The $(x, y)$ position reconstruction method used in the analysis of the complete exposure of the Large Underground Xenon (LUX) experiment is presented. The algorithm is based on a statistical test that makes use of an iterative method to recover the photomultiplier tube (PMT) light response directly from the calibration data. The light response functions make use of a two dimensional functional form to account for the photons reflected on the inner walls of the detector. To increase the resolution for small pulses, a photon counting technique was employed to describe the response of the PMTs. The reconstruction was assessed with calibration data including ${}^{mathrm{83m}}$Kr (releasing a total energy of 41.5 keV) and ${}^{3}$H ($beta^-$ with Q = 18.6 keV) decays, and a deuterium-deuterium (D-D) neutron beam (2.45 MeV). In the horizontal plane, the reconstruction has achieved an $(x, y)$ position uncertainty of $sigma$= 0.82 cm for events of only 200 electroluminescence photons and $sigma$ = 0.17 cm for 4,000 electroluminescence photons. Such signals are associated with electron recoils of energies $sim$0.25 keV and $sim$10 keV, respectively. The reconstructed position of the smallest events with a single electron emitted from the liquid surface has a horizontal $(x, y)$ uncertainty of 2.13 cm.
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