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We consider the metric space of all toric K\"ahler metrics on a compact toricmanifold; when "looking at it from infinity" (following Gromov), we obtain thetangent cone at infinity, which is parametrized by equivalence classes ofcomplete geodesics. In the present paper, we study the associated limit for thefamily of metrics on the toric variety, its quantization, and degeneration ofgeneric divisors.  The limits of the corresponding K\"ahler polarizations become degeneratealong the Lagrangian fibration defined by the moment map. This allows us tointerpolate continuously between geometric quantizations in the holomorphic andreal polarizations and show that the monomial holomorphic sections of theprequantum bundle converge to Dirac delta distributions supported onBohr-Sommerfeld fibers.  In the second part, we use these families of toric metric degenerations tostudy the limit of compact hypersurface amoebas and show that in Legendretransformed variables they are described by tropical amoebas. We believe thatour approach gives a different, complementary, perspective on the relationbetween complex algebraic geometry and tropical geometry.

Toric K\"ahler metrics seen from infinity, quantization and compact tropical amoebas