## CROYDON, David

**CROYDON, David**

My research is based in probability theory, and this year incorporated two main areas, as introduced below.

[Scaling limits for random walks on random graphs]

In describing properties of disordered media, physicists have long been interested in the behaviour of random walks on random graphs that arise in statistical mechanics, such as critical percolation clusters and uniform spanning trees (USTs). Random walks on random graphs are also of interest to computer scientists in studies of complex networks. The models proposed to understand these systems are often simple to define mathematically, but nonetheless can be challenging to analyse. Indeed, many of the canonical examples exhibit large-scale fractal behaviour, which mean it is often a challenge to describe their geometrical properties, let alone that of the associated random walks. In recent years, however, the deep connections between electrical networks and stochastic processes have been advanced so that tackling some of the key examples of random walks on random graphs is now within reach. For instance, for certain classes of fractal-like random graphs, together with my collaborators I have previously shown that the associated random walks can be rescaled to yield a diffusion on the limiting space, where the latter process is described via a so-called ‘resistance form’ [1,2]. (Cf. The convergence of random walks on the integer lattice to Brownian motion.) More recently,
I have applied such an argument and related techniques in order to:

・detail the trapping behavior of biased random walk on the trace of biased random walk [3];

・derive fine properties of the random walk on the two-dimensional UST and its scaling limit [4];

・establish a scaling limit for the random walk on the three-dimensional
UST [5].

In the coming year, together with completing these projects, I envisage working on a variety of further problems in this area.

[A probabilistic approach to the box-ball system]

The box-ball system (BBS) is an interacting particle system introduced in the 1990s by physicists Takahashi and Satsuma as a model to understand solitons, that is, travelling waves. (In particular, it is connected with the Korteweg-de Vries (KdV) equation, which describes shallow water waves.) The BBS is described as follows. Initially, each site of the integer lattice $\mathbb{Z}$ contains a particle or is vacant. (For simplicity at this point, suppose there are only a finite number of particles.) The system then evolves by means of a `carrier', which moves along the integers from left to right (negative to positive). When the carrier sees a ball it picks it up, and when it sees a vacant site it puts a ball down (unless it is not carrying any already, in which case it does nothing). To date, much of the interest in the BBS has come from applied mathematicians/theoretical physicists, who have established many beautiful combinatorial properties of the BBS. What has been less studied are the probabilistic properties of the BBS resulting from a random initial starting configuration, which is a natural starting point. Indeed, one simple example might be to ask if each site contains a particle with probability $p\in (0, 1)$, independently for each site, is it possible to define the action of the carrier, and, if so, how does the system evolve? Moreover, what is the dependence on the parameter $p$ in the model? Together with collaborators, I derived detailed distributional properties of the asymptotic particle flow for this random initial configuration [6]. Moreover, we gave criteria for the invariance under the dynamics of probability measures on configurations, which are satisfied by the aforementioned example. (Other results will soon appear in [7,8,9].) Key to these results is a characterisation of the dynamics of the system in terms of Pitman's famous path transformation (which was used to relate Brownian motion and a Bessel process), and it is an ongoing project to demonstrate that a similar viewpoint is applicable to other integrable systems related to the KdV equation (and also the Toda lattice). Moreover, we anticipate the study will lead to new results for stochastic processes.

- D. A. Croydon, B. M. Hambly and T. Kumagai, Time-changes of stochastic processes associated with resistance forms, Electron. J. Probab. 22 (2017), paper no. 82, 1--41.
- D. A. Croydon, Scaling limits of stochastic processes associated with resistance forms, Ann. Inst. H. Poincar\'{e} Probab. Statist. 54 (2018), 1939--1968.
- D. A. Croydon and M. P. Holmes, Biased random walk on the trace of biased random walk on the trace of ..., preprint appears at arXiv:1901.04673, 2018.
- 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, 2019.
- O. Angel, D. A. Croydon, S. Hernandez-Torresz, and D. Shiraishi, Scaling limit of the three-dimensional uniform spanning tree and the associated random walk, forthcoming, 2019.
- D. A. Croydon, T. Kato, M. Sasada, and S. Tsujimoto, Dynamics of the box-ball system with random initial conditions via Pitman's transformation, preprint appears at arXiv:1806.02147, 2018.
- D. A. Croydon, M. Sasada, and S. Tsujimoto, Dynamics of the ultra-discrete Toda lattice via Pitman's transformation, forthcoming, 2019.
- D. A. Croydon and M. Sasada, Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures, forthcoming, 2019.
- D. A. Croydon and M. Sasada, Duality between box-ball systems of finite box and/or carrier capacity, forthcoming, 2019.