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We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then theOseledets splitting along almost every orbit is either trivial or partiallyhyperbolic. In addition, if $f$ is not Anosov then all the exponents in thecenter bundle vanish. This establishes in full a result announced by R.Ma\~{n}\'{e} in the ICM 1983. The main technical novelty is a probabilisticmethod for the construction of perturbations, using random walks.

$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents