  We present a new study of the high latitude galactic contributions to the
millimeter sky, based on an analysis of the WMAP data combined with several
templates of dust emission (DIRBE/COBE and FIRAS/COBE) and gas tracers (HI and
Halpha). To study the IR to millimeter properties of the diffuse sky at high
galactic latitude, we concentrate on the emission correlated with the HI gas.
We compute the emission spectrum of the dust/free-free/synchrotron components
associated with HI gas from low to large column densities. A significant
residual WMAP emission over the free-free, synchrotron and the dust
contribution is found from 3.2 to 9.1 mm. We show that this residual WMAP
emission (normalised to 10$^{20}$ atoms/cm$^2$) (1) exhibits a constant
spectrum from 3.2 to 9.1 mm and (2) significantly decreases in amplitude when
N$_{HI}$ increases, contrary to the HI-normalised far-infrared emission which
stays rather constant. It is thus very likely that the residual WMAP emission
is not associated with the Large Grain dust component. The decrease in
amplitude with increasing opacity ressembles in fact to the decrease of the
transiently heated dust grain emission observed in dense interstellar clouds.
This is supported by an observed decrease of the HI-normalised 60 microns
emission with HI column densities. Although this result should be interpreted
with care due to zodiacal residual contamination at 60 microns, it suggests
that the WMAP excess emission is associated with the small transiently heated
dust particles. On the possible models of this so-called ``anomalous microwave
emission'' linked to the small dust particles are the spinning dust and the
excess millimeter emission of the small grains, due to the cold temperatures
they can reach between two successive impacts with photons.
