We present improved results of the measurement of the correlation between galaxies and the intergalactic medium (IGM) transmission at the end of reionisation. We have gathered a sample of $13$ spectroscopically confirmed Lyman-break galaxies (LBGs) and $21$ Lyman-$alpha$ emitters (LAEs) at angular separations $20 lesssim theta lesssim 10$ ($sim 0.1-4$ pMpc at $zsim 6$) from the sightlines to $8$ background $zgtrsim 6$ quasars. We report for the first time the detection of an excess of Lyman-$alpha$ transmission spikes at $sim 10-60$ cMpc from LAEs ($3.2sigma$) and LBGs ($1.9sigma$). We interpret the data with an improved model of the galaxy-Lyman-$alpha$ transmission and two-point cross-correlations which includes the enhanced photoionisation due to clustered faint sources, enhanced gas densities around the central bright objects and spatial variations of the mean free path. The observed LAE(LBG)-Lyman-$alpha$ transmission spike two-point cross-correlation function (2PCCF) constrains the luminosity-averaged escape fraction of all galaxies contributing to reionisation to $langle f_{rm esc} rangle_{M_{rm UV}<-12} = 0.14_{-0.05}^{+0.28},(0.23_{-0.12}^{+0.46})$. We investigate if the 2PCCF measurement can determine whether bright or faint galaxies are the dominant contributors to reionisation. Our results show that a contribution from faint galaxies ($M_{rm UV} > -20 , (2sigma)$) is necessary to reproduce the observed 2PCCF and that reionisation might be driven by different sub-populations around LBGs and LAEs at $zsim 6$.
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