  Motivated by the Koml\'os conjecture in combinatorial discrepancy, we study
the discrepancy of random matrices with $m$ rows and $n$ independent columns
drawn from a bounded lattice random variable. It is known that for $n$ tending
to infinity and $m$ fixed, with high probability the $\ell_\infty$-discrepancy
is at most twice the $\ell_\infty$-covering radius of the integer span of the
support of the random variable. However, the easy argument for the above fact
gives no concrete bounds on the failure probability in terms of $n$. We prove
that the failure probability is inverse polynomial in $m, n$ and some
well-motivated parameters of the random variable. We also obtain the analogous
bounds for the discrepancy in arbitrary norms.
  We apply these results to two random models of interest. For random
$t$-sparse matrices, i.e. uniformly random matrices with $t$ ones and $m-t$
zeroes in each column, we show that the $\ell_\infty$-discrepancy is at most
$2$ with probability $1 - O(\sqrt{ \log n/n})$ for $n = \Omega(m^3 \log^2 m)$.
This improves on a bound proved by Ezra and Lovett (Ezra and Lovett,
Approx+Random, 2016) showing that the same is true for $n$ at least $m^t$. For
matrices with random unit vector columns, we show that the
$\ell_\infty$-discrepancy is $O(\exp(\sqrt{ n/m^3}))$ with probability $1 -
O(\sqrt{ \log n/n})$ for $n = \Omega(m^3 \log^2 m).$ Our approach, in the
spirit of Kuperberg, Lovett and Peled (G. Kuperberg, S. Lovett and R. Peled,
STOC 2012), uses Fourier analysis to prove that for $m \times n$ matrices $M$
with i.i.d. columns, and $m$ sufficiently large, the distribution of $My$ for
random $y \in \{-1,1\}^n$ obeys a local limit theorem.
