  The stellar initial mass function (IMF) is playing a critical role in the
history of our universe. We propose a theory that is based solely on local
processes, namely the dust opacity limit, the tidal forces and the properties
of the collapsing gas envelope. The idea is that the final mass of the central
object is determined by the location of the nearest fragments, which accrete
the gas located further away, preventing it to fall onto the central object. To
estimate the relevant statistics in the neighbourhood of an accreting
protostar, we perform high resolution numerical simulations. We also use these
simulations to further test the idea that fragmentation in the vicinity of an
existing protostar is determinant in setting the peak of the stellar mass
spectrum. We develop an analytical model, which is based on a statistical
counting of the turbulent density fluctuations, generated during the collapse,
that are at least equal to the mass of the first hydrostatic core, and
sufficiently important to supersede tidal and pressure forces to be
self-gravitating. The analytical mass function presents a peak located at
roughly 10 times the mass of the first hydrostatic core in good agreement with
the numerical simulations. Since the physical processes involved are all local,
i.e. occurs at scales of a few 100 AU or below, and do not depend on the gas
distribution at large scale and global properties such as the mean Jeans mass,
the mass spectrum is expected to be relatively universal.
