Abstract. 
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. 

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, tunneling 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 a  
"Fourier" uncertainty of the drop's motion. 

Finally, in another set of experiments we demonstate that when the walker 
has an orbiting motion, the possible radii of this orbit are discrete. 

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.