  We introduce the notion of $\epsilon$-irreducibility for arithmetic cycles
meaning that the degree of its analytic part is small compared to the degree of
its irreducible classical part. We will show that for every $\epsilon>0$ any
sufficiently high tensor power of an arithmetically ample hermitian line bundle
can be represented by an $\epsilon$-irreducible arithmetic divisor. Our methods
of proof also allow us to study the distribution of divisors of small sections
of an arithmetically ample hermitian line bundle $\overline{\mathcal{L}}$. We
will prove that for increasing tensor powers $\overline{\mathcal{L}}^{\otimes
n}$ the normalized Dirac measures of these divisors almost always converge to
$c_1(\overline{\mathcal{L}})$ in the weak sense. Using geometry of numbers we
will deduce this result from a distribution result on divisors of random
sections of positive line bundles in complex analysis. As an application, we
will give a new equidistribution result for the zero sets of integer
polynomials. Finally, we will express the arithmetic intersection number of
arithmetically ample hermitian line bundles as a limit of classical geometric
intersection numbers over the finite fibers.
