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in the last decade , the ly@xmath0 forests of qso s absorption spectra at redshifts @xmath8 have played an important role in understanding the diffuse cosmic baryon gas and the uv ionizing photon background , and constraining cosmological models and parameters ( e.g. rauch et al . 1997 ; croft et al . 2002 ; bolton et al . 2005 ; seljak et al . 2005 ; jena et al . 2005 ; viel et al . 2006 ) . recently , more and more uv photon sources , including qsos , grb , lyman - break galaxies , and ly@xmath0-emitters at redshifts @xmath9 have been observed ( see ellis 2007 and reference therein ) . due to the rapidly increase of gunn - peterson ( gp ) optical depth at @xmath10 , their absorption spectra show long dark gaps on scales of tens of mpc separated by tiny transmitted leaks . it has been suggested that we are observing the end stage of reionization ( fan et al . 2006 ) . it has been known that the dark gaps are from the igm where the density of neutral hydrogen are high enough to produce almost complete absorptions , however , where the transmitted leaks come from is still unclear . in photoionization equilibrium , the density of neutral hydrogen @xmath11 , here @xmath0 is the recombination rate and @xmath12 is the density of igm ; therefore , even when most of the igm are neutral enough to produce complete ly@xmath0 absorptions , it is still possible for the lowest density voids to provide prominent transmitted fluxes . on the other hand , the leaks can also come from ionized patches around ionizing sources where the intensity of uv radiation are higher than average . the origin of leaking is crucial to understanding the history of reionization . according to commonly accepted scenario of the reionization , in the early stage , the ionized regions are isolated patches in the neutral hydrogen background ( e.g. , ciardi et al . 2003 ; sokasian et al . 2003 ; mellema et al . 2006 ; gnedin 2006 ; trac & cen 2006 ) . and the subsequent growing and overlapping of the ionized patches lead to the ending of reionization ( e.g. , ciardi et al . 2003 ; sokasian et al . 2003 ; mellema et al . 2006 ; gnedin 2004 ) . thus , if leaks mostly come from ionized patches , reionization should happen in the early stage . in contrast , if they were produced in lowest density voids , the the uv ionizing background might has already underwent an evolution from highly inhomogeneous to uniform distribution . a variety of statistics has been used to study the evolution of reionization , such as the mean and dispersion of gp optical depth , the probability distribution function ( pdf ) of the flux , and the size of dark gaps ( e.g. , fan et al . 2002 , 2006 ; songaila & cowie 2002 ; paschos & norman 2005 ; kohler et al . 2007 ; gallerani et al . 2006 ; becker et al . 2006 ) , but all of them seem to be ineffective to provide the information of leak s origin and the inhomogeneity of uv ionizing background . the gp optical depth is an average , and not sensitive to details of reionization . the statistical properties ( mean and variance ) of the gp optical depth at @xmath5 can be well explained by either the fluctuation of ionizing background ( fan et al . 2006 ) or models with uniform ionizing background ( lidz et al . 2006 liu et al . 2006 , hereafter paperi ) . the pdf of the flux is also insensitive to the geometry of reionization . in addition , the pdf is heavily contaminated by noise and distorted by resolution . dark gaps are defined to be continuous regions with optical depth above a threshold in spectra . intuitively , the statistics of dark gap should contain the same information as leaks . however , the size of dark gaps are sensitive to the instrumental resolution , because higher resolution data contain more small leaks ( e.g. , paschos & norman 2005 ; paperi ) , and they also are contaminated by observational noise . moreover , dark gaps are from saturated absorptions , they are featureless and contains generally less information of non - saturated absorption . in this paper we made a statistical approach to ly@xmath0  leaks . the purpose is to show that the statistical features of ly@xmath0  leaks would be effective tool to reveal the origin of ly@xmath0  leaks , and to probe the evolution of reionization . similar to ly@xmath0  absorption lines , ly@xmath0  leaks have a rich set of statistical properties , such as the width of leak profile . unlike dark gaps , the properties of ly@xmath0  leaks can be defined through integrated quantities , which are less contaminated by resolution and noise . moreover , ly@xmath0 leaks are from regions of non - saturated absorptions and encode more information of reionization ; therefore , the ly@xmath0  statistics would provide more underlying physics of reionization than all the above - mentioned statistics . the paper is organized as follows . 