# Dr David Croydon - Research

**External profiles - Research students - Research articles - Collaborators - Errata - Talks - Probability journals**

**External profiles**

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

**Research students (as first supervisor)**

Adam Bowditch (graduated 2017)

George Andriopoulos (graduated 2019)

Eleanor Archer (graduated 2020)

**Research articles**

*Preprint versions of the following articles are available on arXiv.*

40. *General solutions for KdV- and Toda-type discrete integrable systems based on path encodings* (with M. Sasada and S. Tsujimoto), preprint (2020).

39. *Anomalous scaling regime for one-dimensional Mott variable-range hopping* (with R. Fukushima and S. Junk), preprint (2020).

38. *Detailed balance and invariant measures for systems of locally-defined dynamics* (with M. Sasada), preprint (2020).

37. *Scaling limits of the three-dimensional uniform spanning tree and associated random walk* (with O. Angel, S. Hernandez-Torres and D. Shiraishi), preprint (2020).

36. *The number of spanning clusters of the uniform spanning tree in three dimensions* (with O. Angel, S. Hernandez-Torres and D. Shiraishi), preprint (2020).

35. *Discrete integrable systems and Pitman's transformation* (with M. Sasada), accepted for publication in Advanced Studies in Pure Mathematics (2020).

34. *Generalized hydrodynamic limit for the box-ball system* (with M. Sasada), accepted for publication in Communications in Mathematical Physics (2020).

33. *Dynamics of the box-ball system with random initial conditions via Pitman's transformation* (with T. Kato,
M. Sasada and S. Tsujimoto), accepted for publication in the Memoirs of the American Mathematical Society (2018).

32. *The random conductance model with heavy tails on nested fractal graphs*, accepted for publication in
Fractal Geometry and Stochastics 6 (2019).

31. *Biased random walk on the trace of biased random walk on the trace of...* (with M. P. Holmes), Communications in Mathematical Physics 375 (2020), 1341-1372.

30. *Duality between box-ball systems of finite box and/or carrier capacity* (with
M. Sasada), RIMS Kokyuroku Bessatsu B79 (2020), 63-107.

29. *Dynamics of the ultra-discrete Toda lattice via Pitman's transformation* (with M. Sasada and S. Tsujimoto), RIMS Kokyuroku Bessatsu B78 (2020), 235-250.

28. *An introduction to the trapping experienced by biased random walk on the trace of biased random walk*, RIMS Kokyuroku 2116 (2019), paper no. 28.

27. *Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures* (with M. Sasada), Journal of Mathematical Physics 60, 083301 (2019).

26. *Heat kernel estimates for FIN processes associated with resistance forms* (with B. M. Hambly and T. Kumagai), Stochastic Processes and their Applications 129 (2019), no. 9, 2991-3017.

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), paper no. 1.

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 con**nected component of the critical rando**m 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.

**Collaborators**

O. Angel, M. T. Barlow, P. H. A. Charmoy, Z.-Q. Chen, A. Fribergh, R. Fukushima, B. M. Hambly, S. Hernandez-Torres, M. P. Holmes, S. Junk, T. Kato, T. Kumagai, S. Muirhead, M. Sasada, D. Shiraishi, S. Tsujimoto.

**Errata**

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

**Talks**

*Online videos of talks I have given:*

*Scaling limits of the two- and three-dimensional uniform spanning trees*, Oxford Discrete Mathematics and Probability Seminar, 24th October 2020.*Invariant measures for KdV and Toda-type discrete integrable systems*, Online Open Probability School, 12th June 2020.

*Online videos of talks by some of my collaborators (on projects in which I played a role):*

- S. Hernandez-Torres:
*Scaling limits of uniform spanning trees in three dimensions*, UBC Probability Seminar, 23rd September 2020, and also*Three-dimensional uniform spanning trees*, Joint Israeli Probability Seminar, 19th January 2021. - S. Junk:
*Subdiffusive scaling regime for one-dimensional Mott variable range hopping*, Bernoulli-IMS One World Symposium 2020, 24th-28th August 2020.

*Slides from lecture series I have given:*

*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**

- Advances in Applied Probability
- ALEA
- Annales de l'Institut Henri Poincare (B) Probabilites et Statistiques
- Annals of Applied Probability
- Annals of Probability
- ArXiv Probability
- Bernoulli
- Brazilian Journal of Probability and Statistics
- Electronic Communications in Probability
- Electronic Journal of Probability
- Journal of Applied Probability
- Journal of Theoretical Probability
- Probability Surveys
- Probability Theory and Related Fields
- Statistics and Probability Letters
- Stochastic Analysis and Applications
- Stochastic Processes and their Applications