Context : One of the major challenges in observational cosmology is related to the redshift evolution of the average hydrogen ( HI ) ionization in the Universe , as evidenced by the changing in the ionization level of the intergalactic medium ( IGM ) through cosmic time .
In particular , starting from the first cosmic reionization , the rapid evolution of the IGM physical properties poses severe constraints for the identification of the sources responsible for keeping its high level of ionization up to lower redshifts .

Aims : In order to probe the ionization level of the IGM and the ionization capabilities of bright QSOs at z = 4 , we have selected a sample of 2508 QSOs drawn from the SDSS survey ( DR14 ) in the redshift interval 3.6 \leq z \leq 4.6 and absolute magnitude range -29.0 \lesssim M _ { 1450 } \lesssim - 26.0 .
Particularly , we focus on the estimate of the escape fraction of HI ionizing photons and their mean free path ( MFP ) , which are fundamental in the characterization of the surrounding intergalactic medium .

Methods : Starting from UV/optical rest-frame spectra of the whole QSO sample from the SDSS survey , we estimate the escape fraction and free path individually for each of them .
We calculate the Lyman Continuum escape fraction as the flux ratio blue-ward ( \sim 900 Å rest-frame ) and red-ward ( \sim 930 Å rest-frame ) of the Lyman limit ( LL ) .
We then obtain the probability distribution function ( PDF ) of the individual free paths of the QSOs in the sample and study its evolution in luminosity and redshift , comparing our results with the ones found in literature so far .

Results : We find a lower limit to the mean Lyman Continuum escape fraction of 0.49 , in agreement with the values obtained for both brighter and fainter sources at the same redshift .
We show that the free paths of ionizing photons are characterized by a skewed distribution function peaked at low values , with an average of \sim 49 - 59 proper Mpc at z \sim 4 , after excluding possible associated absorbers ( AAs ) .
This value is larger than the one obtained at the same redshift by many authors in the literature using different techniques .
Moreover , the probability distribution function of free path gives a complementary information w.r.t .
the mean free path derived through the stack technique .
Finally , we also find that the redshift evolution of this parameter results to be possibly milder than previously thought .

Conclusions : Our new determination of the mean free path at z \sim 4 implies that previous estimates of the HI photo-ionization rate \Gamma _ { \mathrm { HI } } available in the literature should be corrected by a factor of 1.2-1.7 .
These results have important implications if extrapolated at the epoch of reionization .