Rates of convergence in the Free Multiplicative Central Limit Theorem

We provide the first quantitative estimates for the rate of convergence in the free multiplicative central limit theorem (CLT), in terms of the Kolmogorov and $r$-Wasserstein distances for $r \geq 1$. While the free additive CLT has been thoroughly studied, including convergence rates, the multiplicative setting remained open in this regard. We consider products of the form $$\pi_n^{g,n^{-1/2}x} := g\left(\frac{x_1}{\sqrt{n}}\right) \cdots g\left(\frac{x_n}{\sqrt{n}}\right),$$ where $x_1, \dots, x_n$ are freely independent self-adjoint operators with common variance $\sigma^2$ and $g \colon \mathbb{R} \to \mathbb{C}$ satisfies certain regularity and integrability conditions. We quantify the deviation of the singular value distribution of $\pi_n^{g,x}$ from the free positive semicircular law, with bounds depending only on the moments of the underlying variables. Additionally, we present a combinatorial proof of the free multiplicative CLT that extends to the unbounded setting.