We report on the first results of the POLAMI program, a simultaneous 3.5 and 1.3mm full-Stokes-polarisation monitoring of a sample of 36 of the brightest active galactic nuclei in the northern sky with the IRAM 30m Telescope. Through a systematic statistical study of data taken from October 2006 (from December 2009 for the case of the 1.3mm observations) to August 2014, we characterise the variability of the total flux density and linear polarisation. We find that all sources in the sample are highly variable in total flux density at both 3.5 and 1.3mm, as well as in spectral index, that is found to be optically thin in general. The total flux-density variability at 1.3mm is found, in general, to be faster, and to have larger amplitude and flatter PSD slopes than 3.5mm. The polarisation degree is on average larger at 1.3mm than at 3.5mm, by a factor of 2.6. The variability of linear polarisation degree is faster and has higher fractional amplitude than for total flux density, with the typical time scales during prominent polarisation peaks being significantly faster at 1.3mm than at 3.5mm. The polarisation angle at both 3.5 and 1.3mm is highly variable. Most of the sources show one or two excursions of >180 deg. on time scales from a few weeks to about a year during the course of our observations. The 3.5 and 1.3mm polarisation angle evolution follow rather well each other, although the 1.3mm data show a clear preference to more prominent variability on the short time scales, i.e. weeks. The data are compatible with multi-zone models of conical jets involving smaller emission regions for the shortest-wavelength emitting sites. Such smaller emitting regions should also be more efficient in energising particle populations. The data also favours the integrated emission at 1.3mm to have better ordered magnetic fields than the one at 3.5mm.
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