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Search for two neutrino double electron capture of $^{124}$Xe and $^{126}$Xe in the full exposure of the LUX detector

Two-neutrino double electron capture is a process allowed in the Standard Model of Particle Physics. This rare decay has been observed in $^{78}$Kr, $^{130}$Ba and more recently in $^{124}$Xe. In this publication we report on the search for this process in $^{124}$Xe and $^{126}$Xe using the full exposure of the Large Underground Xenon (LUX) experiment, in a total of of 27769.5~kg-days. No evidence of a signal was observed, allowing us to set 90\% C.L. lower limits for the half-lives of these decays of $2.0\times10^{21}$~years for $^{124}$Xe and $1.9\times10^{21}$~years for $^{126}$Xe.