  We introduce a new method for determining the influence of galaxies and
active galactic nuclei (AGN) on the physical state of the intergalactic medium
(IGM) at high redshift and illustrate its potential via a first application to
the field of the $z=6.42$ QSO J1148+5251. By correlating the spatial positions
of spectroscopically-confirmed Lyman break galaxies (LBGs) with fluctuations in
the Lyman alpha forest seen in the high signal-to-noise spectrum of a
background QSO, we provide a statistical measure of the typical escape fraction
of Lyman continuum photons close to the end of cosmic reionisation. Here we use
Keck DEIMOS spectroscopy to locate 7 colour-selected LBGs in the redshift range
$5.3\lesssim z\lesssim 6.4$ and confirm a faint $z=5.701$ AGN. We then examine
the spatial correlation between this sample and Ly$\alpha$/Ly$\beta$
transmission fluctuations in a Keck ESI spectrum of the QSO. Interpreting the
statistical HI proximity effect as arising from faint galaxies clustered around
the detected LBGs, we translate the observed mean Ly$\alpha$ transmitted flux
around an average detected LBG into a constraint on the mean escape fraction
$\langle f_{\rm esc}\rangle\geq0.08$ at $z\simeq6$. We also report evidence of
the individual transverse HI proximity effect of a $z=6.177$ luminous LBG via a
Ly$\beta$ transmission spike and two broad Ly$\alpha$ transmission spikes
around the $z=5.701$ AGN. We discuss the possible origin of such associations
which suggest that while faint galaxies are primarily driving reionisation,
luminous galaxies and AGN may provide important contributions to the UV
background or thermal fluctuations of the IGM at $z\simeq6$. Although a limited
sample, our results demonstrate the potential of making progress using this
method in resolving one of the most challenging aspects of the contribution of
galaxies and AGN to cosmic reionisation.
