A phase diagram showing the (complex) roots of a polynomial equation that arises in the study of the spectral asymptotics of a random fractal, as studied in [23].

The range of a random walk on the two-dimensional uniform spanning tree, as studied in [21] (this figure was produced by Sunil Chhita). Related to this, I would strongly recomend a view of the following animations: Wilson's algorithm and Maze tree produced by Mike Bostock.

The local times of a random walk on a Sierpinski gasket graph, as studied in [18].

An example of random media with a boundary, the random walks upon which are studied in [17].

The linked video provides a visualisation of the localisation result for the biased random walk of [14].

Above is a link to an animation showing the convergence of the transition density of a random walk on a critical Galton-Watson tree. The length of time this takes is investigated in [12].

This is a section of the range of a 3-dimensional random walk. Some results for the random walk on higher dimensional versions of this object are proved in [9].

This is a realisation of a critical Galton-Watson tree. A scaling limit for the random walk on this graph is obtained in [7].

This is a random self-similar fractal tree, as studied in [2].

Dr David Croydon - Research

Dr David Croydon - Research

External profiles - Research students - Research articles - Errata - Slides - Probability journals

External profiles

External profiles of mine include: Google Scholar, KAKEN, ResearchGate, researchmap.

Research students (as first supervisor)

Adam Bowditch (graduated 2017)

George Andriopoulos

Research articles

Preprint versions of the following articles are available on arXiv.

28. Biased random walk on the trace of biased random walk on the trace of... (with M. P. Holmes), preprint (2019).

27. Dynamics of the box-ball system with random initial conditions via Pitman's transformation (with T. Kato, M. Sasada and S. Tsujimoto), preprint (2018).

26. Heat kernel estimates for FIN processes associated with resistance forms (with B. M. Hambly and T. Kumagai), accepted for publication in Stochastic Processes and their Applications (2018).

25. Scaling limits of stochastic processes associated with resistance forms, Annales de l'institut Henri Poincare (B) Probabilites et Statistiques 54 (2018), no. 4, 1939-1968.

24. Time-changes of stochastic processes associated with resistance forms (with B. M. Hambly and T. Kumagai), Electronic Journal of Probability 22 (2017), paper no. 82, 1-41.

23. Central limit theorems for the spectra of classes of random fractals (with P. H. A. Charmoy and B. M. Hambly), Transactions of the American Mathematical Society 369 (2017), 8967-9013.

22. Quenched localisation in the Bouchaud trap model with slowly varying traps (with S. Muirhead), Probability Theory and Related Fields 168 (2017), no. 1-2, 269-315.

21. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree (with M. T. Barlow and T. Kumagai), Annals of Probability 45 (2017), no. 1, 4-55.

20. An introduction to stochastic processes associated with resistance forms and their scaling limits, RIMS Kokyuroku 2030 (2017), 1-8.

19. Quenched localisation in the Bouchaud trap model with regularly varying traps (with S. Muirhead), RIMS Kokyuroku Bessatsu B59 (2016), 305-320.

18. Moduli of continuity of local times of random walks on graphs in terms of the resistance metric, Transactions of the London Mathematical Society 2 (2015), no. 1, 57-79.

17. Quenched invariance principles for random walks and elliptic diffusions in random media with boundary (with Z.-Q. Chen and T. Kumagai), Annals of Probability 43 (2015), no. 4, 1594-1642.

16. Functional limit theorems for the Bouchaud trap model with slowly varying traps (with S. Muirhead), Stochastic Processes and their Applications 125 (2015), no. 5, 1980-2009.

15. Slow movement of a random walk on the range of a random walk in the presence of an external field, Probability Theory and Related Fields 157 (2013), no. 3, 515-534.

14. Biased random walk on critical Galton-Watson trees conditioned to survive (with A. Fribergh and T. Kumagai), Probability Theory and Related Fields 157 (2013), no. 1, 453-507.

13. Scaling limit for the random walk on the largest connected component of the critical random graph, Publications of the Research Institute for Mathematical Sciences 48 (2012), no. 2, 279-338.

12. Convergence of mixing times for sequences of random walks on finite graphs (with B. M. Hambly and T. Kumagai), Electronic Journal of Probability 17 (2012), paper no. 3.

11. Spectral asymptotics for stable trees (with B. M. Hambly), Electronic Journal of Probability 15 (2010), 1772-1801.

10. Scaling limits for simple random walks on random ordered graph trees, Advances in Applied Probability 42 (2010), no. 2, 528-558.

9. Random walk on the range of random walk, Journal of Statistical Physics 136 (2009), no. 2, 349-372.

8. Hausdorff measure of arcs and Brownian motion on Brownian spatial trees, Annals of Probability 37 (2009), no. 3, 946-978.

7. Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree, Annales de l'institut Henri Poincare (B) Probabilites et Statistiques 44 (2008), no. 6, 987–1019.

6. Local limit theorems for sequences of simple random walks on graphs (with B. M. Hambly), Potential Analysis 29 (2008), no. 4, 351-389.

5. Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive (with T. Kumagai), Electronic Journal of Probability 13 (2008) 1419-1441.

4. Self-similarity and spectral asymptotics for the continuum random tree (with B. M. Hambly), Stochastic Processes and their Applications 118 (2008), no. 5, 730-754.

3. Volume growth and heat kernel estimates for the continuum random tree, Probability Theory and Related Fields 140 (2008), no. 1-2, 207-238.

2. The Hausdorff dimension of a class of random self-similar fractal trees, Advances in Applied Probability 39 (2007), no. 3, 708-730.

1. Heat kernel fluctuations for a resistance form with non-uniform volume growth, Proceedings of the London Mathematical Society 94 (2007), no. 3, 672-694.

0. Random fractal dendrites. (2006). DPhil thesis. University of Oxford.

00. Markov chains with a large finite state space. Part III essay. University of Cambridge.

Errata

(At least some of) the errata and typos in my published articles that are known to me can be found here.

Slides

An introduction to the scaling limits of random walks via the resistance metric, minicourse given whilst visiting University of Melbourne, School of Mathematics and Statistics, August 2018. NB. These slides cover the first and fourth lectures.
Scaling random walks on critical random trees and graphs, minicourse given whilst visiting Kyoto University, Research Institute for Mathematical Sciences, October 2015. NB. These slides cover the first two lectures. The third lecture was based on material from the article Moduli of continuity of local times of random walks on graphs in terms of the resistance metric listed above.
Scaling limits of random walks on critical random trees and graphs, minicourse given at the meeting Young European Probabilists XII, Eurandom, 23-27 March 2015.

Probability journals