Full Galois groups of polynomials with slowly growing coefficients

Choose a polynomial f uniformly at random from the set of all monic polynomials of degree n with integer coefficients in the box [-L,L]^n. The main result of the paper asserts that if L=L(n) grows to infinity, then the Galois group of f is the full symmetric group, asymptotically almost surely, as n→∞. When L grows rapidly to infinity, say L>n^7, this theorem follows from a result of Gallagher. When L is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if L< 17, it is conditional on the general Riemann hypothesis). Hence the most interesting case of the theorem is when L grows slowly to infinity. Our method works for more general independent coefficients.