New Phenomena For Random Walks In Dynamic Environments

This proposal will be entirely hold in probability theory, as a domain of pure mathematics. It is, however, related to classical models in statistical physics. How to predict the trajectory of particles in a fluid, the movements of an animal searching for food in a changing environment, or the variation of stock prices in an fluctuating market? Modelling dynamic systems straddle several branches of mathematics. Because of the high complexity of real-world instances, using randomness is crucial to make computations tractable, and to focus on essential properties rather than contingent details. In probability theory, random walks are a generic class of models describing a sequence of random steps. Studies often focus on random walks in static environments, and there is today a rich literature since seminal papers by Kestlen, Kozlov, Sinai and Spitzer in the 1970s. However, most of the classical techniques break down when making the environment dynamic - which is problematic if one has realistic applications in mind, but also poses challenging questions from a purely mathematical perspective. In the past two decades, probabilists have intensively investigated random walks in dynamic environments, notably when the later are particle systems. A popular instance is the exclusion process (in which particles move independently under the condition of not superposing, a natural model for a fluid). However, little is still known due to heavy space-time correlations. This makes such models very hard to handle, while giving hope to observe intriguing phenomena such as non-diffusive fluctuations. The aim of this proposal is to deliver pioneering results on random walks on the exclusion process, and to develop powerful tools to understand similar environments. The model is as follows: on top of the environment, which is a one-dimensional symmetric exclusion process, a random walker moves with a bias to the right (resp. to the left) when sitting on a particle (resp. an empty site). I will take advantage of the novel techniques developed in a recent work, where my coauthors and myself show that increasing the density of particles leads to a positive speed increase for the walker. My main objectives are: 1) Investigating the critical density where the speed could be zero: is the speed defined and continuous? What are the fluctuations of the random walker? 2) Adding a current flow: the particles themselves now have a drift. Does the random walker still have a speed? The current complicates the space-time correlations, but our techniques can bypass that. 3) Cooling of the environment: if the activity rate of the particles goes to zero with time, do we start seeing effects from static environments, where the random walker can spend most of its time into traps which now take long to dissolve? 4) Higher dimensions: in dimensions above one, the loss of monotonicity makes the study especially difficult. However, I believe that in dimension at least five, particles get ‘dissipated’ quickly after meeting the walker, so that one can control the correlations to prove a law of large numbers for the walker’s displacement. The findings will hugely advance the understanding of random walks on various dynamic environments. They will also benefit neighbouring areas in probability theory such as percolation or mixing times, where the techniques developed in the proposal will apply. Finally, the results will also deliver valuable insights on conjectures from statistical physics.