Random walks in dynamic random environments
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
Random motions in random media have been intensively studied for over forty years and many interesting features of these models have been discovered. The aim is to understand the motion of a particle in a turbulent media.
Most of the work has been focused on the case where the particle evolves in a static random environment, for which slow-downs and trapping phenomena have been proved.
More recently, mathematicians and physicists have been interested in the case of dynamic random environments, where the media can fluctuate with time. Random walks on the exclusion process is probably the canonical model for the field. Much less is known on this model, but exciting conjectures and questions have been made.
Some of the most challenging questions concern the possibility of super-diffusive regimes, and the existence of effective traps along the trajectory of the walk.
In this project, we aim at, on one hand, adapt the techniques recently developped in one dimension to the multi-dimensional model and, on the other hand, understand the presence or absence of atypical behaviors for this model.
Most of the work has been focused on the case where the particle evolves in a static random environment, for which slow-downs and trapping phenomena have been proved.
More recently, mathematicians and physicists have been interested in the case of dynamic random environments, where the media can fluctuate with time. Random walks on the exclusion process is probably the canonical model for the field. Much less is known on this model, but exciting conjectures and questions have been made.
Some of the most challenging questions concern the possibility of super-diffusive regimes, and the existence of effective traps along the trajectory of the walk.
In this project, we aim at, on one hand, adapt the techniques recently developped in one dimension to the multi-dimensional model and, on the other hand, understand the presence or absence of atypical behaviors for this model.
Organisations
- University of Bath (Lead Research Organisation)
- Leiden University (Collaboration)
- University of Bath (Collaboration)
- Instituto Nacional de Matemática Pura e Aplicada (Collaboration, Project Partner)
- Federal University of Minas Gerais (Collaboration)
- IMPERIAL COLLEGE LONDON (Collaboration)
- Universidade Federal de Minas Gerais (Project Partner)
- University of Paris (Project Partner)
People |
ORCID iD |
| Daniel Kious (Principal Investigator) |
Publications
Holmes M
(2022)
Coexistence of lazy frogs on
in Journal of Applied Probability
Kious D
(2022)
The trace-reinforced ants process does not find shortest paths
in Journal de l'École polytechnique - Mathématiques
| Description | With my research assistant, paid on this grant, we have two main worked that have been achieved. One is on the arXiv, and the other one is in the final writing phase. First, we proved a challenging result about scaling limits of large critical random trees in random environments, extending the current literature on the model. Second, we managed to use recent results on correlated percolation in order to prove strict monotonicity results for random walks in dynamic random environments. This second paper will be submitted to a leading journal in Mathematics and I consider it as the major scientific achievement on this grant. Moreover, my reasearch assistant, who was trained in France, moved to the UK for this grant and just accepted a Lectureship at King's College London. I believe this is the proof that this project has had an impact on our community and is relevant is the broader scientific picture in the UK since it brought talented researcher from abroad who will stay in the UK in the longer term. Finally, I am well into the organisation of a workshop that will take place in Bath in June. The reaction from the community has been excellent and we have currently over 70 participants registered, including world leading researchers. This event will have certainly an impact and will participate to the strong scientific reputation of the UK and of the University of Bath. |
| Exploitation Route | The research papers will be useful to the experts in Probability. I believe that our conference in June will create new collaboration among the participants. As said above, my research assistant, trained abroad, just accepted a Lectureship at King's College London. I created two long term collaborations with this project, and this will definitely have an impact on my future research. |
| Sectors | Other |
| Description | Fluctuation bounds for symmetric random walks on dynamic environments via Russo-Seymour-Welsh |
| Organisation | Federal University of Minas Gerais |
| Country | Brazil |
| Sector | Academic/University |
| PI Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). I contribute by my expertise and research work. |
| Collaborator Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). They contribute by their expertise and research work. |
| Impact | No output yet, but we are in the final phase of writing a paper, which will hopefully be available in the next month. |
| Start Year | 2021 |
| Description | Fluctuation bounds for symmetric random walks on dynamic environments via Russo-Seymour-Welsh |
| Organisation | Instituto Nacional de Matemática Pura e Aplicada |
| Country | Brazil |
| Sector | Public |
| PI Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). I contribute by my expertise and research work. |
| Collaborator Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). They contribute by their expertise and research work. |
| Impact | No output yet, but we are in the final phase of writing a paper, which will hopefully be available in the next month. |
| Start Year | 2021 |
| Description | Fluctuation bounds for symmetric random walks on dynamic environments via Russo-Seymour-Welsh |
| Organisation | Leiden University |
| Country | Netherlands |
| Sector | Academic/University |
| PI Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). I contribute by my expertise and research work. |
| Collaborator Contribution | I work on a paper in collaboration with Augusto Teixeira (IMPA, Brazil), Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). They contribute by their expertise and research work. |
| Impact | No output yet, but we are in the final phase of writing a paper, which will hopefully be available in the next month. |
| Start Year | 2021 |
| Description | Random walks on the exclusion process: a pertubative result in high dimensions |
| Organisation | Federal University of Minas Gerais |
| Country | Brazil |
| Sector | Academic/University |
| PI Contribution | I work in collaboration with Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). I contribute by my expertise and my research work. |
| Collaborator Contribution | I work in collaboration with Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). They contribute by their expertise and my research work. |
| Impact | No output yet, but we are in the process of writing up a paper, which will hopefully be available in the next six months. |
| Start Year | 2021 |
| Description | Random walks on the exclusion process: a pertubative result in high dimensions |
| Organisation | Leiden University |
| Country | Netherlands |
| Sector | Academic/University |
| PI Contribution | I work in collaboration with Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). I contribute by my expertise and my research work. |
| Collaborator Contribution | I work in collaboration with Marcelo Hilario (UFMG, Brazil) and Rangel Baldasso (Leiden). They contribute by their expertise and my research work. |
| Impact | No output yet, but we are in the process of writing up a paper, which will hopefully be available in the next six months. |
| Start Year | 2021 |
| Description | Scaling limit of critical random trees in random environment |
| Organisation | University of Bath |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | I and my research assistant (paid on this grant) are collaborating with Cecile Mailler (University of Bath). We contribute by our expertise and research work. |
| Collaborator Contribution | I and my research assistant (paid on this grant) are collaborating with Cecile Maill (University of Bath). She contributes by her expertise and research work. |
| Impact | The paper is on the arXiv and has been submitted to a leading journal in Probability. |
| Start Year | 2021 |
| Description | Strict monotonicity for random walks in dynamic random environment |
| Organisation | Imperial College London |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | I and my research assistant (paid on this grant) are doing research work with Pierre-Francois Rodriguez at Imperial, London. We contribute by our expertise and work. |
| Collaborator Contribution | I and my research assistant (paid on this grant) are doing research work with Pierre-Francois Rodriguez at Imperial, London. He is contributing through his expertise and research work. |
| Impact | No output yet, but we are in the final writing phase of a paper that will be submitted in a top journal. |
| Start Year | 2021 |