Context : The interstellar medium ( ISM ) is now widely recognized to display features ascribable to magnetized turbulence .
With the public release of Planck data and the current balloon-borne and ground-based experiments , the growing amount of data tracing the polarized thermal emission from Galactic dust in the submillimetre provides choice diagnostics to constrain the properties of this magnetized turbulence .

Aims : We aim to constrain these properties in a statistical way , focusing in particular on the power spectral index \beta _ { B } of the turbulent component of the interstellar magnetic field in a diffuse molecular cloud , the Polaris Flare .

Methods : We present an analysis framework which is based on simulating polarized thermal dust emission maps using model dust density ( proportional to gas density n _ { \mathrm { H } } ) and magnetic field cubes , integrated along the line of sight , and comparing these statistically to actual data .
The model fields are derived from fractional Brownian motion ( fBm ) processes , which allow a precise control of their one- and two-point statistics .
The parameters controlling the model are ( 1 ) - ( 2 ) the spectral indices of the density and magnetic field cubes , ( 3 ) - ( 4 ) the RMS-to-mean ratios for both fields , ( 5 ) the mean gas density , ( 6 ) the orientation of the mean magnetic field in the plane of the sky ( POS ) , ( 7 ) the dust temperature , ( 8 ) the dust polarization fraction , and ( 9 ) the depth of the simulated cubes .
We explore the nine-dimensional parameter space through a Monte-Carlo Markov Chain analysis , which yields best-fitting parameters and associated uncertainties .

Results : We find that the power spectrum of the turbulent component of the magnetic field in the Polaris Flare molecular cloud scales with wavenumber as k ^ { - \beta _ { B } } with a spectral index \beta _ { B } = 2.8 \pm 0.2 .
It complements a uniform field whose norm in the POS is approximately twice the norm of the fluctuations of the turbulent component , and whose position angle with respect to the North-South direction is \chi _ { 0 } \approx - 69 ^ { \circ } .
The density field n _ { \mathrm { H } } is well represented by a log-normally distributed field with a mean gas density \left \langle n _ { \mathrm { H } } \right \rangle \approx 40 \mathrm { cm } ^ { -3 } , a fluctuation ratio \sigma _ { n _ { \mathrm { H } } } / \langle n _ { \mathrm { H } } \rangle \approx 1.6 , and a power spectrum with an index \beta _ { n } = 1.7 ^ { +0.4 } _ { -0.3 } .
We also constrain the depth of the cloud to be d \approx 13 \mathrm { pc } , and the polarization fraction p _ { 0 } \approx 0.12 .
The agreement between the Planck data and the simulated maps for these best-fitting parameters is quantified by a \chi ^ { 2 } value that is only slightly larger than unity .

Conclusions : We conclude that our fBm-based model is a reasonable description of the diffuse , turbulent , magnetized ISM in the Polaris Flare molecular cloud , and that our analysis framework is able to yield quantitative estimates of the statistical properties of the dust density and magnetic field in this cloud .