2 describes the method to produce ly@xmath0  absorption samples .  3 presents the statistical properties of ly@xmath0 leaks with a uniform ionizing background . 4 analyzes the effect of inhomogeneous ionizing background . conclusion and discussion are given in  5 . we simulate lyman series absorption spectra of qsos between @xmath13 and 6.5 using the same lognormal method as those for low redshifts @xmath14 ( e.g. , bi et al . 1995 ; bi & davidsen 1997 ) . in this model , the density field @xmath15 of the igm is given by an exponential mapping of the linear density field @xmath16 as @xmath17,\ ] ] where @xmath18 is the variance of the linear density field on scale of the jeans length . obviously , the 1-point pdf of @xmath15 is lognormal . in this model , the velocity field of baryon gas is produced by considering the statistical relation between density and velocity field ( bi & davidsen , 1997 ; choudhury et al . 2001 ; veil et al . 2002 ) . the dynamical bases of the lognormal model have gradually been settled in recent years . first , although the evolution of cosmic baryon fluid is governed by the naiver - stokes equation , the dynamics of growth modes of the fluid can be sketched by a stochastic force driven burgers equation ( berera & fang 1994 ) . on the other hand , the lognormal field is found to be a good approximation of the solution of the burgers equation ( jones 1999 ) . the one - point distribution of the cosmic density and velocity fields on nonlinear regime are consistent with lognormal distribution ( e.g. yang et al . 2001 , pando et al . especially , it has been shown recently that the velocity and density fields of the baryon matter of the standard @xmath19cdm model is well described by the so - called she - lvque s universal scaling formula , which is given by a hierarchical process with log - poisson probability distribution ( he et al . 2006 , liu & fang 2007 ) . the simulation is performed in the concordance @xmath19cdm cosmological model with parameters @xmath20 , @xmath21 , @xmath22 , and @xmath23 . the thermodynamic evolution in the igm is actually a rather complex process , because the nonlinear clustering leads to a multi - phased igm . as shown in cosmological hydrodynamic simulations ( e.g. , he et al . 2004 ) , for a given mass density , the temperature of the igm could have large scatters with differences up to two orders . nevertheless , the equation of state in ly@xmath0  clouds can be well approximated by a polytropic relation with @xmath24 ( hui & gnedin ; he et al . the neutral fraction @xmath25 is obtained by solving the photoionization equilibrium equation . the photoionization rate @xmath26 will be given in  2.2 . we then construct synthetic absorption spectra by convoluting the neutral hydrogen density field with voigt profiles . for each given @xmath27 , the size of the simulation samples is @xmath28 , and there are @xmath29 pixels in each simulation box . if the distribution of the igm is uniform and the uv ionizing background is independent of redshift , the mean gp optical depth of ly@xmath0  absorption should approximately increase with redshift as @xmath30 . the observations of dark gaps directly show that the gp optical depth undergoes a stronger evolution at @xmath31 , and consequently , the uv ionizing background would decrease rapidly with redshift at @xmath5 . the strong evolution scenario of the uv ionizing background is supported by a number of simulations or semi - analytical models of reionization ( e.g. , razoumov et al . 2002 ; gnedin 2004 ; oh & furlanetto 2005 ; pascho & norman 2005 ; wyithe & loeb 2005 ; kohler et al . 2007 ; gallerani et al . 2006 ) . it has been found that an evolution of photoionization rate as follows can fit the strong redshift evolution of the gp optical depth ( paper i ) : @xmath32^{2.4 } \},\ ] ] which is in units of 10@xmath33 s@xmath34 . note that the power index 2.4 in equation ( 2 ) is little different from the one used in paper i because we use a different @xmath35 in this paper ( also see below ) .    with eq.