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In this paper we consider the Cauchy problem for gravity water waves, in adomain with a flat bottom and in arbitrary space dimension. We prove that ifthe data are of size $\varepsilon$ in a space of analytic functions which havea holomorphic extension in a strip of size $\sigma$, then the solution existsup to a time of size $C/\varepsilon$ in a space of analytic functions having attime $t$ a holomorphic extension in a strip of size $\sigma - C'\varepsilon t$.

Cauchy theory for the water waves system in an analytic framework