In the first part of the talk, we recall the concept of "amoebas" 

associated to Laurent polynomials and their counterpart in tropical geometry, and review some facts about the space of compatible Kähler metrics on a smooth compact toric manifold. Then we show how to obtain a compact variant of the amoebas in this context by studying the behaviour of complex hypersurfaces along infinite geodesics in the space of metrics. (Based on joint work with C.Florentino, J.M.Mourão, and J.P.Nunes)