LLN and CLT in high dimensions for the random walk on the symmetric exclusion process

Rémy Poudevigne
LPSM

Date(s) : 14/10/2025 iCal
14h30 - 15h30

The random walk on the exclusion process
is an example of random walk in dynamic environment.
In this specific model,
the law of the movement of the walker depends
on whether its current position is occupied
by a particle or not.
Furthermore, the position of the particles
is given by a symmetric exclusion process (on $\Z^d$).
While this dynamic is quite simple,
it has the inconvenient of being both
conservative and slowly mixing
which will cause strong correlations
in the position of the particles.
In dimension 1, thanks to a monotonicity property,
the model is somewhat well understood
but in higher dimensions
only perturbative results are known.
For a wide range of parameters,
we show that the walker satisfies
a law of large number in dimensions 5 and higher
and a CLT in dimensions 9 and higher.
For this we look at the law of the environment
knowing the trajectory of the walker
and show that it is not too far
from the initial measure, for any trajectory,
and conclude with standard tools for Markov chains.
Joint work with Guillaume Conchon-Kerjan and Daniel Kious.