  We study asymptotic zero distribution of random Laurent polynomials whose
support are contained in dilates of a fixed integral polytope $P$ as their
degree grow. We consider a large class of probability distributions including
the ones induced from i.i.d. random coefficients whose distribution law has
bounded density with logarithmically decaying tails as well as moderate
measures defined over the projectivized space of Laurent polynomials. We obtain
a quantitative localized version of Bernstein-Kouchnirenko Theorem.
