Random walk on a perturbation of the infinitely-fast mixing interchange process

We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $\gamma>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $\gamma$ large enough. This statement is thus a perturbation of the case $\gamma =+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i)$ We deal with any dimension $d\geq 1$; $(ii)$ Each particle of the interchange process carries a transition vector chosen according to an arbitrary law $\mu$; $(iii)$ We show that $X_t/t$ is not only in the same direction of the annealed drift, but that it is also close to it.