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We study the behavior of the tail of a measure $\mu^{\boxtimes t}$, where$\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq1$. We focus on the case where $\mu$ is a probability measure on the positivehalf-line with a regularly varying tail i.e. of the form $x^{-\alpha} L(x)$,where $L$ is slowly varying. We obtain a phase transition in the behavior ofthe tail of $\mu^{\boxplus t}$ between regimes $\alpha<1$ and $\alpha>1$. Ourmain tool is a description of the regularly varying tails of $\mu$ in terms ofthe behavior of the corresponding $S$-transform at $0^-$. We also describe thetails of $\boxtimes$ infinitely divisible measures in terms of the tails ofcorresponding L\'evy measure, treat symmetric measures with regularly varyingtails and prove the free analog of the Breiman lemma.

A phase transition for tails of the free multiplicative convolution powers