A phase transition for tails of the free multiplicative convolution powers

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 $t\geq 1$. We focus on the case where $\mu$ is a probability measure on the positive half-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 of the tail of $\mu^{\boxplus t}$ between regimes $\alpha<1$ and $\alpha>1$. Our main tool is a description of the regularly varying tails of $\mu$ in terms of the behavior of the corresponding $S$-transform at $0^-$. We also describe the tails of $\boxtimes$ infinitely divisible measures in terms of the tails of corresponding L\'evy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.