  We study the evolution of a random walker on a conservative dynamic random
environment composed of independent particles performing simple symmetric
random walks, generalizing results of [16] to higher dimensions and more
general transition kernels without the assumption of uniform ellipticity or
nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers,
a functional central limit theorem and large deviation estimates for the
position of the random walker under the annealed law in a high density regime.
The main obstacle is the intrinsic lack of monotonicity in higher-dimensional,
non-nearest neighbour settings. Here we develop more general renormalization
and renewal schemes that allow us to overcome this issue. As a second
application of our methods, we provide an alternative proof of the ballistic
behaviour of the front of (the discrete-time version of) the infection model
introduced in [23].
