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This threshold is defined as the supremum of constants $c>0$ such that $e^{-2c\\varphi}$ is integrable on a neighborhood of $0$. We relate $c(\\varphi)$ with the intermediate multiplicity numbers $e_j(\\varphi)$, defined as the Lelong numbers of $(dd^c\\varphi)^j$ at $0$ (so that in particular $e_0(\\varphi)=1$). Our main result is that $c(\\varphi)\\ge\\sum e_j(\\varphi)/e_{j+1}(\\varphi)$, $0\\le j\\le n-1$. 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