On the statistical characterization of the synchrotron multi-zone polarization of blazars

Multi-wavelength polarimetric observations of blazars reveal complex energy-dependent polarization behavior, with a decrease in polarization fraction from X-ray to millimeter bands and significant variability in the electric vector position angle (EVPA). These trends challenge simple, single-zone synchrotron models and suggest a more intricate, turbulent jet structure with multiple emission zones. This work aims to develop a statistical framework to model the energy-dependent polarization patterns observed in blazars, particularly focusing on the behavior captured by IXPE in the X-ray and RoboPol in the optical. The goal is to determine the statistical characterization of multi-zone model, in terms of the cell size distribution, and of the distribution of the physical parameters of the electron energy distribution (EED). A Monte Carlo simulation approach was employed to generate synthetic multi-zone synchrotron emission, using the JetSeT code, from a spherical region populated by turbulent cells with randomly distributed physical properties. Simulations were run across various scenarios: from identical cells to power-law distributed cell sizes and EEDs with different cut-off and low-energy slope distributions. Simulation results were compared to observed IXPE and RoboPol polarization trends. Our analysis demonstrates that the observed energy-dependent polarization patterns can be explained by a purely turbulent, multi-zone model without requiring a correlation between the cell size and the EED parameters. The key determinant of polarization is the effective number (flux-weighted) of emitting cells, which is significantly modulated by the dispersion in cell properties, especially the EED cut-off energy, at higher frequencies, and the dispersion in the EED low-energy spectral index, at lower frequencies. Using a fractional dispersion on the EED cutoff of the order of 90\%, and a dispersion of the EED low-energy spectral index between $\approx 0.5$ and $\approx 1.5$, our model reproduces both the mm-to-X-ray polarization trends, observed in IXPE multiwavelength campaigns, and the optical polarization dependence on synchrotron peak frequency, observed in the RoboPol dataset.