We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{k=0}^\infty a_k z^k$, for $a_k$ independent, with finite first moment, and anti-concentrated. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a (strong) natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ when $n\to \infty$ and $m$ stays fixed, and we provide almost sure rates of convergence of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_k)$ with not necessarily independent coordinates. This is joint work with S. Dostoglou (University of Missouri).