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

Scaling limits of Maximal Entropy Random Walks on a non-negative integer lattice perturbed at the origin