We have shown recently that a droplet bouncing on a vertically vibrated liquid interface can become dynamically coupled to the surface waves it excites. It thus becomes a self-propelled ``walker'', a symbiotic object formed by the droplet and its associated wave. \par Through several experiments we will address one central question. How can a continuous and spatially extended wave have a common dynamics with a localized and discrete droplet? We will show that in all cases (diffraction, interference, tunnelling, etc.) where the wave is split, a single droplet has an apparently random response but that a deterministic behaviour is statistically recovered when the experiment is repeated. The truncation of the wave is thus shown to generate an uncertainty in the drop's motion. \par Finally, in another set of experiments analogous to Landau experiments, we demonstrate that when the walker has an orbiting motion, the possible radii of the orbit are discrete. \par We will show how these properties result from what we call the walker's ``path-memory''. The limits in which these results can be compared to those at quantum scale will be discussed.