A phase transition for tails of the free multiplicative convolution powers

We study the behavior of the tail of a measure $μ^{\boxtimes t}$, where $\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq 1$. We focus on the case where $μ$ is a probability measure on the positive half-line with a regularly varying tail i.e. of the form $x^{-α} L(x)$, where $L$ is slowly varying. We obtain a phase transition in the behavior of the tail of $μ^{\boxplus t}$ between regimes $α<1$ and $α>1$. Our main tool is a description of the regularly varying tails of $μ$ 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 Lévy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.

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