(2 ) , we calculate the redshift dependencies of neutral hydrogen fraction @xmath36 and the effective optical depth , @xmath37 , where @xmath38 is the mean of transmitted flux . the results are plotted in figure 1 . the data points for @xmath39 are taken from ly@xmath0  observations of fan et al . ( 2006 ) . for best fitting , the amplitude @xmath40 is in the range 5 - 7 , which can be considered as the allowed range of @xmath41 . in this paper , we will use @xmath42 as the fiducial photoionization rate . it is interesting to note that @xmath36 approaches to @xmath43 at redshift @xmath44 , which is consistent with the electron scattering optical depth given by the data of cmb polarization of wmap iii ( page et al . 2007 ) . it should be pointed out that the assumption of @xmath45 k (  2.1 ) is well reasonable at @xmath46 ( e.g. hui & gnedin 1997 ; he et al 2004 ) and may still be applicable at @xmath1 if the mass averaged neutral fraction of hydrogen is not larger than 10@xmath47 , and the photon heating rate is small . however , at higher redshift , say @xmath48 , the temperature @xmath35 might be redshift - dependent . yet , no proper information on @xmath49 is available at high redshift , and this leads to uncertainty of the model . fortunately , in photoionization equilibrium , the neutral fraction @xmath36 depends mainly on a degenerate factor @xmath50 . thus , the problem with the uncertainty of @xmath49 can be overcame if we use the combined parameter of @xmath51^{0.75}$ ] to fit the data . in the range @xmath52 , this parameter actually is @xmath53 ; in the range of @xmath54 , it is different from @xmath53 by a factor of @xmath55^{0.75}$ ] . thus , the redshift - evolution of @xmath41 would be slower than eq.(2 ) if @xmath49 is less than @xmath56 k at higher redshifts .      as an example of ly@xmath0 leaks , we plot a simulated sample of ly@xmath0  absorption spectrum at @xmath57 with a uniform ionizing background in figure 2 , which shows the transmitted flux @xmath58 , the density @xmath12 of baryon gas , the bulk velocity @xmath59 , and the neutral hydrogen fraction @xmath36 . as expected , the mean of transmitted flux is very small , about 0.004 , and corresponds to an effective optical depth 5.5 . nevertheless , we see spiky features with the transmitted flux @xmath58 as large as @xmath60 . they are leaks .    at low redshifts @xmath61 , the ly@xmath0  forests in qso s spectra have a spectral filling factor significantly less than one and can be decomposed into individual ly@xmath0  absorption lines . at redshifts @xmath9 , it is meaningless to decompose the spectra into individual lines since almost the whole spectra are absorbed completely . we note , however , the transmitted leaks look like emission features upon the dark background , and the absorption spectra can be decomposed into individual `` emission lines '' , i.e. , ly@xmath0  leaks .    comparing the top , the second and bottom panels , we see that all the leaks comes from the regions with mass density less than 0.3 of the mean mass density of baryon gas . the neutral fraction for leaks is @xmath62 , which yields a gp optical depth @xmath63 for overdensity 0.25 and a @xmath64 , while the mean neutral fraction of the entire example is about @xmath65 , which is high enough to produce dark gaps . the column density of neutral hydrogen of the leaking features is mainly in the range of 10@xmath66 - 10@xmath67 @xmath68 , which are the non - saturated absorption regions , and therefore , leaks can come from regions where no enough neutral hydrogen to produce complete absorptions . similar to very high density clouds , @xmath69 , the regions with very low density @xmath70 are rare events in the cosmic clustering . therefore , leaks may provide valuable test on models of clustering . for instance , the lognormal distributions are long tailed in both high and low density sides , and it contains more high density events as well as low density events than gaussian model . it should be emphasized that , once the photoionization rate @xmath26 is determined from the gp optical depth , the statistical property of the sample shown in figure 2 does nt contain free parameters . these samples have been successfully used to explain the following observations : 1 . ) the large dispersion of the gp optical depth ; 2.)the pdf of the flux , and 3 . ) the evolution of the size of dark gaps ( paper i ) . now we use them to study the statistical properties of ly@xmath0 leaks . the ly@xmath0  absorption lines at low redshifts are described by well defined quantities , such as the equivalent width , fwhm ( full width half maximum ) , and the voigt profile , all of which are easily related to physical interpretations . since the transmitted leaks look