:2003.09055, 2020.
- O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraishi, The number of spanning clusters of the uniform spanning tree in three dimensions, arXiv:2003.04548, 2020.
- M. T. Barlow, D. A. Croydon, and T. Kumagai, Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree, forthcoming, 2020.
- D. A. Croydon and M. Sasada, Generalized hydrodynamic limit for the box-ball system, arXiv:2003.06526, 2020.
- D. A. Croydon, M. Sasada, and S. Tsujimoto, General solutions for KdV- and Toda-type discrete integrable systems based on path encodings, forthcoming, 2020.