Constraints on the mean free path of ionising photons at $z\sim6$ using
  limits on individual free paths
  The recent measurement of an ionising mean free path $\lambda_{\text{mfp}}<1$
pMpc at $z=6$ challenges our understanding of the small-scale structure of the
intergalactic medium (IGM) at the end of reionisation. We introduce a new
method to constrain \mfp at $z=6$ by using lower limits on the individual free
paths of ionisation around quasars. Lyman-limit absorbers with a density
sufficient to halt ionising photons produce strong absorption in the 6
lowest-energy Lyman transitions, in the absence of which a robust lower limit
can be placed on the individual free path. Applying this method to a set of
$26$ quasars at $5.5<z<6.5$, we find that $80\%$ of bright quasars
($M_{1450}<-26.5$) require individual free paths larger than $2$ pMpc. We model
the relation between opacity $\kappa$ and photo-ionisation rate $\Gamma$ via
the parameter $\xi$ such that $\kappa\propto\Gamma^{-\xi}$, and pose joint
limits on \mfp and $\xi$. For the nominal value of $\xi=2/3$, we constrain
$\lambda_{\text{mfp}}>0.31 \ (0.18)$ pMpc at $2\sigma \ (3\sigma)$: a much
tighter lower bound than obtained through traditional stacking methods. Our
constraints get significantly stronger for lower values of $\xi$. New
constraints on \mfp and $\xi$ are crucial to our understanding of the
reionisation-era IGM.