like emission features , one can decompose the spectra into individual `` emission lines '' and describes it by quantities similar to the ly@xmath0  absorption lines . in this section we study the statistical distribution of ly@xmath0  leaks in the model with a uniform ionizing background . the effects of inhomogeneous ionizing background will be discussed in next section . the center of a leak is identified as the maximum of transmitted flux , and the boundaries are two nearby positions around the center where the flux falls down to zero or to noise level ( we take it to be @xmath71 in this paper ) . the properties of leaks can be measured by two quantities : an equivalent width @xmath2 , which is the total area under the profile of a leak , and a half width @xmath72 , which is defined as the width around center within which the area under the profile of the leak is equal to half of the total area of the leak .    the equivalent width @xmath2 and the half width @xmath3 are two independent measurements of ly@xmath0  leaks . this point can be seen in figure 3 , which gives the value of @xmath2 and @xmath3 for each leak at @xmath57 . it does not indicate a significant correlation between @xmath2 and @xmath3 , especially in the region of @xmath73   and @xmath74  . figure 3 also shows that the distribution of @xmath72 has a lower limit @xmath75  , which is due to the jeans scale used for smoothing the sample . on the other hand , the equivalent width @xmath2 distributes in the range from @xmath76   to @xmath77  . obviously , one is unable to introduce two independent quantities for characterizing dark gaps .    the profiles of 100 randomly sampled leaks with @xmath2 = 0.01 , 0.1 , and 1    are displayed in fig.[profile ] . the tails of the profiles in the three panels of fig.[profile ] look like the lorentz profile , and of course , they do not have the meaning of the natural width of an absorption line . the profiles of leaks for a given equivalent width @xmath2 have very large dispersions ; for example , for @xmath2=0.1 , the flux covers a range from @xmath78 to 0.15 , and @xmath3 can be 0.4 to 4  . as the leaks are formed out of the two neighboring complete absorption troughs , which depend on inhomogeneities of density , velocity , and temperature fields , the large scatter of the profiles is expected . figure 4 also shows that some leaks may have multiple local maximums above the noise level . for clarity and easy - operating , we treat them as one leak . the current observational resolution is of the order of 10km / s , which corresponds to @xmath79  in observer s frame , or about 13 pixels of simulation . therefore , the observed ly@xmath0  leaks would have a resolvable width . similar to ly@xmath0 forests , we can define the cumulative number densities @xmath80 and @xmath81 of leaks as the number of leaks with widths larger than a given @xmath2 and @xmath3 at @xmath27 per unit @xmath27 , respectively . the differential number densities are @xmath82 and @xmath83 . it should be pointed out that statistics of @xmath2 and @xmath84 are not the same as the largest peak width statistics proposed by gallerani et al . ( 2007 ) , which considered only the largest peak width . a peak may contains more than one leaks , i.e. leak statistics describe the details of the leaking area . the mean transmitted flux at @xmath27 within @xmath85 is @xmath86 . we calculate the number densities of leaks in redshift range @xmath87 . in each redshift region we produce 100 light - of - sight samples to calculate the density functions . the results are shown in figure 5 . the errors are estimated by jackknife method , i.e. , the variance over 5 subsamples , each of which contains 20 light - of - sight samples . the number density @xmath88 shown in the up - left panel of figure 5 are similar to a schechter function : they follow a power law at small @xmath2 and have a cut - off at large @xmath2 . the distribution of @xmath89 at @xmath90  declines with decreasing @xmath91 , this is because of the jeans length smoothing . the slope of @xmath88 and @xmath89 are smaller for small redshift . it means the lack of low density voids with small size . that is , the increase of voids of small size is less than voids of large size . this trend can also be seen from the flattening of the cumulative density distributions @xmath92 and @xmath80 . the redshift - evolution of the number densities @xmath80 and @xmath93 of leaks for @xmath2 , @xmath3 = 0.5 , 1.0 , and 3.0   are shown in figure [ zdis ] . as has been seen in figure [ ldis ] , the number densities dramatically decrease at higher redshifts . the evolution is more rapidly for large leaks : the number density @xmath94,@xmath95 drops by a factor of @xmath96 5 when redshift increasing from 5 to 5.8 , while @xmath97,@xmath98 drops by a factor of @xmath96 60 . the evolution trend is the same for the number density @xmath89 . from the error bars of figures [ ldis ] and [ zdis ] we see that the predicted features of leaks would be able used to compare with observed data set containing 20 or more light - of - sight samples with the similar quality as simulation .      