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.
People |
ORCID iD |
| Daniel Kious (Principal Investigator) |
Publications
Conchon-Kerjan G
(2024)
Scaling limit of critical random trees in random environment
in Electronic Journal of Probability
Conchon-Kerjan G
(2023)
Anatomy of a Gaussian giant: supercritical level-sets of the free field on regular graphs
in Electronic Journal of Probability
Croydon D
(2025)
Aging and sub-aging for one-dimensional random walks amongst random conductances
in Stochastic Processes and their Applications
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
Kious D
(2022)
Finding geodesics on graphs using reinforcement learning
in The Annals of Applied Probability
| Description | With my former research assistant Guillaume Conchon--Kerjan, now lecturer at King's College London and who used to be on this grant, we have two main works that have been achieved. One is published and the other one is on the arXiv and submitted. First, we proved a challenging result about scaling limits of large critical random trees in random environments, extending the current literature on the model (Conchon--Kerjan, Kious and Mailler). G. Conchonc--Kerjan and I are now writing a follow-up paper, which will open a new direction of research on scaling limit of random trees (defining a new limiting object). Second, with G. Conchon--Kerjan (King's College London) and P.F. Rodriguez (Imperial College), 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 has been submitted to a leading journal in Mathematics and I consider it as the major scientific achievement on this grant. I have moreover written another paper, with R. Baldasso (PUC Rio), M. Hilario (UFMG) and A. Teixeira (IMPA) about the fluctations of random walks in dynamic random envrionments, developping a new approach, inspired by Russo-Seymour-Welsh techniques from percolation, in order to tackle this challenging question (which is the main conceptual difficulty remaining in the problem). The final scientific contribution of the project is a paper I wrote with D. Croydon (RIMS Kyoto) and C. Scalio (TU Munich) about aging and sub-aging of random walks among having tailed random conductances. From a different perspective, my reasearch assistant, who was trained in France, moved to the UK for this grant and is now Lectuer 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. We are still working together, and build a longterm scientific relationship with Dr. P.F. Rodriguez at Imperial College. Within the context of this grant, I have organised a conference in June 2023 at the University of Bath. This conference had a highly international profile and we had a full room, with 85 participants, which is a large audience for our field! I received many very positive comments after the conference about its quality and about encouraging the participants to interact together and take advantage of this conference to start new projects. Finally, I have co-organised a second conference in Mexico in January 2024, together with CIMAT, UNAM and the University of Warwick. One of the main speakers, giving a mini-course, was Pierre-Francois Rodriguez who was my collaborator on this award. My research assistant also gave a talk at this conference, on our work together. All of this participates to the international recognition of the project that have be funded. The impact of this award is lasting much later than the duration of the award. I have many projects following up from it and it enabled me to start organising international conferences, with two more currently in the pipeline. |
| Exploitation Route | The research papers will be useful to the experts in Probability. Our conference in June 2023 created new collaborations among the participants. As said above, my research assistant, trained abroad, is now Lecturer at King's College London. He is continuing his work on the topic we have investigated during our work on this award. In the main two topics we have worked on, we have created, for ourselves and for the community, a new line of research which will certainly allow us and other to dig more into several sets of problems and write high level, internationally visible research in the years to come. I created three long term collaborations with this project, and this will definitely have an impact on my future research. |
| Sectors | Other |
| URL | https://sites.google.com/view/danielkious |
| Description | Sharp threshold for the ballisticity of the random walk on the exclusion process |
| Organisation | Imperial College London |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | G. Conchon--Kerjan (now at King's College London, former RA on the grant), P.F. Rodriguez (Imperial) and I wrote a preprint, now submitted. Most of the work was done over the duration of the award. |
| Collaborator Contribution | Both P.F. Rodriguez and G. Conchon--Kerjan are expert on the topic and brought an invaluable expertise to towards the success of this project. While Conchon--Kerjan was RA on this award, we visited many times PF. Rodriguez at Imperial, where we worked together. After the award was over and G. Conchon--Kerjan started as Lecturer at King's College London, we also met there a few times in order to finalise the paper and submit it. |
| Impact | One preprint submitted to a top journal: https://arxiv.org/abs/2409.02096. We are currently working on the subsequent questions and plan to have more papers out of this collaboration. |
| Start Year | 2021 |
| Description | Sharp threshold for the ballisticity of the random walk on the exclusion process |
| Organisation | King's College London |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | G. Conchon--Kerjan (now at King's College London, former RA on the grant), P.F. Rodriguez (Imperial) and I wrote a preprint, now submitted. Most of the work was done over the duration of the award. |
| Collaborator Contribution | Both P.F. Rodriguez and G. Conchon--Kerjan are expert on the topic and brought an invaluable expertise to towards the success of this project. While Conchon--Kerjan was RA on this award, we visited many times PF. Rodriguez at Imperial, where we worked together. After the award was over and G. Conchon--Kerjan started as Lecturer at King's College London, we also met there a few times in order to finalise the paper and submit it. |
| Impact | One preprint submitted to a top journal: https://arxiv.org/abs/2409.02096. We are currently working on the subsequent questions and plan to have more papers out of this collaboration. |
| Start Year | 2021 |