The tail distribution of the Derivative martingale and the global minimum of the branching random walk

Abstract: In a seminal paper Biggins and Kyprianou \cite{BKy04} proved the existence of a non degenerate limit for the {\it Derivative martingale} of the branching random walk. As shown in \cite{Aid11} and \cite{Mad11}, this is an object of central importance in the study of the extremes of the branching random walk. In this paper we investigate the tail distribution of the limit of the {\it Derivative Martingale} by mean of the study of the global minimum of the branching random walk. This new approach leads us to extend the results of \cite{Gui90} and \cite{Bur09} under slighter assumptions.