Antimatter cosmic rays from dark matter annihilation: First results from an N-body experiment

[Abridged]. We take advantage of the galaxy-like 3D dark matter map extracted from the HORIZON Project results to calculate the positron and antiproton fluxes from dark matter annihilation, in a model-independent approach as well as for dark matter particle benchmarks relevant at the LHC scale (from supersymmetric and extra-dimensional theories). Such a study is dedicated to a better estimate of the theoretical uncertainties affecting predictions, while the PAMELA and GLAST satellites are currently taking data which will soon provide better observational constraints. We discuss the predictions of the antiproton and positron fluxes, and of the positron fraction as well, as compared to the current data. We finally discuss the limits of the Nbody framework in describing the dark matter halo of our Galaxy.