It is well known that a random polynomial of degree n with i.i.d. coefficients has most of its roots clustering near the unit circle (at scale 1/n) as n grows large. Moreover, the roots tend to repel from one another. I will present a recent result showing that the minimal separation distance between the roots, when normalized by n^{-5/4}, converges in distribution to a non-trivial limit law.  Based on a joint work with Marcus Michelen.