in this section , we study the observational and instrumental effects on the statistics of leaks . since both @xmath2 and @xmath3 are defined through the area under the profile of leaks , the effects of resolution and noise would be small . to simulate the observational effects , we bin the original data to a coarse grid corresponding to a resolution of 20000 , and we add gaussian noises with signal - to - noise ratio s / n=3 on binned pixels . the number densities of leaks for the noisy binned samples are shown in figures [ noise ] . the effect of binning and noise is very small for @xmath2 : the original plot of @xmath2 actually is the same as the plot of binned @xmath2 , and the distribution of noisy @xmath2 is affected only when @xmath99 . the effect of binning for @xmath91 is also small on scales larger than the binning length . as expected , the noise effects for @xmath91 are significant for @xmath100  . the noise effects for @xmath91 are even smaller if we smooth the noisy sample . this is very different from the pdf of the flux and the size of dark gaps , both of which are heavily contaminated by instrumental resolution and observational noises . one can compare the uncertainty of @xmath40 with the effect of noises . figures [ gamma ] shows the number densities for the uv background amplitude @xmath40 = 5 , 6 , and 7 ( eq.2 ) , which represent the allowed range of @xmath40 . different from noises , the difference of @xmath40 will cause uncertainty in the whole ranges of @xmath2 and @xmath3 . the uncertainties of number densities are within a factor of 2 when the amplitude @xmath40 changes from 7 to 5 . these uncertainties essentially are from the mass density perturbations with long wavelengths ( paper i ) . the error bars from the scattering of light - of - sight samples is also shown in figure [ gamma ] . therefore , the scattering of @xmath101 light - of - sight samples actually is less than the uncertainty of @xmath40 . in the early stage of reionization , ionizing photons are mainly in ionized patches around uv ionizing sources , and therefore , the spatial distribution of ionizing background is highly inhomogeneous and has patchy structures . when the ionized patches spread over the whole space , the ionizing background become uniform or quasi - uniform , and so the ionizing background underwent an inhomogeneous - to - uniform transition during the reionization . in this section we study the effects of inhomogeneous ionizing background on ly@xmath0  leak statistics . the first question is how to model the inhomogeneous spatial distribution of the ionizing photon field . to our problem , the most important property is the relation between the fields of mass density and ionizing photon background or photoionization rate . if the spatial fluctuation of photoionization rate @xmath102 is statistically uncorrelated with the density field @xmath103 , the reionization of igm will statistically be the same as a uniform ionizing background , regardless of the details of the fluctuation of @xmath102 . in this case , the only effect of the fluctuations would be to yield a larger variance in relevant statistics . however , as shown in last sections , the uncertainty of the leak statistics is already large even when the ionizing background is uniform , and therefore , one would not be able to distinguish the fluctuating photon field from inhomogeneous density field if they are uncorrelated . although many simulations on the uv photon field at the epoch of reionization have been done , there still lack the results of the correlation between photon and density fields . in this context , we will consider the following four models on the statistical relation between the inhomogeneous fields of photon and density , which are mainly based on physical consideration of different stage of reionization . _ model 0_. the photoionization rate is spatially uniform . it corresponds to the post - overlapping stage of reionization . this model has been used in last two sections . _ model i_. the photoionization rate at a give redshift is assumed to be proportional to the density field of igm , @xmath104 , @xmath105 being a constant . this model is motivated by the so - called inside - out scenario : high density regions around uv sources are ionized first , and is most probable at the early stage of reionization .    _ model ii_. just opposite to model i , the photoionization rate at a give redshift is assumed to be inversely proportional to the density field of igm , @xmath106 , @xmath107 being a constant . this model comes from the so - called outside - in scenario : under - dense regions are ionized first ( miralda - escude et al . 2000 ) , which is applicable at the late stage of reionization ( furlanetto & oh 2005 ) .    in order to fit the observed effective optical depth @xmath39 at redshift @xmath57 , we take the following parameters : @xmath108 , @xmath109 where @xmath40 is the photoionization rate at @xmath57 for uniform ionizing background [ eq.(2 ) ] . _ model iii or patch model_. this model corresponds to the stage before the overlapping of ionized patches . the igm is almost fully neutral except for some isolated ionizing patches . we model the ionized patch as a stromgren sphere : the neutral hydrogen fraction is equal to 0 within the patch and equal to 1 outside the patch . the mean radius of the stromgren sphere is assumed to be @xmath110 comoving mpc , which corresponding to an uv photon source with luminosity @xmath111 erg s@xmath34 with active time @xmath112 year and a @xmath113 spectrum . to fit the observed optical depth , we found there should be 2 patches every simulated box ( @xmath114 ) . this number is actually consistent with the following estimation @xmath115 where @xmath116 the comoving luminosity function , i.e. the 3-d number density of sources with luminosity @xmath117 . using @xmath118 mpc@xmath47 ( bouwens et al . 2006 ) , we have @xmath119 and @xmath120 .      for the four models described in the last section , we recalculate the transmitted flux with the same underlying density and velocity fields of fig2 . the results are shown in figure 9 . as mentioned above , the mean of transmitted flux for all models takes approximately the same value of 0.004 , which corresponds to an effective optical depth @xmath121 . however , figure 9 shows clearly different behaviors of ly@xmath0  leaks in various models . for models i and ii , the leaks appear exactly at the same positions as model 0 except some small leaks , which are more prominent for model i. in other words , all the leaks of models 0 , i and ii are from lowest density voids . therefore , the three models have the same distribution of dark gaps , and it is impossible to discriminate among these models with the dark gap statistics .    however , the profiles of the leaks of model 0 , i and ii are statistically different from each other . for model i , the leaks generally have larger width and lower height than model 0 , while for model ii , the width of leaks generally is narrower than model 0 , and the height of leaks is larger than model 0 . the reason is straightforward . comparing with model 0 , model i gives a higher @xmath36 at low density and lower @xmath36 at high density this leads to lower amplitude and broader width . for model ii , the effect is just opposite to model i and yields higher amplitude and narrower width .    in the patch model , all the leaks come from ionized patches , within which ionizing sources are enclosed , and so the internal information of ionized patches can be inferred from the leak statistics . the size of dark gaps is actually given by the distance between ionized patches . generally , ly@xmath0  leaks in the patch model have a maximum flux @xmath122 , which is higher than the maximum flux of other models , 0.2 . this is because the neutral fraction within the stromgren sphere is much less that than other models . this behavior is similar to the so - called proximity effect of qsos at low redshift ( e.g. , rauch 1998 ) . thus , the statistics of the maximum flux of ly@xmath0  leaks may be used to reveal the patchy origin of ly@xmath0  leaks .    