  We investigate random walks on the integer lattice perturbed at the origin
which maximize the entropy along the path or equivalently the entropy rate.
Compared to usual simple random walks which maximize the entropy locally, those
are a complete paradigm shift. For finite graph, they have been introduced as
such by Physicists in [BDLW09] and they enjoy strong localization properties in
heterogeneous environments. We first give an extended definition of such random
walks when the graph is infinite. They are not always uniquely defined contrary
to the finite situation. Then we introduce our model and we show there is a
phase transition phenomenon according with the magnitude of the perturbation.
We obtain either Bessel-like or asymmetric random walks and we study there
scaling limits. In the subcritical situation, we get the classical
three-dimensional Bessel process whereas in the supercritical one we get a
recurrent reflected drifted Brownian motion. We produce a unify proof relying
on the equivalence between submartingale problems and reflected stochastic
differential equations obtained in [KR17]. Finally, we remark that applying our
results to a couple of particles subject to the famous Pauli exclusion
principle, we can recover the electrostatic force assuming only the maximal
entropy principle. Also, we briefly explain how to retrieve those results in
the macroscopic situation without go through the microscopic one by using the
Kullback-Leibler divergence and the Girsanov theorem.
