  We study the role of small-size instantons in the determination of the
topological susceptibility of the 2-d $O(3) \: \sigma $ model on the lattice.
In particular, we analyze how they affect the non-perturbative determination,
by Monte Carlo techniques, of the renormalizations on the lattice. As a result,
we obtain a high-precision non-perturbative determination of the mixing with
the unity operator, finding good agreement with perturbative computations. We
also obtain the size distribution of instantons in the physical vacuum up to
very small values of the size in physical units, without observing any
ultraviolet cut-off. Moreover, we show by analytical calculation that the
mixing of the topological susceptibility with the action density is a
negligible part of the whole non-perturbative signal.