as we sample the size of patches along the sight of light according to the impact probability , the size of patches is smaller than @xmath123 . if the size is too small , the ionized patch will be opaque to ly@xmath0  photons due to the damping wing of the surrounding neutral hydrogen absorption ( miralda - escude 1998 ) . obviously , it explains why only one leak is apparent in the patch model as displayed in figure 9 . we now calculate the number densities @xmath124 , @xmath80 , @xmath125 and @xmath126 for three inhomogeneous ionizing background models at @xmath57 , and the results are shown in figures 10 . we can see from figure 10 that the effects of different models on the number densities of @xmath2 and @xmath91 are different . that is , although all the models of 0 , i , ii and iii give the same mean effective optical depth , their leak distributions are different . for clarifying , in figures 10 we show only error bars for the curves of model 0 . it would be enough to show that these results can be tested with data set containing about 20 light - of - sight data . first , for model i , both of the number densities @xmath88 and @xmath80 have only small deviations from model 0 . however , @xmath125 are significantly different from their counterparts of model 0 . the number density of leaks with @xmath127  is much more than model 0 . the basic feature of model i is to increase the number of leaks with large @xmath3 as shown in figure 9 . second , for model ii , we see once again that the number densities @xmath88 and @xmath80 have only small deviations from model 0 . yet , the number densities @xmath89 and @xmath81 of the model ii are systematically lower than model 0 . therefore , the basic feature of model ii is to keep the total area under the profile of leaks almost unchanged , but the widths of leaks are significantly narrowed .    finally , the behavior of patch model is very different from models 0 , i and ii . as expected , the patch model yields more leaks with large @xmath2 , and @xmath88 shows a bump around @xmath128   , which characterize the area of the ionized patches . on the other hand , the number densities of @xmath89 and @xmath81 are lower than model 0 because the characteristic size of ionized patches is less than that of low density voids . it is interesting to note that for the patch model , the tails of @xmath88 , @xmath80 and @xmath129 , @xmath126 are quite different from each other . generally , the tails of @xmath88 and @xmath80 can extend to as large as @xmath130 , while for @xmath125 and @xmath81 , there are no tails higher than 3  . it results partially from the damping wing effect ( miralda - escude 1998 ) .    in summary , the statistical properties of leaks with respective to @xmath2 and @xmath91 are sensitive to the details of ionizing photon field . combining the distribution of @xmath2 and @xmath91 , the ly@xmath0  leaks would be able to probe the origin of themselves , and thus reveal the ionization state of igm , the inhomogeneity of ionizing background , and the evolution stage of reionization . it should be pointed out that we considered only the patches of the hi regions around high redshift galax ies . the hi regions around quasars or the proximity effect would also be the patches of leaking . the mean luminosity of quasars probably is higher than galaxies , and therefore , the above - mentioned feature of @xmath89 and @xmath81 would be more prominent for the patches of quasars . the damped ly@xmath0 absorption system is important for modeling low redshift ly@xmath0 absorption . since these systems have high column density of neutral hydrogen , it will not contribute to leaking either in density void models or patch model . we show that the ly@xmath0 absorption spectra of uv photon emitters at redshifts around @xmath131 can be decomposed into ly@xmath0  leaks , which come from either lowest density voids or ionized patches containing ionizing sources . the ly@xmath0  leaks are well defined and described by the equivalent width @xmath2 and the width of half area @xmath91 . since both @xmath2 and @xmath91 are defined through integrated quantities , the distributions of ly@xmath0  leaks in terms of @xmath2 and @xmath91 are stable with respect to observational noises and instrumental resolution . although the number densities @xmath88 , @xmath80 , @xmath125 , and @xmath81 evolve very rapidly at redshift @xmath5 , these statistics are measurable up to @xmath132 . if the ly@xmath0  leaks come from lowest density voids , the igm should be still highly ionized and the ionizing background is almost uniform ; in contrast , if the leaks come from isolated ionized patches , the ionizing background should be inhomogeneous , and the reionization is still in the overlapping stage . therefore , the origin of ly@xmath0  leaks is crucial to understand the history of reionization . we show that the statistics with @xmath2 and @xmath91 are sensitive to the