  The role of the charm quark in the dynamics underlying the \Delta I = 1/2
rule for kaon decays can be understood by studying the dependence of kaon decay
amplitudes on the charm quark mass using an effective \Delta S = 1 weak
Hamiltonian in which the charm is kept as an active degree of freedom. Overlap
fermions are employed in order to avoid renormalization problems, as well as to
allow access to the deep chiral regime. Quenched results in the GIM limit have
shown that a significant part of the enhancement is purely due to low-energy
QCD effects; variance reduction techniques based on low-mode averaging were
instrumental in determining the relevant weak effective lowenergy couplings in
this case. Moving away from the GIM limit requires the computation of diagrams
containing closed quark loops. We report on our progress to employ a
combination of low-mode averaging and stochastic volume sources in order to
control these contributions. Results showing a significant improvement in the
statistical signal are presented.