origin of ly@xmath0  leaks , because the ly@xmath0  leaks are sensitive to the correlation between photon and density fields . based on physical consideration of reionization , we studied four phenomenological models of the photon field : the uniform ionizing background ( model 0 ) , the photoionization rate @xmath4 is proportional to the density @xmath12 ( model i ) , @xmath4 and @xmath12 are anti - correlated ( mode ii ) , and ionized regions only given by stromgren sphere around ionizing photo sources ( patch model ) . we found that , although all the four models can fit the observed mean and variance of optical depth at @xmath5 , the width distribution of ly@xmath0  leaks show very different behaviors . for model 0 , i , and ii , most of ly@xmath0  leaks are from lowest density voids , and the distribution of dark gaps are similar . however , the properties of individual ly@xmath0  leaks are different . model i gives broader width @xmath91 than model 0 ; model ii gives narrower width @xmath91 than model 0 . for patch model , the ly@xmath0  leaks are mainly from ionized sphere , and they generally have higher maximum of transmitted flux than other models . there is a bump in the distribution of the equivalent width @xmath2 , which characterize the intensity of uv photons within ionized patches . ly@xmath0  leaks from ionized patches will provide the information of ionized patches , such as their intensity and size , and constrain the properties of uv sources that contribute most of ionizing photons of the reionization .    finally , we point out that similar analysis is also applicable to the reionization of heii . the optical depth of heii ly@xmath0  evolves rapidly at @xmath133 . it reaches @xmath134 at @xmath135 . that is , the evolution of heii reionization at @xmath135 would be similar to that of h at @xmath5 . observed sample of heii ly@xmath0  absorption indeed show structures similar to the leaks of hydrogen absorption spectrum ( zheng et al . one can also draw the information of background photons at the energy band of heii ionization . this work is supported in part by the us nsf under the grant ast-0507340 . llf acknowledges support from the national science foundation of china ( nsfc ) .
ly@xmath0 absorption spectra of qsos at redshifts @xmath1 show complete gunn - peterson absorption troughs ( dark gaps ) separated by tiny leaks . 
 the dark gaps are from the intergalactic medium ( igm ) where the density of neutral hydrogen are high enough to produce almost saturated absorptions , however , where the transmitted leaks come from is still unclear so far . 
 we demonstrate that leaking can originate from the lowest density voids in the igm as well as the ionized patches around ionizing sources using semi - analytical simulations . 
 if leaks were produced in lowest density voids , the igm might already be highly ionized , and the ionizing background should be almost uniform ; in contrast , if leaks come from ionized patches , the neutral fraction of igm would be still high , and the ionizing background is significantly inhomogeneous . 
 therefore , the origin of leaking is crucial to determining the epoch of inhomogeneous - to - uniform transition of the the ionizing photon background . 
 we show that the origin could be studied with the statistical features of leaks . 
 actually , ly@xmath0 leaks can be well defined and described by the equivalent width @xmath2 and the full width of half area @xmath3 , both of which are less contaminated by instrumental resolution and noise . 
 it is found that the distribution of @xmath2 and @xmath3 of ly@xmath0 leaks are sensitive to the modeling of the ionizing background . 
 we consider four representative models : uniform ionizing background ( model 0 ) , the photoionization rate of neutral hydrogen @xmath4 and the density of igm are either linearly correlated ( model i ) , or anti - correlated ( model ii ) , and @xmath4 is correlated with high density peaks containing ionizing sources ( model iii ) . although all of these models can match to the mean and variance of the observed effective optical depth of the igm at @xmath5 , their distribution of @xmath2 and @xmath3 are very different from each other . 
 consequently , the leak statistics provides an effective tool to probe the evolutionary history of reionization at @xmath6 . 
 similar statistics would also be applicable to the reionization of he ii at @xmath7    = 7.4truecm = 17.7truecm = 6.0truecm    [ firstpage ]    cosmology : theory - intergalactic medium - large - scale structure of the universe